II.1-1

Slepian Functions – Why, What, How? Christopher Jekeli Division of Geodetic Science School of Earth Sciences Ohio State Univeristy e-mail: jekeli.1@osu.edu Lecture 1 (8 June 2017): Basics of Fourier and Fourier-Legendre Transforms The Slepian functions were introduced in a series of papers by Slepian, Landau, and Pollak in the 1960s in connection with applications in communications theory. More recently they have surfaced in geodesy (Albertella et al. 1999), as well as geophysics and related sciences (Dahlen and Simons 2008). In preparation for a discussion of the so-called Slepian functions this lecture presents a brief introduction to Fourier spectral analysis with a lead into the spatiospectral concentration problem and further applications. A function, ( )g x , defined everywhere on the continuous domain of real numbers, ℝ , and

satisfying general properties (Jekeli 2017) may be represented in terms of a Fourier transform,

( )( ) ( )g x G f≡F ,

( ) ( ) 2 ,i f xg x G f e df xπ∞

−∞

= − ∞ < < ∞∫ , (1.1)

where

( ) ( ) 2 ,i f xG f g x e dx fπ∞

−

−∞

= − ∞ < < ∞∫ . (1.2)

We also write for the Fourier transform pair: ( ) ( )g x G f↔ . The variable, f , is cyclical

frequency, distinguished from radian frequency, 2 fω π= , which is commonly used for more compact notation. The cyclical frequency is used here to emphasize the reciprocal relationship between frequency and period and for added symmetry in the Fourier transform pair. We will assume generally that the function is real (although almost all formulas hold equally for complex functions) and that it is square-integrable,

( ) 2g x dx

∞

−∞

< ∞∫ . (1.3)

The theory of Fourier transforms can be developed with the less restrictive condition of absolute integrability, and we also use some functions that have Fourier transforms, but are not square-

II.1-2

integrable (or even absolutely integrable). However, the square-integrability permits interpretation of the energy of a function. A finite sequence of function values, ( )g g x∆=

ℓɶ ℓ , 0, , 1N= −ℓ … , where x∆ is a constant

sampling interval, may be represented in terms of the discrete Fourier transform (DFT), kGɶ ,

1 2

0

1, 0 1

Ni k

Nk

k

g G e NN x

π

∆

−

=

= ≤ ≤ −∑ℓ

ℓɶɶ ℓ , (1.4)

where

1 2

0

, 0 1N

i kN

kG x g e k Nπ

∆−

−

=

= ≤ ≤ −∑ℓ

ℓ

ℓ

ɶ ɶ . (1.5)

The tilde notation, ~, signifies that, in fact, both gℓɶ and kGɶ are periodic with period N in their

respective indices. The periodicity must usually be assumed whenever the DFT is implemented.

Again, the Fourier transform pair is denoted kg G↔ℓ

ɶɶ .

Other Fourier transform pairs can be formulated for periodic functions (period, P ) and infinite sequences,

( ) ( )2 2

0

1P

i kx i kxP P

k k

k

g x G e G g x e dxP

π π∞−

=−∞

= ↔ =∑ ∫ɶ ɶ , (1.6)

( ) ( )2 2

f

i x f i x f

f

g G f e df G f x g eπ∆ π∆∆∞

−

=−∞−

= ↔ = ∑∫ ℓ ℓ

ℓ ℓ

ℓ

ɶ ɶ

N

N

, (1.7)

where ( )1 2f x∆=N is called the Nyquist frequency and defines the principal spectral domain

of the periodic spectrum, ( )G fɶ (period, 1 x∆ , f f≤ N ). The DFT and its inverse are the only

transforms that are implemented numerically (typically as the Fast Fourier Transform (FFT)), but the other three types of transform pairs have important theoretical value. The Fourier transform in its various incarnations satisfies many useful properties – too many to list here; see (Jekeli 2017). However, one conceptually important property is the similarity (scaling) property,

( )1 2

1 2

1 fa g ax G

aa

↔

, (1.8)

which shows that a scale increase in the function implies a corresponding scale decrease in the spectrum; and, an expansion of the spatial domain implies a contraction of the frequency domain.

II.1-3

This property is readily proved from the transform definitions with appropriate changes in variables. Another important property is Parseval’s theorem,

( ) ( )2 2g x dx G f df

∞ ∞

−∞ −∞

=∫ ∫ , (1.9)

which is also easily proved by substituting (1.1) for one of the g ’s on the left and treating it as a complex function. Parseval’s theorem says that the total energy of a function is equal to the total energy of its Fourier spectrum. In other words, the Fourier transform does not lose any information of the function (provided (1.1) and (1.2) hold for all x and f ). A function is called band-limited if its Fourier transform is non-zero on a finite interval, i.e., ( ) 00,G f f f= > ; (1.10)

and, the bandwidth is defined by 02W f= . Analogously, a function is called space-limited, or

extent-limited, if it is non-zero on a finite interval, T , i.e., ( ) 0, 2g x x T= > . (1.11)

Consider the simple example of the rectangle function,

( ) 1, 1 2

0, 1 2

xb x

x

≤= > (1.12)

with Fourier transform,

( ) ( ) ( )sinsinc

fB f f

f

ππ

= ≡ , (1.13)

which also defines the sinc (cardinal sine) function. The rectangle function clearly is extent-limited (Figure 1.1); equally evident, its Fourier transform is non-zero almost everywhere. This demonstrates a fundamental theorem that a non-trivial function cannot be both space-limited and band-limited. A formal proof of this theorem follows along the argument (Slepian 1983) that one can extend a band-limited function, being smooth in all its derivatives, analytically everywhere to the complex plane, and thus, all its derivatives exist and its Taylor series converges everywhere. As such, if it vanishes on some interval (i.e., if it is extent-limited), then its Taylor series must also vanish everywhere, leaving only the trivial, zero-valued function as being both band-limited and space-limited. This basic property is analogous to the “uncertainty principle” (of Heisenberg fame) by which two complementary variables, such as position and momentum, have variances (energy concentrations) whose product is always greater than some positive constant (Slepian 1983).

II.1-4

Figure 1.1: The rectangle function (left) and its Fourier transform (right). Here, ( )b x is defined

more correctly to equal 1 2 at 1 2x = ± , which is the value of convergence of (1.1). In most applications the definition (1.12) is adequate. The rectangle function can be used to derive the Fourier transform of a particularly pathological function that nevertheless is quite useful in functional analysis, especially from the point of view of distribution theory. This is the Dirac delta function, ( )xδ , which may be

defined as

( )0

1limT

xx b

T Tδ

→

=

. (1.14)

In view of (1.12), as the base, T , of the rectangle shrinks, the rectangle function becomes more extent-limited, but the area of the rectangle, ( )b x T T, is always unity. In the limit, of course,

the function is not defined, becoming infinite at the origin. Thus, we have

( ) 0, 0x xδ = ≠ , and ( ) 1x dxδ∞

−∞

=∫ . (1.15)

Its Fourier transform is then ( ) ( )( ) ( ) ( )( )

0 0lim lim sinc 1, for all T T

f x B Tf Tf f∆ δ→ →

≡ = = =F . (1.16)

Another interesting example is the Gaussian function, with parameter, 0β > ,

II.1-5

( ) ( ) ( )21 xx eβ π βγβ

−= , (1.17)

and Fourier transform,

( ) ( )2

( ) ff eβ π βΓ −= . (1.18) For 1β = , the Gaussian function and its Fourier transform have exactly the same form. As β

becomes small, the “equivalent width” of ( ) ( )xβγ expands, but the “equivalent width” of ( ) ( )fβΓ shrinks; and, vice versa as β becomes large (Figure 1.2). The qualifier, “equivalent”,

indicates that a proper definition is required to define the width of a function that attenuates for large absolute arguments, but is never exactly zero. It is a concept we leave heuristic and do not pursue here. However, it is again a nice demonstration of the trade-off between (equivalent) extent and (equivalent) bandwidth, already noted by the similarity property (1.8). From a practical standpoint, one might argue that the Gaussian function is both band-limited and extent-limited since for large x and large f both the function and its transform are virtually equal to zero. In fact, this is the case for all functions that have a Fourier transform since they both must attenuate to zero to be integrable. And, Slepian (1976) has argued that any practical (measurable) signal is both band-limited and extent-limited since its energy in both domains at some level cannot be distinguished from the energy of signals with infinite bandwidth or extent. The inverse relationship between domain extents, however, characterized by the Shannon number or Shannon limit, 2WT (see Lecture 3), plays an extremely important role in communications theory.

Figure 1.2: Gaussian functions and their Fourier transforms for 0.5β = and 2β = .

