Gegenbauer Transforms via the Radon Transform

SlAM J. MATH. ANAL.Vol. 10, No. 3, May 1979

1979 Society for Industrial and Applied Mathematics0036-1410/79/1003--0014 $01.00/0

GEGENBAUER TRANSFORMS VIA THE RADON TRANSFORM*

STANLEY R. DEANS"

Abstract. Use is made of the Radon transform on even dimensional spaces and Gegenbauer functions ofthe second kind to obtain a general Gegenbauer transform pair. In the two-dimensional limit the pair reducesto a Chebyshev transform pair.

1. Introduction. Gegenbauer polynomials of the first kind appear in a natural waywhen studying the Radon transform of functions which have certain spherical sym-metry. We shall make use of this property of the Radon transform to obtain a newGegenbauer transform pair. Although the final result does not contain Gegenbauerfunctions of the second kind, these functions are important in the derivation and theiruse here supplements the informative recent study of these particular special functionsby Durand [1] and by Durand, Fishbane, and Simmons [2].

The work which follows serves a threefold purpose. First, we are able to demon-strate an important use of the Radon transform as a tool. Second, more insight isobtained regarding the use of Gegenbauer functions of the second kind. Useful materialon these functions is contained in the Appendix. Finally, we derive a set of equationswhich constitute a Gegenbauer transform pair with a fundamental connection to thedimensionality of the space

2. The Radon transform. Let x (xl, X2, Xn) be a point in R (n -> 2) and letF 6 be a function of the n real variables xl, x2, , x,,. The properties of the space ,which consists of all rapidly decreasing C functions on R", are developed by Schwartz[3]. The reason for working in a space with such nice properties will be clear when itbecomes necessary to make changes in the order of integration and to perform repeatedintegrations by parts.

Given F 6e, the Radon transform of F is given by [4],

(1) f(cs’ P)= I F(x)(p- x) dx,

where p is real, is an arbitrary unit vector in [n, sc. x Y’.= sCKx, is the Dirac deltafunction, dx dxa dx2 dx,,, and the integral is over the entire space. It is importantto observe that the symmetry condition

(2) f(d, -p)= f(-, p)

follows directly from the definition (1).Following the initial work by Radon [5], many of the technical properties of the

Radon transform were worked out by several authors [4], [6]-[ 10]. Among other thingsthese authors develop a formal expression for inversion of the transform, valid forfunctions in 5, and it turns out that the inversion formula for even n is considerablymore complicated than the formula for odd n. There is a Hilbert transform associatedwith the even case which remains unevaluated for the most general functions. Ourconcern here is with this even n case exclusively and involves defining F in such afashion that it is possible to perform the Hilbert transform.

* Received by the editors January 27, 1978, and in revised form June 8, 1978. This work was partiallysupported by the U.S. Department of Energy.

t Department of Physics, University of South Florida, Tampa, Florida 33620, and Lawrence BerkeleyLaboratory, University of California, Berkeley, California 94720.

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578 STANLEY R. DEANS

3. Decomposition of F. We consider those functions F which may be decomposedeither as

(3) F(x) Gl(r)Slm(..),

or as a linear combination of terms of this form. Here, x I", r Ix I, x/r, and thedoubly subscripted St,() is a real spherical (or surface) harmonic [11], [12] of degreewhich comes from an orthonormal set with N(n, l) members. That is, members of theset {S/l(.), S/2(.),""", SIN(n,l)()} satisfy

(4) In Slrn ()Sl’m’() d ll’mm’

where d is understood to be the surface element in hyperspherical polar coordinates,andn designates an integral over the unit sphere. A very useful property of the Sl() isthat they satisfy the symmetry condition

(5) Sl(-) (--)lSl().Further properties, including an explicit expression for N(n, l), can be found inHochstadt 12].

4. Radon transform of the decomposed function. When the Radon transform (1)is applied to (3) the result is

(6) [(, p)= Gl(r)S()(p-. x) dx.

Without loss of generality we may assume that p 0 since one may always calculate[(,-p) from (2). If we convert (6) to spherical coordinates (dx r- dr d) andobserve that 8(p-r. )= (p/(. )-r)/l" 1 for purposes of doing the r integra-tion we obtain

(7) f(, p): Slm() 2 al(p/(" ))l" "Application of the Hecke-Funk theorem [12] yields

(8) f(’ P))n-lSlm()Iol()n-lc7(1) ()Gt C (t)(1 t2)-l/2dtt’

where C’ (t) is a Gegenbauer polynomial of the first kind, ton is the surface area of a unitsphere in Rn, and v 1/2(n 2). The ratio tOn-x/C’ (1) may be written in terms of Gammafunctions,

(4,r)F(/+ 1)F(v)(9) Mr ton_,/ C7 (1)

F(/+2v)

Equation (8) may be converted to the desired form by making the change ofvariables r p t,

(10) f(sc, p)= MS,.(j) rG(r)Cl 1 dr.

It will be especially useful to write this equation as

(11) f(:, p)= gt(p)Slm(),

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GEGENBAUER TRANSFORMS VIA THE RADON TRANSFORM 579

where

(12) g(p) M r2G(r)C 1- dr.

The symmetry conditions on/c and St, yield the defining equation for gl(-p),

(13) g(-p) (-1)g(p).5. The inversion. We now turn our attention to inverting the Radon transform

when F is given by (3) and f is given by (11). The inversion may be written as anintegration over a unit sphere in : space [9]

(14) F(x) Ia f*(j’ x)d,

where

(15) f*(, x)= Yf(, p).

(Keep in mind that sc is already a unit vector. We have used the notation ? in (14) toemphasize that the integration is over the unit sphere.) For even n the operator isdefined by

2(2r)._H f(, p)

and H designates the Hilbert transform

(17) H{q(p)} --1 [’ q(P) dp.7r ._p-t

After inserting the decompositions for F and " in (14) we have

Gl(r)Sm() [.. Slm()gf (r .) d"(18)

M[Stm(;) 1 gf (rt)C (t)(1 t2) -1/2 dt,.1_

where the second step was obtained by applying the Hecke-Funke theorem again [12].By inspection of (18) it is clear that

(19) Gl(r) M| g’{ (rt)C (t)(1 t2) ’-1/2 dt3--

and g must be calculated from g Ygl.

6. Improvement on the inversion formula. For even n (2, 4, 6,...) and u1/2(n- 2) it is possible to modify (19) considerably by actually doing the integration.Explicitly, g{ (rt) is given by

(--1)n/2 1 I_(20) gf (rt)2(2r),,_ --r g"-x) (P)(P rt)- dp,

where gl"-)(p)=(d/dp)"-gt(p). If (20) is substituted into (19) and the order ofintegration reversed, the equation for G(r) becomes

(--1)n/2 Mr g,,_) p(21) at(r) 2(2r)._a rr (p)Ir dp,

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where

(22) I’(;)= 1C(t)(-t) (1-t2)-l/Edt.

At this point it is clearly desirable to require that r > 0. The r 0 case may be doneseparately.-rStartingo w)th (ool 9i The integration in (22) may be taken over four separateregions, -oo +-r+0 +jr we observe that I(-p/r) (-1)’/1I’(p/r) then by achange of variable p -p over the negative p region in (21) it follows that

(_1)-/ M’ gn-1) (p)! dp + gn-X) (p)I dp(23) G(r) (2,a.),_------r rr--7The reason for writing Gt(r) in this form is to enable us to evaluate the I’ integrals.

Notice that in (23) the o integral forces p/r _-< 1 and the , integral forces p/r >= 1.For convenience we momentarily designate

P=x if P< 1,

P=z ifP>l.r r

(Unlike prior usage of x, here x is a real variable rather than a vector.) This establishescontact with the usage of x and z in the Appendix and in [1], i2] where the Gegenbauerfunctions of the second kind D are discussed.

From (A.1) and (A.3) we immediately obtain

I’ (x) rr(1 X2)v-1/2D’[(X)(24)

and

(25) I’[ (z) 2rr e-i’’(z 2-1)"-*/2D’[(z).These results, combined with (23) give

M’ {(-1)"/2 gn-1)(p)Dr(x)(1-x2)-l/2 dpG(r)(2) r

(26)

2 g,-x) (p)Dr(z)(z2_ 1)-x/2 dp

By use of (A.4) and (A.14) the two integrals may be combined, and after somesimplification we find

-v-1

r(/+2)rg (p)N__ dp

=F(/+ 1)F(v) g"-) (p)C 12r F(/+2v)r

The o integral can be shown to vanish. To see this, first perform n 1 integrationsby parts to obtain an integral of the form

where Ol-2(z) is a polynomial of degree I-2, and Ol_2(-z)=(-1)lOl_2(z). (The

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integrated parts always vanish by symmetry.) Next, make use of (12) to replace gl(p).This leads to an integral of the form

A change in the order of integration over the indicated region of the pt plane leads to

I dtt2VGl(t) fot dp QI-2() C[() (1- ()2]v-lNow the p integration can be shown to vanish. If the variable change p yt is made, andthe symmetry of the functions in the integrand taken into account the p integralbecomes (aside from a constant factor)

I_ QI-2()C7 (y)(1-y2)U-1/2 dy.

Since Q-2 is a polynomial of degree I-2 in y it follows by orthogonality that thisintegral vanishes.