II.1-6

It is, therefore, an interesting problem (solved by D. Slepian) to determine the function with a given limited extent that is maximally concentrated in terms of its energy to a given bandwidth. And, because of the obvious duality between a function and its Fourier transform, (1.1) and (1.2), there is a dual concentration problem: find the band-limited function that is maximally concentrated in a given extent. Both of these problems find utility in the geosciences (and elsewhere). They are formulated precisely and solved in the next lectures. In preparation for these subsequent presentations, an important result is the convolution theorem. A convolution of two functions, ( )g x and ( )h x , is defined by

( ) ( ) ( ) ( ) ( )*y x g x h x g x h x x dx

∞

−∞

′ ′ ′= = −∫ . (1.19)

The theorem says that the Fourier transform of the convolution is the product of the Fourier transforms of the convolved functions, ( ) ( )( ) ( ) ( )*g x h x G f H f=F . (1.20)

Moreover, there is a dual convolution theorem that says the Fourier transform of a product of functions is the convolution of their Fourier transforms, ( ) ( )( ) ( ) ( )*g x h x G f H f=F . (1.21)

Analogous definitions of convolution and corresponding convolution theorems hold for the other types of functions and sequences. It is now also easy to derive

( ) ( ) ( )g x x x dx g xδ∞

−∞

′ ′ ′− =∫ , (1.22)

in view of (1.16) (i.e., ( )( ) 1xδ =F ).

Two fundamental errors are committed when analyzing a discrete sample of a function over a finite sub-domain. The first is due to the non-zero sampling interval, which means that the very high-frequency spectrum of the parent function cannot be recovered. But, more than that, the recoverable spectrum is corrupted because each sample contains the full spectrum, which has to be accommodated somehow in the recovered spectrum – this is called the aliasing error. The second error arises because the truncation of the spatial domain prevents a determination of the longer wavelengths. But, again, the recovered spectrum is corrupted even at higher frequencies because the discontinuity in the available function created by the truncation causes a ripple effect (Gibbs’s effect) in the spectrum. This is called spectral leakage and is associated most clearly with the sinc function that is the Fourier transform of the rectangle function. We do not pursue aliasing error in these lectures. The reduction of spectral leakage, however, is a particular application of the Slepian functions and is discussed in the last lecture.

II.1-7

The Fourier transform pairs are readily generalized to domains of higher dimensions in Euclidean space. For example, in two dimensions,

( ) ( ) ( )1 1 2 221 2 1 2 1 2, , i f x f xg x x G f f e df dfπ

∞ ∞+

−∞ −∞

= ∫ ∫ , (1.23)

( ) ( ) ( )1 1 2 221 2 1 2 1 2, , i f x f xG f f g x x e dx dxπ

∞ ∞− +

−∞ −∞

= ∫ ∫ . (1.24)

Also, the various properties, special functions, convolution, etc., are readily formulated for these extensions. Of greater interest in geodesy and geophysics is the extension to the spherical domain,

( ) , | 0 ,0 2Ω θ λ θ π λ π= ≤ ≤ ≤ ≤ , to which much (but not all) of the Fourier analysis in the

Cartesian domain carries over. The analogous transform is the Fourier-Legendre transform,

,n mG ; and, a function, ( ),g θ λ , on the sphere can be expressed as

( ) ( ) ( ), ,

0

, , , ,n

n m n m

n m n

g G Yθ λ θ λ θ λ Ω∞

= =−

= ∈∑∑ , (1.25)

where

( ) ( ), ,

1, , , , 0

4n m n mG g Y d n m n nΩ

θ λ θ λ Ωπ

= − ≤ ≤ ≥∫∫ . (1.26)

Here, as in the previous lectures, we have a Hilbert space of functions and a countable, but infinite set of basis functions, the spherical harmonics,

( ) ( ), ,

cos , 0, cos

sin , 0n m n m

m m nY P

m n m

λθ λ θ

λ≤ ≤

= − ≤ < (1.27)

where the associated Legendre functions, ,n mP , given by

( ) ( )( ) ( ), ,

!2 1

!n m n mm

n mnP y P y

n mε−+=+

, (1.28)

1 2, 0

1, 0m

m n

mε

< ≤= =

(1.29)

are normalized so that

II.1-8

( ) ( ), ', ' ' '

1, ,

4 n m n m n n m mY Y dΩ

θ λ θ λ Ω δ δπ − −=∫∫ , (1.30)

and where nδ is the Kronecker delta, equal to one if 0n = and zero otherwise. The domain for

the frequencies, or wave numbers, ,n m, is discrete, analogous to the periodic case as by (1.6). A useful property among spherical harmonics is the addition theorem,

( ) ( ) ( ), ,

1cos , ,

2 1

n

n n m n m

m n

P Y Yn

ψ θ λ θ λ=−

′ ′=+ ∑ , (1.31)

where ψ is the central angle between points ( ),θ λ and ( ),θ λ′ ′ on the unit sphere, and nP is the

thn -degree Legendre polynomial. Due to the particular topology of the sphere, however, many of the special functions and operations require isotropy, or invariance with respect to orientation, in order to obtain practical results. For such functions, depending only on the angle of a great circle arc, ψ , 0 ψ π≤ ≤ , one has the Legendre transform pair,

( ) ( ) ( ) ( ) ( )0 0

12 1 cos cos sin

2n n n n

n

g n G P G g P d

π

ψ ψ ψ ψ ψ ψ∞

=

= + ↔ =∑ ∫ , (1.32)

where ,0 2 1n nG G n= + . For example, the analogy of the rectangle function is a circular cap

function. For a practical convolution theorem, the convolution is defined for functions, ( ),g θ λ and

( )h θ ,

( ) ( ) ( ) ( ) ( )1, , * , sin

4y g h g h d d

Ω

θ λ θ λ θ θ λ ψ θ θ λπ

′

′ ′ ′ ′ ′= = ∫∫ , (1.33)

where ( )cos cos cos sin sin cosψ θ θ θ θ λ λ′ ′ ′= + − . The corresponding convolution theorem is

( ) ( ) ,, * n m ng h G Hθ λ θ ↔ . (1.34)

There is no dual convolution theorem in this case. The same concentration problems can be formulated also for functions on the sphere where band-limited functions are defined by a maximum degree, n , and extend-limited functions are defined over some patch on the sphere. Most of the practical developments and advancements for this domain where achieved recently by F. Simons and his colleagues (Simons et al. 2006).

II.2-1

Lecture 2 (12 June 2017): Spatiospectral Concentration Problem – Cartesian and Spherical Cases The concentration problem noted in the previous lecture has two versions that with an appropriate notation can be solved simultaneously with a single solution if both the function and its Fourier transform are defined on the continuous domain, ℝ . In the first case, consider the class of functions that are extent-limited, ( ) 0Tg x = , 2x T> . We seek such an extent-limited

function whose energy is concentrated maximally within a given spectral band, 0f f≤ . The

energy of a function is defined by the integral of its square or of the square of its transform, according to Parseval’s theorem (1.9). Thus, consider the ratio of the spectral energy of the extent-limited function in the given spectral band to its total energy,

( )( )

( )

0

0

2

02

f

T

fT

T

G f df

f

G f df

λ −∞

−∞

=∫

∫, (2.1)

where ( )TG f is the Fourier transform of ( )Tg x . Substituting (1.2), we get

( )( ) ( ) ( )

( )

( ) ( ) ( )

( )

0

0

2 2

2

0

2 20 2

2 2

2

;

f T T

i x x fT T T T

f T TT T

T T

T

g x g x e df dxdx g x g x D x x f dxdx

f

g x dx g x dx

π

λ

∞ ∞′−

−∞ −∞ − − −∞

−∞ −

′ ′ ′ ′ ′− = =

∫ ∫ ∫ ∫ ∫

∫ ∫,(2.2)

where

( ) ( ) ( )( )( )

0

0

020

sin 2;

f

i x x f

f

x x fD x x f e df

x xπ π

π′−

−

′−′− = =

′−∫ . (2.3)

We wish to find the ( )Tg x that maximizes the concentration ratio, ( )0T fλ . The solution may

be obtained by the method of variations. In order to avoid confusion with the delta function, we use the less conventional symbol, ∂ , instead of δ , to denote the variation, emphasizing that it does not mean partial differentiation. Slightly re-writing (2.2), we have

( ) ( ) ( ) ( ) ( )2 2 2

20 0

2 2 2

;

T T T

T T T T

T T T

f g x dx g x g x D x x f dxdxλ− − −

′ ′ ′∂ = ∂ −

∫ ∫ ∫ , (2.4)

II.2-2

and, applying the product rule for variations and noting the symmetry of the kernel function, D ,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2

20 0 0

2 2 2 2

2 2 ;

T T T T

T T T T T T T

T T T T

f g x dx f g x g x dx g x g x D x x f dxdxλ λ− − − −

′ ′ ′∂ + ∂ = ∂ −∫ ∫ ∫ ∫ ,(2.5)

or

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

2 2 2 2

20 0 0

2 2 2 2

2 2 2 2

0 0

2 2 2 2

2

0 0

2

2 ; 2

2 ; 2

2 ;