Hence we finally have the desired result, which consists of the Gegenbauertransform pair,

-r(/+ )r() g"-’ (p)C[ 1 dp(28) G(r)27r/1F(/+ 2u)r

and

(29) gt(p)(4"rr)"F(/+ 1)F(u) r2Gt(r)C[ 1

p 2 ,-1/2

F(l+2u)dr,

where , 1/2(n- 2) and the dimensionality n is even (n 2, 4, 6,. .).

7. Limiting ease n = 2. It is especially interesting to examine the n 2 limitingcase of the above transform pair since that corresponds to the Radon transform on aplane. The result is straightforward if one first multiplies by u/u and then lets u 0. Theresult is the Chebyshev transform pair [13]

(30) G,(r) =-1 g(p)T 1 dp

and

(31) gt(p) 2IfGt(r)Tl()[1 ()21-1/2 dl?’o

8. Closing remarks. The connection between Radon and Gegenbauer transformswas observed by Ludwig I-9] in his discussion of the Radon transform on Euclideanspace where it was pointed out that results about the Radon transform imply cor-responding results for Gegenbauer transforms and conversely. (Apparently the Radontransform--Gegenbauer transform connection was known prior to Ludwig’s work, butnot published. See the remark in the introduction to Ludwig’s paper.) For spaces ofeven dimension the inversion formula obtained in [9] involved both a Hilbert transformand an integration with a Gegenbauer polynomial. The work here shows that after theHilbert transform is performed the resulting inversion formula (28) still may beexpressed as one member of a Gebenbauer transform pair. Recent work with odd

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dimensional spaces [14] shows that with minor modification the inversion formula (28)also holds for odd n. The only modification required is that the overall minus sign in (28)must be changed to plus. Finally, it should be pointed out that the transform pair (28),(29) is not the only possible transform pair involving Gegenbauer polynomials, and themethod here using the Radon transform is not the only way to obtain such pairs. Forexample, another transform pair was obtained by Higgins [15] using a very differentapproach, and a still different procedure developed by Sneddon 16] utilizing the Mellintransform could be adapted to solving the inversion problem discussed here.

Appendix. In this appendix we collect several formulas which are needed in thepreceding work. Some of these are included for convenience and may be found in [2].Others, notably those involving the Chebyshev functions, do not seem to be available inthe standard sources. Our notation and conventions conform to that used by Durand,Fishbane, and Simmons [2], since their treatment of the Gegenbauer functions is thebest available source for the type of results needed here. These authors derive manyproperties of the Gegenbauer functions of the first kind C (z) and second kind D(z)for general values of a, A, and z. Our concern here is primarily with the restricted casewhere both a and A are nonnegative integers (designated by writing a v and A l)and z is real. We use x (in place of z) to emphasize that the argument lies on the interval[-1, + 1] or [0, 1] and z whenever the argument is complex or greater than unity.

For integral h and Re a >-1/2, D and C are related by [2]

(A.1) D’(z) ei"’(z 2 1)1/2_, 1 [ C (t)(z t)-l(1 t2)-/2 dt.2r J_l

To obtain D7 (x) we make use of the general prescription

D(x)= lime [ei,m(x+ie)_e-i,D(x_ie)]e-0

(A.2)i=

(z2-1)+ (1 -x2) e for (x + ie),(1 x) e-i for (x ie ).

This yields

(A.3) D{(x) (1_X2)1/2_, 1 I C{ (t)(x t)-l(1 t2)’-1/2 dt."/r

For integral a v the following relation holds

(A.4) z2- 1)v-ll2DT(z)=1/2(Z2- 1)v-112C7 (z)---7-TEI+2v-1 (z),ltv)

where El+2,,_(z) is a polynomial of degree l+2v-1. These polynomials can beexpressed in terms of associated Legendre functions of the first kind,

(A.5) E"I+2v -1 (Z) () 1/2

(Z 2 1) 1/2(v--1/2)DV--1/2_r l+v_l/2 (Z)

Explicitly, in terms of Chebyshev polynomials of the first kind T and second kind Ul(the argument may be either z or x),

1EP_, 7 U,_,

(A.6) E+I T/+I,

EL3 =1/2[(/+ 1)Tt+3- (1 + 3) T/+I].

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In general,

_2- v-1 (l+v-k-1)!(/+2v-1)!(A.7) E’+_- 2 (-1) r+_._,

k=O k l (l+2-k- 1)

where we have used the standard symbols for factorials and binomial coecients. TheE’s satisfy the recursion relation

(A.8) El+2v-1 (z) (I + 1)zET_2(z) (I + 2V --2)El+2v-3(z)V-1and the symmetry property

(A.9) E+2-1 (-z)= (-1)/+1E+2_ (z).

Explicit results for the functions D may be written conveniently in terms of thefunction V(z) where

(A.10)

with recursion relation

(A.11)

We have

Vl(Z) Tl(Z)-(Z2- 1)’/2Ul-l(Z),

gl+z(Z) 2z gl+l(Z)- gl(z).