T T T T

T T T T T T T

T T T T

T T T T

T T T T T

T T T T

T

T T T T T

T

f g x dx g x g x D x x f dxdx f g x g x dx

g x g x D x x f dxdx f g x x x dx g x dx

g x g x D x x f f g x x x g x dx dx

λ λ

λ δ

λ δ

− − − −

− − − −

− −

′ ′ ′∂ = ∂ − − ∂

′ ′ ′ ′ ′ ′= ∂ − − − ∂

′ ′ ′ ′ ′= ∂ − − − ∂

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫

( ) ( ) ( ) ( ) ( )( ) ( )

2

2

2 2

0 0

2 2

2 ;

T

T

T T

T T T T

T T

g x D x x f f g x x x dx g x dxλ δ− −

′ ′ ′ ′ ′= − − − ∂

∫

∫ ∫

(2.6)

where ( )xδ is the Dirac delta function and (1.22) is applied. If ( )0T fλ is maximum then the

variation, ( )0T fλ∂ , vanishes in the limit for arbitrarily small variations, ( )Tg x′∂ ; and, this can

only happen if the inner integral vanishes, or

( ) ( ) ( ) ( )2

0 0

2

;

T

T T T

T

g x D x x f dx f g xλ−

′ ′ ′− =∫ . (2.7)

The solution to the concentration problem, ( )0 maxT fλ → , is thus reduced to the problem of

finding the solution to the integral equation (2.7) that is associated with the maximum ( )0T fλ .

The complementary concentration problem is to find the band-limited function, ( )0f

g x , for a

given 0f that has maximally concentrated energy in a given spatial extent. In this case the

concentration ratio is

( )( )

( )

( ) ( ) ( )

( )

0 0

0 00

0 0

0 0

0 0

0

2

*2

2

22

; 2

f fT

f ff

f fTf f

f f

f

G f G f D f f T dfdfg x dx

T

g x dx G f df

µ − −−∞

−∞ −

′ ′ ′−

= =∫ ∫∫

∫ ∫, (2.8)

II.2-3

where ( )

00fG f = for 0f f> . The corresponding integral equation is

( ) ( ) ( ) ( )0

0 0 0

0

; 2

f

f f f

f

G f D f f T df T G fµ−

′ ′ ′− =∫ . (2.9)

With a change in integration variable, 2y x T= , and

0c Tfπ= , ( ) ( ) ( )2cTy g Tyψ = , ( ) ( )0

cT fλ λ= , (2.10)

the integral equation (2.7) becomes

( ) ( ) ( ) ( ) ( ) ( ) ( )1

1

c c c cy D y y dy yψ λ ψ−

′ ′ ′− =∫ , (2.11)

where

( ) ( ) ( )( ) ( )( )( )

sin; 2c c y y

D y y D y y cy y

ππ

′−′ ′− = − =

′−. (2.12)

Analogously changing the integration variable, 0f fν = , transforms (2.9) to the same integral

equation,

( ) ( ) ( ) ( ) ( ) ( ) ( )1

1

c c c cD dΨ ν ν ν ν µ Ψ ν−

′ ′ ′− =∫ , (2.13)

where ( ) ( ) ( )0 0

cfG fΨ ν ν= and ( ) ( )

0

cf Tµ µ= .

The solutions to (2.11) and (2.13) can be interpreted either as extent-limited functions or band-limited functions, depending on the problem at hand. The integral equation is a

homogeneous Fredholm integral equation of the second kind and a solution, ( ) ( )c yψ or ( ) ( )cΨ ν , is called an eigenfunction of the integral, with corresponding eigenvalue, ( )cλ or ( )cµ .

The functionally independent eigenfunctions are countable and form a basis of a Hilbert space for the particular type of functions under consideration (extent-limited or band-limited). Each corresponding eigenvalue is unique and, by (2.1) or (2.8), less than unity. Thus, the concentration problem is solved by the eigenfunction with the maximum eigenvalue. For reasons that become evident in the next lecture, the eigenfunctions, or solutions to the integral equation, are also called prolate spheroidal wave functions.

II.2-4

In view of the definition of convolution, equation (1.19), the integral (2.11), interpreted for

the spectral concentration problem, is a truncated convolution of ( ) ( )c yψ and the sinc function

(1.13), since ( )( ) ( )( ); 2 sincD y y c c c y yπ π π′ ′− = − . That is, it is only an approximate

convolution of these two functions. Therefore, by the dual convolution theorem (1.21) one can expect on the right side a function whose spectrum is only approximately truncated (windowed) by the rectangle function in the frequency domain, i.e., it is an approximately band-limited function. It turns out, however, that the concentration within this bandwidth is quite good (as long as the given values of T and 0f are not unreasonable). Examples are given below

corresponding to the analogous problems for sequences instead of functions on the continuous domain. Indeed, consider the sequence, ( )Ng

ℓ, that is uniformly sampled from an extent-limited

function, with ( ) 0Ng =ℓ

for 2N< −ℓ or 2 1N> −ℓ . The same concentration problems may

be formulated, except that integrals are replaced where appropriate by summations. For example, for the spectral concentration problem, the concentration ratio analogous to (2.2) is

( )( )

( )

( ) ( ) ( )( )

( )

( )

0

0

2 1 2 12

0T

2 20 2 1 T

22

2

;

fN N

N N cf N N

N f N

N

Nf

G f df x g g D x f

f

gG f df

∆ ∆λ

− −

′′− =− =−

−

=−−

′−

= = =∫ ∑ ∑

∑∫

Dg gg g

ℓ ℓ

ℓ ℓ

ℓ

ℓ

ɶℓ ℓ

ɶ

N

N

, (2.14)

where the N N× matrix, ( )cD , has elements,

( ) ( ) ( )( )( ),

sin 2c c c ND

π′

′− = = ′−

Dℓ ℓ

ℓ ℓ

ℓ ℓ, (2.15)

and where in this case, 0c N xfπ ∆= . Maximizing (2.14) means solving the corresponding matrix

equation,

( ) ( )c cλ=D g g , (2.16) for the eigenvector with maximum eigenvalue. The elements of the eigenvector form a sequence that is also called the discrete prolate spheroidal sequence (dpss). Figure 2.1 shows the solution for the maximum eigenvalue if 1x∆ = , 50N = , and

( )0 1 0.02f N x∆= = . The sequence is normalized such that

2 1

2

2

1N

N

g−

=−

=∑ ℓ

ℓ

. (2.17)

II.2-5

The maximum eigenvalue in this case is ( )0 0.9811cλ = . The numerical method to find this

solution is discussed in the next lecture (standard methods can easily fail since the matrix is strongly ill-conditioned because many eigenvalues are close to zero). The amplitude of the Fourier transform (1.7) of the sequence is also shown and compared to the transform corresponding to the rectangle sequence,

( ) 1, 12 2

0, otherwise

NN N

b − ≤ ≤ −=

ℓ

ℓ (2.18)

and both transforms are scaled such that they are unity at the origin. The transform of the rectangle sequence is the sinc function whose first zero is at ( )1f N x∆= . The spectrum of the

dpss is concentrated to near this bandwidth, with the “side lobes” much reduced compared to those of the sinc function.

Figure 2.1: Left: Discrete prolate spheroidal sequence (dpss) for 1x∆ = , 50N = , and

( )0 1f N x∆= corresponding to the maximum eigenvalue, ( )0 0.9811cλ = . Right: Its Fourier

transform (amplitude spectrum) compared to the transform of the rectangle sequence. Both transforms are normalized to unity at the origin. One can also formulate an equation analogous to (2.9) for the complementary problem of finding the band-limited sequence that has the maximum energy concentrated over a finite number of values. The domain of the dpss in this case is infinite. The sequences approach the prolate spheroidal wave functions as 0x∆ → and N → ∞ . These complementary problems are left to the interested reader (Percival and Walden 1993). Finally, however, we formulate and solve the problem for the band-limited function,

( )max

,ng θ λ , on the sphere that is most concentrated in a particular region, RΩ . The application

is the optimal local modeling of a truncated spherical harmonic series, as might be determined

II.2-6

from the GOCE or GRACE satellite missions. Truncating the spherical harmonic expansion (1.25) for ( ),g θ λ at maxn K n= ≡ , the concentration ratio, defined in this case by

( )( )

( )

2

2

1,

4

1,

4

R

K

K R

K

g d

g d

Ω

Ω

θ λ Ωπ

µ Ωθ λ Ω

π

=∫∫

∫∫, (2.19)

is the same as

( )( ) ( ), , , ,

0 0

2,

0

1, ,

4R

K n K n

n m n m n m n m

n m n n m n

K R K n

n m

n m n

G G Y Y d

G

Ω

θ λ θ λ Ωπ

µ Ω

′

′ ′ ′ ′′ ′ ′= =− = =−

= =−

=∑∑ ∑∑ ∫∫

∑∑, (2.20)

where the denominator is obtained from Parseval’s theorem, which is proved by the orthogonality (1.30). Denote the integral in the numerator as