1D(z) .-olim 1D’}’(z)=--[a V(z),(A.12) D (z)=-(z2- 1)-1/2Vl+1(Z),

D (z) -1/4(z- 1)-3/ 1/2[(/+ 1) V+(z)-(l + 3) V+(z)].

In general,

(A.13)

D’{(z) -(z2-1)1/2-vv-1E (-1)k( )v-122v-iF(v) k=0 k

(l+v+k-1)! (l+2v-1)!l!(l+2v-k-1)!

By application of (A.2) and (A.4),

(-1)"+(1-X2)1/2-’’

(A. 14) D’{(x) 2._r() ElV+2v-1 (X).

The Chebyshev expansion for the C’ is given by

2C (z)= .-olim --al C7 (z) - Tl(Z),

(A.15) c (z)= U(z),

In general,

(A.16) C’ (z)

C (z)=1/4(z2- 1)-1[(/+ 1)Ut+2(z)-(l + 3) Ut(z)].

(Z2--1)1-vv-14-’r’(v) E= (_1)k u-k i (1 +v-k-1)!(l+2v-1)!

l!(l+2v-k-1)!

These results also hold for z x.

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For future reference, we examine another form for D’ which is especially valuablefor large values of or z. We first define r (z - 1) 1/2 and observe that

(z + ,)-1 z " and (z + r)- 1 2sr(z + r).

In terms of these variables we have

1D(z) =-[(z )-l,

-1(A. 17) D (z)= --(z + st)--1,

D(z)= --(z + .)--/--2 1 +

Higher terms have the form

(A.I8) D[(z)(-l)/’-1(z +

Or, in general,

(A.19)

v(v- 1). z + 2_______ O(/_2)}.2l "DT(z)

(_l),,l,,-l(z + ()-l-, v-1

2"F(v)sr"(2/st)l-" =o2 (-1) V-k 1

(l+k)!(l+2v-1)!l!(l+v+k)!

Acknowledgments. It is a pleasure to thank the Theoretical Division, Los AlamosScientific Laboratory, University of California, Los Alamos, New Mexico, for theirhospitality during the summer of 1977, when this work originated. Discussions withProf. T. F. Budinger and G. T. Gullberg are gratefully acknowledged along with helpfulsuggestions from the referee.

REFERENCES

[1] L. DURAND, Nicholson-type integrals for products of Gegenbauer functions and related topics, Theoryand Application of Special Functions, R. A. Askey, ed., Academic Press, New York, 1975, pp.353-374.

[2] L. DURAND, P. M. FISHBANE AND L. M. SIMMONS, JR., Expansion formulas and addition theoremsfor Gegenbauerfunctions, J. Math. Phys., 17 (1976), pp. 1933-1948.

[3] L. SCHWARTZ, Mathematics ]’or the Physical Sciences, Addison Wesley, Reading, MA, 1966.[4] I. M. GEL’FAND, M. I. GRAEV AND N. YA. VILENKIN, Geneneralized Functions, vol. V, Academic

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keiten, Ber. Verh. Siichs. Akad., 69 (1917), pp. 262-277.[6] F. JOHN, Bestimmung einer Funktion aus ihren Integralen iiber gewisse Manningfaltigkeiten, Math.

Ann., 109 (1934), pp. 488-520.[7] ., Plane Waves and Spherical Means Applied to Differential Equations, Interscience, New York,

1955.[8] S. HELGASON, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and

Grassman manifolds, Acta Math., 113 (1965), pp. 153-180.[9] D. LUDWIG, The Radon transform on Euclidean space, Comm. Pure. Appl. Math., 19 (1966), pp.

49-81.10] K. T. SMITH, D. C. SOLOMON AND S. L. WAGNER, Practical and mathematical aspects of the problem

of reconstructing objects from their radiographs, Bull. Amer. Math. Soc., 83 (1977), pp. 1227-1270.[11] A. ERDILYI, W. MAGNUS, F. OBERHETTINGER AND F. G. TRICOMI, Higher Transcendental

Functions, vols. I, II, McGraw-Hill, New York, 1953.

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[12] H. HOCHSTADT, The Functions ol Mathematical Physics, John Wiley, New York, 1971.13 S. R. DEANS, The Radon transform: Some remarks andformulas ]:or two dimensions, Lawrence Berkeley

Laboratory Report LBL-5691, 1977.[14],A unified Radon inversion formula, J. Math. Phys., 19 (1978), pp. 2346-2349.15] T. P. HIGGINS, An inversion integral for a Gegenbauer transformation, J. Soc. Indust. Appl. Math., 11

(1963), pp. 886-893.16] I.N. SNEDDON, A procedure ]:or deriving inversion formulae for integral transform pairs o] a general kind,

Glagow Math. J., 67 (1968), pp. 67-77.

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Documents

Gegenbauer Transforms via the Radon Transform