( ) ( ) ( ), , , , ,

1, ,

4R

n m n m R n m n mD Y Y dΩ

Ω θ λ θ λ Ωπ′ ′ ′ ′= ∫∫ , (2.21)

and the concentration ratio becomes

( )( ) ( ), , , , , T

0 0T

2,

0

R

K n K n

n m n m n m n m R

n m n n m nK R K n

n m

n m n

G G D

G

ΩΩ

µ Ω

′

′ ′ ′ ′′ ′= =− = =−

= =−

= =∑∑∑∑

∑∑DG GG G

, (2.22)

where the elements of the ( ) ( )2 21 1K K+ × + matrix, ( )RΩD , are given by (2.21) and the elements

of the vector, G , are ,n mG . By an analogous method of variations, it is easily shown that this

ratio is maximized by a solution to

( ) ( )R RΩ Ωµ=D G G ; (2.23)

specifically, the eigenvector of ( )RΩD with the maximum eigenvalue. The subscript, K , in the notation of (2.23) is omitted for convenience, but is implied. For the simple case that RΩ is a circular cap, especially a polar cap,

( ) , | 0 2 ,0R s sΩ Ω θ λ λ π θ θ= = ≤ ≤ ≤ ≤ , the elements (2.21) are

II.2-7

( ) ( ) ( )

( ) ( )

2

, , , , ,

0 0

, ,

0

1, , sin

4

cos cos sin2

0,

s

s

n m n m s n m n m

mn m n m

D Y Y d d

P P d m m

m m

θ π

θ λ

θ

θ

θ θ λ θ λ θ θ λπ

ε θ θ θ θ

′ ′ ′ ′

= =

′

=

=

′==

′≠

∫ ∫

∫ (2.24)

Thus, for , ,m K K= − … , let

( ) ( ) ( ) ( ), , ,

0

cos cos sin2

s

m mn n s n m n mD P P d

θ

θ

εθ θ θ θ θ′ ′

=

= ∫ ; (2.25)

and the matrix, ( ) ( )R

sΩ θ≡D D , becomes a block-diagonal matrix, with 2 1K + blocks, ( ) ( )m

sθD ,

each having dimension, ( ) ( )1 1K m K m− + × − + . It is noted that if sθ π= then

( ) ( ),m

n n n nD π δ′ ′−= . Each block, ( ) ( )msθD , 0m≠ , occurs twice; that is, ( ) ( ) ( ) ( )m m

s sθ θ−=D D . The

elements (2.25) can be evaluated using (Simons et al. 2006, eq.5.5)

( ) ( ) ( ) ( )( ) ( ) ( )( ), 1 1

2 1 2 11

0 0 0 02

n nmm

n n s s s

n n

n n n n n nD P y P y

m mθ

′+

′ − +′= −

′+ + ′ ′ = − − −

∑ ℓ ℓ

ℓ

ℓ ℓ,(2.26)

where coss sy θ= and the parenthetical quantities are Wigner 3 j− symbols (Varshalovich et al.

1988). This solution is a finite set of spherical harmonic coefficients of the band-limited function whose energy is maximally concentrated on the polar cap. With a rotation of coordinates, one can create a maximally concentrated function over a circular cap situated anywhere on the globe. With even greater generalization, the domain of concentration, RΩ , can be any shape (for

example, outlining a particular continent). The numerical challenge is the integration of the product of spherical harmonic functions over this domain, equation (2.21) (Simons and Dahlen 2007). Without going into further details here, the analogous comparison is made in Figure 2.2 between a simple band-limited function on the sphere and the band-limited function, ( )g θ ,

maximally concentrated over the polar cap of radius, 10sθ = ° . In this case, the “simple

function” is the truncated Dirac delta function, corresponding to a rectangular Legendre spectrum in the frequency domain,

( ) ( ) ( )0

2 1 cosK

K n

n

n Pδ θ θ=

= +∑ . (2.27)

II.2-8

Also shown in Figure 2.2 is the spectrum of the solution to the optimal spatial concentration problem. Details on the computation of ( )g θ and its spectrum are given in the next lecture.

polar angle, θ [deg]

( )Kδ θ

( )g θ

polar angle, θ [deg]

( )Kδ θ

( )g θ

Figure 2.2: Left: The truncated Dirac delta function and the solution, ( )g θ , to the spatial

concentration problem on the sphere for a polar cap of radius, 10sθ = ° . Both function are band-

limited to 60K = and normalized to unity at the pole. Right: The absolute Legendre spectrum of ( )g θ (non-zero only for degrees, 0 n K≤ ≤ , and order, 0m= ).

The discussions in this lecture have set up the spatiospectral concentration problem and only solved it from the most elementary viewpoint. The theory, however, is much deeper and applications flow in many directions. The following lecture(s) attempt to elucidate the theory with a view toward the practical side.

II.3-1

Lecture 3 (14 June 2017): Slepian Functions and Sequences From the theory of integral equations (specifically the homogeneous Fredholm equation of the second kind), it is known (Courant and Hilbert 1966, pp.122-135) that for a symmetric, real, kernel, such as (2.3), the eigenfunctions of the integral equation (2.11),

( ) ( ) ( ) ( ) ( ) ( ) ( )1

1

c c c cy D y y dy yψ λ ψ−

′ ′ ′− =∫ , (3.1)

are infinite in number, but countable, and associated with real and positive eigenvalues. For the kernel (2.3) and by virtue of (2.1) and (2.8) they are distinct and also less than one, and thus may be ordered,

( ) ( )0 11 c cλ λ> > >⋯ (3.2)

We denote corresponding eigenfunctions by ( ) ( )ck yψ , 0,1,2,k = … .

Normalizing the limited extent to 2T = , we may assume that the extent-limited functions are square-integrable on [ ]1,1− and are elements of a Hilbert space with inner product defined by

( ) ( )1

1

,g h g x h x dx−

= ∫ . (3.3)

As such there is a complete sequence of orthogonal basis functions. Indeed, we may take the

eigenfunctions, ( ) ( )ck yψ , as the basis since they are orthogonal. This follows immediately from

(3.1) since their eigenvalues are distinct. That is, multiply (3.1) for ( ) ( )ck yψ on both sides by

( ) ( )c yψℓ

, k≠ℓ , then integrate and substitute (3.1) for ( ) ( )c yψℓ

,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 1

1 1 1

1 1

1 1

1

1

,

,

c c c c c ck k k

c c ck

c c ck

y y dy y y D y y dy dy

y y D y y dydy

y y dy

λ ψ ψ ψ ψ

ψ ψ

λ ψ ψ

− − −

− −

−

′ ′ ′=

′ ′ ′=

′ ′ ′=

∫ ∫ ∫

∫ ∫

∫

ℓ ℓ

ℓ

ℓ ℓ

(3.4)

Hence, ( ) ( )( ) ( ) ( ), 0c c c ck kλ λ ψ ψ− =

ℓ ℓ; and, since ( ) ( ) 0c c

kλ λ− ≠ℓ

, this implies ( ) ( ), 0c ckψ ψ =

ℓ. Now,

any constant times an eigenfunction is also an eigenfunction with the same eigenvalue.

II.3-2

Therefore, we may assume that ( ) ( )ck yψ is normalized such that it has (squared-) norm,

( ) ( ), 1c ck kψ ψ = , and thus the eigenfunctions are orthonormal on [ ]1,1− ,

( ) ( ),c ck kψ ψ δ −=

ℓ ℓ. (3.5)

The eigenfunction, ( ) ( )0c yψ , with maximum eigenvalue is maximally concentrated in terms of its

energy in the given bandwidth. The next eigenfunction, ( ) ( )1c yψ , with the next largest

eigenvalue, being orthogonal to ( ) ( )0c yψ , is the limited-extent function that is next best

concentrated in the given bandwidth; and, so on. It is interesting that, like the Gaussian function, these eigenfunctions share the same form with their Fourier transforms. To show this, consider the function operators,

( ) ( )1

1

icyyc y y e dyψ ψ ′−

−

′ ′= ∫P , (3.6)

( ) ( ) ( ) ( )1

1

cc y y D y y dyψ ψ

−

′ ′ ′= −∫Q , (3.7)

which, in view of (2.3), are related according to

( ) ( ) ( ) ( ) ( )

( )

1 1 1

*

1 1 1

2

2

cicy y icyyc c

c

y y e dy e dy y D y y dyc

yc

πψ ψ ψ

π ψ

′′ ′ ′′−

− − −

′ ′ ′′ ′ ′ ′= = −

=

∫ ∫ ∫P P

Q

(3.8)

where the second equality results with a change in variables, ( )2u cy π′′= . Hence, it is easy to

verify that a real eigenfunction of the operator, cP , with eigenvalue, ( )cnα , is also a solution of

(3.1), being an eigenfunction of the operator, cQ . Indeed, one has,

( ) ( ) ( ) ( ) ( ) ( )2

*

2 2c c

c c c

c cy y y yα ψ ψ ψ λ ψ

π π= = =P P Q . (3.9)

Hence, since the solutions of (3.1) are ( ) ( )cn yψ , one may set ( ) ( )2c c

n n cα πλ= . Now, the

extended functions defined by

II.3-3

( ) ( )( )

,

2,

2

0,2

cn

T cn

x Tx

Tx

Tx

ψψ

≤ = >

(3.10)

have Fourier transforms, given with an appropriate change in integration variable, by

( ) ( ) ( ) ( ) ( ) ( ) ( )2

, , 2 2

02

2

2

T

T c T c c c ci fx i fxn n n n n

T

x T ff x e dx e dx

T fπ πΨ ψ ψ α ψ

∞− −

−∞ −

= = = ∫ ∫ . (3.11)

This shows that the Fourier transform of ( ) ( ),T cn xψ is proportional to ( ) ( )0

cn f fψ , defined for

f−∞ < < ∞ . Analogous to a matrix linear operator, the sum of the eigenvalues is the trace of the kernel (Gohberg et al., Theorem 8.1), defined by

( ) ( )1 1

0

1 1

22c c c

D y y dy dy f Tπ π

− −

− = = =∫ ∫ . (3.12)

This is the so-called extent-bandwidth product (more often the time-bandwidth product since most applications refer to time as the independent variable). That is,

( )0

0

2ck

k

f Tλ∞

=

=∑ . (3.13)

Furthermore, Landau and Pollak (1962) show that the eigenvalues are either close to 1 (( ) 1cnλ ≈ )

or close to 0 ( ( ) 0cnλ ≈ ). The transition from one extreme to the other is rather sharp and occurs

near 02 1n f T= − . The quantity, 02E f T= , is also called the Shannon number. Therefore,

(3.13) shows that the Shannon number is approximately equal to the number of significant

eigenvalues. In other words, among all the extent-limited functions, for which the set, ( ) 0

ck

kψ

∞

=,

is a basis, the subset, ( ) 1

0

Ec

kk

ψ−

=, is the basis for a subspace of extent-limited functions whose

energy is well-concentrated in the given bandwidth. As indicated, the dimension of this subspace is E (Simons 2009, p.3). Slepian (1983, p.384) notes that the solutions to the Fredholm integral equation (3.1) are, by a “seemingly lucky accident,” also special solutions to the Helmholz wave equation,

2 2 0F k F∇ + = , in prolate spheroidal coordinates (Morse and Feshbach 1953, p.1502). When solving the Helmholz equation by separation of variables, the relevant solution for one of the variables is given by the so-called angular prolate spheroidal wave function, u , that satisfies the differential equation,

II.3-4

( ) ( ) ( )2

2 2 22

1 2 0d u du

y y y u ydy dy

ζ χ− − + − = , (3.14)

where χ , ζ are constants. The solutions to this equation are also solutions to the Fredholm integral equation (3.1) (Slepian and Pollak 1961, p.57). The idea behind proving this is to show that if (3.1) and (3.14) are viewed as operators applied to a function, ( )h y , i.e.,

( ) ( ) ( ) ( ) ( ) ( )1

1

c ch y D y y h y dy h yλ−

′ ′ ′= − =∫D , (3.15)

( ) ( ) ( ) ( ) ( )2 2 21 cd dhh y y c y h y h y

dy dyλ

= − − + =

L . (3.16)

with ( )cζ λ= and 2 2cχ = , then the operators, D and L , commute, ( )( ) ( )( )h y h y=D L L D . (3.17)

To prove this we start with the left side and find

( )( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 1

2 2 2

1 1

1

1

1c c

cy

d dhh y D y y y dy D y y c y h y dy

dy dy

h y D y y dy

− −

′

−

′ ′ ′ ′ ′ ′ ′= − − − + − ′ ′

′ ′ ′= −

∫ ∫

∫

D L

L

(3.18)

which is readily established by integrating the first integral twice by parts. Then, the right side of (3.17) follows since

( ) ( ) ( ) ( )c cy yD y y D y y′ ′ ′− = −L L , (3.19)

which is also easily derived using the facts that ( ) ( ) ( ) ( )c cD y y y D y y y′ ′ ′∂ − ∂ = −∂ − ∂ and ( ) ( ) ( ) ( )2 2 2 2c cD y y y D y y y′ ′ ′∂ − ∂ = ∂ − ∂ .

Because of the commutativity of D and L , they share eigenfunctions. Indeed, if ( )0h y is

an eigenfunction of D , then ( ) ( ) ( )( ) ( ) ( )( ) ( )0 0 0 0 0 0 0 0 0h y h y h y h y h y h yλ λ λ= ⇒ = ⇒ =D L D L D L L , (3.20)

II.3-5

which means that ( )0h yL is also an eigenfunction of D with the same eigenvalue as for

( )0h y . But this implies that the two eigenfunctions, ( )0h yL and ( )0h y , are linearly dependent

and so ( )0h y is also an eigenfunction of L (but different eigenvalue).

Eigenfunctions of L , being now also eigenfunctions of D , are solutions to the differential equation (3.14), which are called prolate spheroidal wave functions that are well studied in

connection with the wave equation. Given an eigenfunction, the eigenvalue, ( )cλ , with respect to D can be determined from (3.15). We pursue this no further, but instead delve into the analogous and more practical situation involving sequences (sampled functions). The concentration problem for sequences is solved with the eigenvector that has maximum eigenvalue for the matrix equation (2.16). Slepian (1978) gives the following corresponding difference equation, adapted to present notation,

( ) ( )( )2

1 1

2 1 1cos 1 1 0

2 2 2n n n

n c NN n g n g n N n g

Nζ− +

− − + − − + + − − = , (3.21)

which in vector-matrix form is

( )( )

( )( )

( )( )

( )( )

2

0 0

1 12

2 2

1 1

2

2 1 1cos 1 1 0 0

2 2

11 1

2

2 10 cos 0

2

11 1

2

1 2 10 0 1 1 cos 1

2 2

N N

N N

c NN

N

g gNg g

c Nn

Ng g

N g g

c NN N

N

ζ

− −

− −

− −

− − =− −

− − − +

⋯

⋱ ⋱ ⋱ ⋮

⋮ ⋮⋱ ⋱

⋮ ⋱ ⋱ ⋱

⋯

(3.22) or,

( )c ζ=S g g . (3.23)

The N N× matrices, ( )cD (equation (2.15)) and ( )cS , commute, as is readily shown by

comparing elements of ( ) ( )c cD S and ( ) ( )c cS D . Indeed, these elements,

II.3-6

( ) ( )( ) ( )( ) ( )( )

( )( )( )

( )( )

( )( )( )( )

1

,, ,

0

22 2

sin 1 sin2 1

cos2 1 2

2sin 1

11 1

2 1

Nc c c c

n kj k j n

n

c cj k j k

j c NN NN j j

j k N j k

cj k

Nj N j

j k

π π

π

−

=

=

− − − − = − + − − − −

− + + + − −

− +

∑S D S D

(3.24)

( ) ( )( ) ( )( ) ( )( )

( )( )( )

( )( )

( ) ( )( )( )

1

,, ,

0

22 2

sin 1 sin2 1

cos2 1 2

2sin 1

11 1

2 1

Nc c c c

j nj k n k

n

c cj k j k

k c NN NN k k

j k N j k

cj k

Nk N k

j k

π π

π

−

=

=

− + − − = − + − − + −

− − + + − −

− −

∑D S D S

(3.25)

when differenced are shown to be identical to zero for all ,j k using symbolic simplification (for

example, available in the programming language, MathCad). Thus, ( ) ( ) ( ) ( )c c c c=D S S D , and ( )cD

and ( )cS share eigenvectors (but not eigenvalues). Unlike the matrix, ( )cD , the calculation of

eigenvectors, ( )ckg , 0, , 1k N= −… , for the tri-diagonal matrix, ( )cS , is relatively stable. Then,

equipped with these, the eigenvalues of ( )cD are easily computed from (2.16),

( )( ) ( )

( ) , 0, , 1

c ckc

k ck

k Nλ = = −D g

g… . (3.26)

Figure 3.1 shows the eigensequences (elements of the eigenvectors), ( )( )ck

jg ,

2, , 2 1j N N= − −… , also known as discrete prolate spheroidal sequences, for 0,2,4,6k = ,

and for the case: 1x∆ = , 60N = , 0 0.05f = . Also shown are the eigenvalues, ( )ckλ , 0, ,20k = … .

The amplitudes of the Fourier spectra of the eigensequences are displayed in Figure 3.2 for the frequencies, 0 0.2f≤ ≤ (being less interesting for the remainder of the domain up to the Nyquist frequency).

II.3-7

Figure 3.1: Discrete prolate spheroidal sequences, ( )( )c

kj

g , 0,2,4,6k = , for the case: 1x∆ = ,

60N = , 0 0.05f = ( 3c π= ) are shown on the left. The sequences are normalized such that ( ) 1ck =g . On the right are the eigenvalues, ( )c

kλ , 0, ,20k = … .

Figure 3.2: Fourier spectra, ( ) ( )c

kG f , 0,2,4,6k = , of the discrete prolate spheroidal sequences

shown in Figure 3.1 (left). The concentration half-bandwidth is 0 0.05.f =

The eigenvalues (Figure 3.1, right) are close to unity up to near the Shannon number,

02 6E N xf∆= = , and then decrease rapidly. And, as indicated in Figure 3.2, the eigensequences, ( )( )ck

jg , 0, , 1k E= −… , having significant eigenvalues, are well concentrated in the spectral

domain, 0f f≤ . The seventh eigensequence, ( )( )6c

jg , associated with eigenvalue, ( )

6 0.289cλ = ,

already is much less well concentrated in this domain.

II.3-8

These concepts extend to spherical functions as shown in the previous lecture. Returning to the spatial concentration problem on the sphere, the eigenvectors, G , of the general matrix equation (2.23) represent the spectra of band-limited functions, where the eigenvector with maximum eigenvalue contains the spectrum of the band-limited function that is maximally concentrated on the region, RΩ .. These eigenvectors are orthogonal since the eigenvalues are

distinct. That is, for two eigenvectors, αG and βG , and corresponding eigenvalues, ( )RΩαµ and

( )RΩβµ ,

( )( ) ( )( ) ( )( ) ( )( )T T T TR R R RΩ Ω Ω Ωα β β α β β α β β αµ µ= = =D DG G G G G G G G ; (3.27)

hence,

( ) ( )( ) T T0 0R RΩ Ωβ α α β α βµ µ− = ⇒ =G G G G . (3.28)

We may choose to normalize the eigenvectors so that they are orthonormal, T

α β α βδ −=G G . (3.29)

Denoting the elements of the eigenvectors by ( ) ( )( )T

0,0 ,K KG Gα α α=G ⋯ , the corresponding

eigenfunctions,

( ) ( ) ( ), ,

0

, ,K n

n m n m

n m n

g G Yα αθ λ θ λ

= =−

=∑∑ , (3.30)

are also orthonormal on Ω , since

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

, , , ,

0 0

, ,

0 0

, ,

0

1 1, , , ,

4 4

K n K n

n m n m n m n m

n m n n m n

K n K n

n m n m n n m m

n m n n m n

K n

n m n m kn m n

g g d G Y G Y d

G G

G G

α β α βΩ Ω

α β

α βα

θ λ θ λ Ω θ λ θ λ Ωπ π

δ δ

δ

′

′ ′ ′ ′′ ′ ′= =− = =−

′

′ ′ ′ ′− −′ ′ ′= =− = =−

−= =−

=

=

= =

∑∑ ∑∑∫∫ ∫∫

∑∑ ∑∑

∑∑

(3.31)

They are also orthogonal on RΩ ,

II.3-9

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

, , , ,

0 0

, , , , ,

0 0

T

1 1, , , ,

4 4R R

R R

K n K n

n m n m n m n m

n m n n m n

K n K n

n m n m n m n m R

n m n n m n

g g d G Y G Y d

G G D

α β α βΩ Ω

α β

Ω Ωα β α α β

θ λ θ λ Ω θ λ θ λ Ωπ π

Ω

µ δ

′

′ ′ ′ ′′ ′ ′= =− = =−

′

′ ′ ′ ′′ ′ ′= =− = =−

−

=

=

= =

∑∑ ∑∑∫∫ ∫∫

∑∑ ∑∑DG G

(3.32)

where the last equality follows from (3.27) and (3.29). Hence, the eigenfunctions, ( ),gα θ λ ,

form an orthogonal basis for band-limited functions the are maximally concentrated on RΩ .

The total number of eigenvalues, ( ) ( ),

R R

n mΩ Ω

αµ µ≡ , of the matrix, ( )RΩD , is ( )21K + ; and, the

sum of the eigenvalues is also the trace of the matrix, given with (2.21) by

( ) ( )( ) ( ) ( ) ( ), , , , , ,

0 0 0

1tr , ,

4R R

R

K n K n K n

n m n m n m R n m n m

n m n n m n n m n

E D Y Y dΩ Ω

Ω

µ Ω θ λ θ λ Ωπ

= =− = =− = =−

= = = =∑∑ ∑∑ ∑∑∫∫D .(3.33)

Now, using (1.31), this simplifies to

( ) ( ) ( ) ( )

( ) ( )

, ,

0 0

2

1 1, , 2 1 cos0

4 4

14

R R

K n K

n m n m n

n m n n

R

E Y Y d n P d

AK

Ω Ω

θ λ θ λ Ω Ωπ π

Ωπ

′ ′= =− =

= = +

= +

∑ ∑ ∑∫∫ ∫∫ (3.34)

since ( )1 1nP = and where ( )RA Ω is the area of RΩ . Thus, if the significant eigenvalues are

close to one and the remaining (insignificant) eigenvalues are close to zero, we may identify the Shannon number with E . For the spherical cap, sΩ , of radius, sθ , this is

( ) ( )2 11 1 cos

2 sE K θ= + − . (3.35)

The numerical facility of calculating the solutions to the corresponding concentration problem, in particular for the circular cap on the sphere, was achieved by F.J. Simons and colleagues who discovered, developed, and implemented an operator corresponding to a difference equation that commutes with the block-diagonal matrix, ( )sθD . Specifically, the

Grünbaum matrices, ( ) ( )msθT , that respectively commute with the matrices, ( ) ( )m

sθD , having

elements (2.25), is given by

( ) ( ) ( ), 1 cosmn n s sT n nθ θ= − + , , ,n m K= … (3.36)

II.3-10

( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )( )

2 2

, 1 1,

12 2

2 1 2 3m m

n n s n n s

n mT T n n K K

n nθ θ+ +

+ −= = + − +

+ +, , , 1n m K= −… (3.37)

( ) ( ), 0mn n sT θ′ = , otherwise. (3.38)

The matrices, ( ) ( )msθT , , ,0, ,m K K= − … … , are tri-diagonal, symmetric, and have dimensions,

( ) ( )1 1K m K m− + × − + . The proof that

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )m m m ms s s sθ θ θ θ=T D D T (3.39)

is rather long and may be found in (Simons et al. 2006, see their eq.(5.34)). The eigenvectors,

( ) ( )mj sθG , 1, , 1j K m= − +… , of ( ) ( )m

sθT are easily determined using standard algorithms.

The corresponding eigenvalues, ( ) ( )mj sµ θ , are obtained by

( ) ( )( ) ( ) ( ) ( )

( ) ( )

m ms j sm

j s mj s

θ θµ θ

θ=

D G

G. (3.40)

where ⋅ denotes the magnitude of a vector, and the elements of ( ) ( )msθD can be computed by

(2.26). We may re-order the eigenvalues in descending rank, ( ) ( ) ( ) ( ) ( )20 1 1

1 s s s sKαµ θ µ θ µ θ µ θ+

> > > > > >⋯ ⋯ , (3.41)

where each α is associated with a particular order, m , and eigenvector, j , of ( ) ( )msθD .

The complete Fourier-Legendre spectrum of the thα basis function of the space of band-limited spherical functions is the eigenvector, ( )sα θG , of the total block-diagonal matrix,

( )sθD , which is also the thα eigenvector of the block-diagonal matrix,

II.3-11

( )

( ) ( )( )

( ) ( )( )( ) ( )( )

( ) ( )( )

( ) ( )( )

1 1

1

0

1 1

1

1 1

0

0

Ks

sK K

ssK K

sK K

Ks

θ

θ

θθ

θ

θ

−

−

+ +

=

T

T

TT

T

T

⋱

⋱

(3.42) Clearly, in view of (3.39), also ( ) ( ) ( ) ( )s s s sθ θ θ θ=T D D T ; and, the thα eigenvector of ( )sθD

is given by the ( )21 1K + × vector,

( ) ( ) ( )( )TT

T Tms j sα θ θ =

0 0G G⋯ ⋯ , (3.43)

where the only non-zero elements are those of the ( )1 1K m− + × vector,

( ) ( ) ( )( ) ( )( )( )T

,,m

j s s K m sm m jjG Gθ θ θ=G ⋯ . (3.44)

All eigenvectors, 1, , 1j K m= − +… , for a particular order, m , occupy the same location in the

corresponding vectors, ( )sα θG . The eigenvalues for the band limit, 60K = , and polar cap

radius, 10sθ = ° , are shown in Figure 3.3 (left) with respect to their rank, and in Figure 3.3 (right)

with respect to the corresponding order, m . The eigenvectors (and eigenvalues) for m and m− are the same due to the axial symmetry of the polar cap. There is a reasonably sharp break in values between near unity and near zero at the rounded Shannon number of 28E = (equation (3.35)). Also, note that there are multiple significant eigenvalues for some orders, and the maximum order of significant eigenvalues in this case is 7m= .

II.3-12

Figure 3.3: Eigenvalues in the spherical spatial concentration problem for the polar cap with radius, 10sθ = ° , and for the band-limited basis functions with maximum degree and order,

60K = . The left panel shows the first 101 ranked eigenvalues (out of 3721) and the Shannon number (27 + 1) indicates the number of significant eigenvalues. The right panel shows that the 28 significant eigenvalues are associated with harmonic orders, 7m ≤ .

The eigenfunction with spectrum given by the elements of ( )sα θG is

( ) ( ) ( ) ( ) ( )

( ) ( )

, , , ,

, ,

, , ,

cos , 0cos

sin , 0

K K K

n m n m n m n mj jm K n m n m

K

n m n mjn m

g G Y G Y

m mG P

m m

α θ λ θ λ θ λ

λθ

λ

=− = =

=

= =

≥ = <

∑∑ ∑

∑ (3.45)

where the second and third equations follow from (3.43), and the order, m , and the eigenvector, j , are associated with the particular α . Figure 3.4 (left) shows the eigenfunctions (for 0λ = )

for the most significant eigenvalues associated with the orders, 0, ,7m= … . On the right side of Figure 3.4 are the eigenfunctions for the first few eigenvalues associated with the order, 0m= . The eigenfunctions with 1Eα ≤ − are well concentrated in the polar cap, where the optimally concentrated function is associated with 0α = ( 0m= ).

II.3-13

Figure 3.4: Some eigenfunctions corresponding to the eigenvalues in Figure 3.3 evaluated on the zero meridian. The left panel shows well concentrated functions within the polar cap radius,

10sθ = ° for 19α ≤ and this deteriorates already at 23α = (the Shannon number is

1 27 1 28α + = + = ). All functions on the left correspond to different orders (being the lowest harmonic degree of their spectrum that is also limited at high degree, 60K = ). The right panel shows 4 of the 61 eigenfunctions associated with order, 0m= , where 29g is not well

concentrated, corresponding to the th30 eigenvalue that is, indeed, beyond the Shannon number.

II.4-1

Lecture 4(3a) (14 June 2017): Applications The estimation of the power spectral density (PSD) of a geophysical signal has several applications, for example, in defining covariance models for the geopotential fields, or in designing surveys compatible with instrumentation resolution and accuracy. Formally, the PSD is the Fourier transform of the correlation function. The correlation function, in turn, may be defined for strictly deterministic functions, but usually is defined for stochastic processes, in which case, “covariance” function may be a more appropriate appellation (zero-mean processes are usually assumed). Nevertheless, we briefly start with deterministic functions that are so-called finite-power functions, ( )g x , which, as opposed to finite-energy functions, satisfy

( )2

2

2

1lim

T

TT

g x dxT→∞

−

< ∞∫ . (4.1)

That is, these functions may not attenuate to zero as x → ∞ , but are still bounded in the sense of

(4.1) (compare this to the finite-energy functions that satisfy (1.3)). The correlation function of g is then defined by

( ) ( ) ( )2

,

2

1lim

T

g gT

T

x g x g x x dxT

φ→∞

−

′ ′ ′= +∫ , (4.2)

where the subscripts on φ allow for the definition of a cross-correlation function for two finite-

power functions, ( )g x and ( )h x .

A stochastic process is a function both of a deterministic variable, such as time or a spatial coordinate, as well as an event in a sample space, so that at any point of the deterministic variable, the process is a random variable. As such, there is a probability function associated with each deterministic point and the statistical operator of expectation may be applied as needed. We assume, as is required when formulating corresponding Fourier transforms, that the process is stationary (homogeneous), which means that the probabilities are translation-invariant, i.e., independent of the origin of the deterministic variable. To keep the notation simple, we denote a stochastic process on the real line also by ( )g x , where x is the deterministic variable,

and for fixed jx x= , ( )j jg g x≡ is a random variable.

If ( )gp g is the probability density function of the random variable (the same for every x

under stationarity), then the expected value of ( )g x is

( ) ( )g

gg g p g dgΣ

= ∫E , (4.3)

II.4-2

where gΣ is the sample space. The expected value is also called the mean value; and, again, to

keep things simple, we assume that the mean value is zero. The covariance function of the process is then also the correlation function (but we continue to refer to the covariance function in order to emphasize its stochastic foundation),

( ) ( ) ( ) ( )1 2, 1 2 1 2 1 2 , 1 2 1 2cov , ,

g g

g g g gc g g g g g g p g g dg dgΣ Σ

ξ ≡ = = ∫ ∫E , (4.4)

where

1 2,g gp is the joint probability density function of 1g and 2g .. Because of the translation-

invariance, the covariance function depends only on the difference, 2 1x xξ = − .

It is possible to define the Fourier transform of a stochastic process with respect to the deterministic variable using the theory of distributions (note that the integral of a stochastic process is not defined in the usual Riemannian sense since for every x , ( )g x is a random

variable). We avoid this complication simply by not venturing into it and only assume that the covariance function has a Fourier transform. Indeed, it can be shown (Priestley 1981) that if the process is “continuous in the mean,” then the covariance function is continuous everywhere in the usual sense. Moreover, it is assumed that it attenuates to zero sufficiently fast as ξ → ∞ so

that it has a Fourier transform. Thus, let

( ) ( ) 2, ,

i fg g g gC f c e dπξξ ξ

∞−

−∞

= ∫ . (4.5)

This is also called the power spectral density (PSD) of g because, in principle, being a stationary process, g is a finite-power function of x Under certain conditions, an unbiased and consistent estimator of the covariance function is given by

( ) ( ) ( )2

,

2

1ˆ

T

g g

T

c g x g x dxT

ξ ξ−

= +∫ , (4.6)

where ( )g x is a realization of the process. Being unbiased means that the expected value of the

covariance estimator equals the true covariance function, which is easy to see. Consistency means that the variance of the estimator vanishes in the limit, T → ∞ . We can estimate the covariance function from a realization of the process at a finite number of discrete points, nx , 2, , 2 1n N N= − −… , with constant interval, x∆ , by discretizing (4.6),

( )2 1

,

2

1ˆ , 0, , 1

N

g g n n

n N

c g g NN

− −

+=−

= = −− ∑

ℓ

ℓℓ

ℓ …ℓ

. (4.7)

II.4-3

This is a random variable since the ng are random variables. Thus, one can take its expectation

and it is easy to see that this estimator is also unbiased. However, the estimates for larger ℓ are not reliable ( max 10N=ℓ is sometimes recommended), since they are based on fewer and fewer

products in the summation. Furthermore it can be shown that the estimates may generate a non-positive-definite covariance. To remedy this situation, it is generally preferred to use the biased estimator,

( )2 1

,

2

1, 0, , 1

N

g g n n

n N

c g g NN

− −

+=−

= = −∑ℓ

ℓℓ

ℓ … , (4.8)

which, however, is asymptotically unbiased, but still gives unreliable estimates for larger ℓ . A corresponding estimator of the PSD for frequencies, f f≤ N , is then

( ) ( ) ( )

( )

2 1 2 11 12 2 2

, ,

1 2 1 2

2 1 2 12

2 2

N NN Ni x f i x f

g g g g n n n n n

N n N n N

N Ni x j k f

j k

j N k N

xC f x c e g g g g g e

N

xg g e

N

π∆ π∆

π∆

∆∆

∆

− − −− −− −

+ +=− + =− = =−

− −− −

=− =−

= = + +

=

∑ ∑ ∑ ∑

∑ ∑

ℓ

ℓ ℓ

ℓ ℓℓ

ℓ ℓ

(4.9)

For discrete frequencies, ( )kf f k N x∆= = , this is also

( ) ( )2 2

,

1 1DFTg g k k n k

C f G gN x N x∆ ∆

= =

ɶ , (4.10)

and is known as the periodogram of the finite sequence, ng . It is a biased estimator of the PSD,

not only because of the aliasing associated with the sampling interval, but also because of the spectral leakage due to the finite extent of the data. There is not much one can do about aliasing except to sample the signal at smaller interval, x∆ , or apply a low-pass filter that attenuates the high-frequency content of the signal, which is usually responsible for the aliasing error. On the other hand, the bias due to spectral leakage, especially at the high frequencies can be reduced by applying an appropriate window function or data taper. A function limited to an interval is equivalent to the product of that function and the rectangle function,

( ) ( ) ( ), 2 2

0, otherwiseT

g x T x T xg x b g x

T

− ≤ ≤ = =

,

By the dual convolution theorem (1.21), the Fourier transform of the extent-limited function is, therefore, the convolution of ( )G f and ( ) ( )sincB Tf Tf= . The spectral leakage error is caused

by the main lobe of the sinc function, and significantly by its side lobes, that multiply the true

II.4-4

spectrum and thus smear it out or blur it – the spectral energy at each frequency leaks into its neighbors. The large oscillations of the sinc function are a manifestation of a Gibbs’s effect associated with the discontinuity of the rectangle function at the edge of the domain of the truncated function. Although the main lobe is unavoidable, options exist to minimize or reduce the size of the side lobes (but at the expense of widening the main lobe). Multiplying the original function by a window function that is continuous, or nearly so, at the edges of the limited domain reduces these side lobes and the corresponding spectral leakage. We consider only sequences of data and design window sequences that have the effect of reducing spectral leakage due to truncation. Popular window sequences are the cosine tapers, such as the Hann window sequence

( )( )Hann

1 1 2cos , 1

2 2 2 2

0, otherwise

N

N N

u N

π + − ≤ ≤ − =

ℓ

ℓℓ

(4.11)

and, there are many others (Harris 1978). The discrete prolate spheroidal sequence (dpss) that solves the spectral concentration problem, equation (2.16), is also ideally suited as a taper sequence since it concentrates spectral energy of an extent-limited data sequence in a given spectral band. As an example, Figure 4.1 shows a profile of 600N = magnetic anomalies, sampled at 1.33 kmx∆ = , and its tapered version obtained using the dpss with parameter,

( )0 1f N x∆= , that is the first zero of the transform of the rectangle “taper” (i.e., no taper).

Figure 4.2 then shows the Fourier spectra of the three discrete tapers, rectangle, Hann, and dpss, with all spectra normalized to unity at the origin for ease of comparison. Finally Figure 4.3 compares the unmodified periodogram with the modified versions based on the two tapers, Hann and dpss.

Figure 4.1: Magnetic anomaly data along a profile in longitude in the Dakotas/Minnesota area. The red curve represents the original data compared to the blue curve that is the product of the data and the normalized discrete prolate spheroidal taper (dps taper, also shown). The data and taper sequences are connected by lines for clarity.

II.4-5

Figure 4.2: Amplitudes of energy spectral densities, ( )o

,u u fΦɶ , for several window sequences

with 600N = , 1.33 kmx∆ = , and normalized to unity at the origin for ease of comparison. The half bandwidth for the discrete prolate spheroidal sequence is chosen as ( )0 1f N x∆= . Only

part of the total frequency domain, 0 0.376 cy/kmf f≤ ≤ =N , is shown.

Figure 4.3: Periodograms of magnetic anomaly data (Figure 4.1) according to various normalized tapers. The red line represents the un-tapered data, while the green line corresponds to the discrete prolate spheroidal (dps) taper. The blue line is the periodogram of the data tapered by the discrete Hann window. Although the Hann taper has a significantly larger main lobe than the dps taper, the corresponding modified periodograms are nearly indistinguishable. However, the periodogram

II.4-6

of the un-tapered data appears to have substantially greater power at the higher frequencies, which may be attributed to spectral leakage. More detailed analyses using simulated signals for which the true PSD is known can be found in (Percival and Walden 1993). A second application of Slepian functions, in this case, for the spatial concentration problem, is the optimally efficient representation of a band-limited function within a finite extent. We briefly introduce the problem of representing a band-limited gravitational field in a circular cap on the sphere. Let this gravitational signal be denoted by ( ),s θ λ and given by a finite spherical

harmonic series, as might be determined from a satellite mission such as GRACE or GOCE Barthelmes and Köhler 2012),

( ) ( ), ,

0

, ,K n

n m n m

n m n

s S Yθ λ θ λ= =−

=∑∑ , (4.12)

where the band-limit is defined by maxn K= .

A more efficient representation on a limited domain, RΩ , uses the basis functions that are

optimally concentrated on RΩ , which, if the area of RΩ is small, also are much fewer in number

than ( )21K + , according to (3.34). We thus wish to find the representation,

( ) ( )( )

( )2

1

1 1

, , ,K E

s S g S gα α α αα α

θ λ θ λ θ λ+

= =

= ∑ ∑≃ . (4.13)

From (3.30) and (4.13) we have the equivalent representations,

( )( )

( )2

1

, ,

1 0

, ,K K n

n m n m

n m n

S g S Yβ ββ

θ λ θ λ+

= = =−

=∑ ∑∑ . (4.14)

Multiplying both sides of this by ( ),gα θ λ and integrating over Ω ,

( ) ( )( )

( ) ( )2

1

, ,

1 0

1 1, , , ,

4 4

K K n

n m n m

n m n

S g g d S Y g dβ β α αβ Ω Ω

θ λ θ λ Ω θ λ θ λ Ωπ π

+

= = =−

=∑ ∑∑∫∫ ∫∫ , (4.15)

which implies with the orthonormality (3.31),

( ), ,

0

K n

n m n m

n m n

S S Gα α= =−

=∑∑ . (4.16)

This provides a way to obtain the coefficients, Sα , in the representation (4.13) of the signal in

terms of the Slepian functions, ( ),gα θ λ . The interested reader may refer to Simons et al.

II.4-7

(2006), Simons and Dahlen (2007), Dahlen and Simons (2008) for additional details and applications, as well as to Wang (2012) who, using these Slepian functions, obtained an efficient representation of the GRACE gravitational model to characterize the co-seismic gravitational effect in the local region affected by the 2010 Chile earthquake.

II.4-8

References Albertella, A., Sansò, F., Sneeuw, N. (1999): Band-limited functions on a bounded spherical

domain:the Slepian problem on the sphere. Journal of Geodesy, 73:436-447. Barthelmes, F., Köhler, W. (2012): International Centre for Global Earth Models (ICGEM).

Journal of Geodesy, 86(10):932-934. Courant, R., Hilbert, D. (1966): Methods of Mathematical Physics, Volume I. Interscience

Publishers, Inc., New York. Dahlen, F.A., Simons, F.J. (2008): Spectral estimation on a sphere in geophysics and cosmology.

Geophysical Journal International, 174:774–807. Gohberg, I., Goldberg, S., Krupnik, N. (2000): Traces and Determinants of Linear Operators,

Birkhäuser Verlag, Basel. Harris, F.J. (1978): On the use of windows for harmonic analysis with the discrete Fourier

transform. Proceedings of the IEEE, 66(1):51-83. Jekeli, C. (2017): Spectral Methods in Geodesy and Geophysics. CRC Press, Boca Raton,

Florida. Landau, H.J., Pollak, H.O. (1962): Prolate spheroidal wave functions, Fourier analysis and

uncertainty-III: The dimension of the space of essentially time- and band-Limited signals. Bell Syst. Tech. J., 41(4):1295-1336.

Morse, P.H., Feshbach, H. (1953): Methods of Theoretical Physics, Parts I and II. McGraw-Hill Book Co., Inc., New York.

Percival, D.B, Walden, A.T. (1993): Spectral Analysis for Physical Applications. Cambridge University Press, Cambridge, U.K.

Priestley, M.B. (1981): Spectral Analysis and Time Series, Vols. 1 and 2. Academic Press, London.

Simons, F.J., Dahlen, F.A. (2007): A spatiospectral localization approach to estimating potential fields on the surface of a sphere from noisy, incomplete data taken at satellite altitudes. In Van de Ville, D., Goyal, V.K., Papadakis, M. (eds.), Wavelets XII, Vol. 6701, pp. 670117, doi:10.1117/12.732406, Proc. SPIE.

Simons, F.J., Dahlen, F.A., Wieczorek, M.A. (2006): Spatiospectral concentration on a sphere. SIAM Review, 48(3):504-536.

Simons, F.J., Hawthorne, J.C., Beggan, C.D. (2009): Efficient analysis and representation of geophysical processes using localized spherical basis functions. In: Goyal, V.K., Papadakis, M., Van de Ville, D. (eds.), Wavelets XIII, Vol. 7446, doi:10.1117/12.825730, Proc. SPIE.

Slepian, D. (1976): On bandwidth. Proceedings of the IEEE, 64(3):292-300. Slepian, D. (1978): Prolate spheroidal wave functions. Fourier analysis, and uncertainty – V: The

Discrete Case. Bell Syst. Tech. J., 57(5):1371–1430. Slepian, D. (1983): Some comments on Fourier analysis, uncertainty, and modeling. SIAM

Review, 25(3):379-393. Slepian, D., Pollak, H.O. (1961): Prolate spheroidal wave functions, Fourier analysis and

Uncertainty – I. Bell Syst. Tech. J., 40(1):43–63. Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K. (1988): Quantum Theory of Angular

Momentum. World Scientific Publishing Co., Singapore. Wang, L. (2012): Coseismic deformation detection and quantification for great earthquakes

using spaceborne gravimetry. Report no.498, Geodetic Science, Ohio State University, Coumbus, Ohio, https://earthsciences.osu.edu/sites/earthsciences.osu.edu/files/report-498.pdf.