On the Expansion of Harmonic Functions in Terms of Harmonic PolynomialsAuthor(s): J. L. WalshSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 13, No. 4 (Apr. 15, 1927), pp. 175-180Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/84872 .

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PROCEEDINGS OF THE

NATIONAL ACADEMY OF SCIENCES Volume 13 April 15, 1927 Number 4

ON THE EXPANSION OF HARMONIC FUNCTIONS IN TERMS OF HARMONIC POLYNOMIALS

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY

Communicated March 4, 1927

Fourier's Series is an expansion of an arbitrary function f(O) of period 2Ir in terms of trigonometric functions:

co

f(0) = E (ak cos kO + bk sin kO). k=O

If (r, 0) are considered polar co6rdinates in the plane and (x, y) the cor- responding rectangular co6rdinates, the function f(O) may be considered defined on the unit circle r = 1, and on that circle expanded in terms of trigonometric functions. But if the series be written in the form

oo

f(r, 0) = > rk(ak cos kO + bk sin kO), k=O

we have the same expansion for r = 1, yet each term of the series is a har- monic function of (x, y), and indeed a harmonic polynomial in those var- iables. Moreover, if the series converges uniformly in 0 for r = 1, the series converges likewise uniformly in the closed region r ? 1. The sum of the series is, therefore, harmonic for r < 1 and continuous for r _ 1, although the original function f(O) is defined merely on the circle r = 1.

Little seems to have been done hitherto in the literature in consideration of expansions in terms of harmonic polynomials-expansions of arbitrary functions defined on given curves, or expansions in regions of harmonic functions d'efined in those regions-beyond the illustration just given.' The object of the present note is to give an example of such expansions, generalizations of Fourier's Series, by proving the following theorem:

THEOREM 1. Let C be a simple closed finite analytic curve in the (x, y)- plane. Then there exist harmonic polynomials

Pi(x, y), p2(x, y), p3(x, y), ....

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176 MATHEMATICS: J. L. WALSH PROC. N. A. S.

such that if f(x, y) is defined and continuous on C and on C is of bounded

variation,2 then f(x, y) can be developed into a series

f(x, y) = alp,(x, y) + a2p2(x, y) + ap(x, y) + ...., (1)

where the series (1) converges uniformly in the closed region consisting of C and its interior. The series (1) thus represents a function harmonic interior to C, continuous in the corresponding closed region, and having the value f(x, y) on C. There exist continuous functions qn(x, y), n = 1, 2, 3, ...., defined on C, such that the coefficients of (1) are given by the formulas

an = f(x, y)qn(x, y)ds, n = 1, 2, 3, ... ; JC

the functions qn(x, y) depend on C but not on f(x, y). Theorem 1 is an almost direct application of the following theorem.3 THEOREM 2. Suppose that {un(Q) } is a set of uniformly bounded con-

tinuous normal orthogonal functions in the interval 0 ? sp _ 2Ir, and that in this interval {un(op)} is a set of uniformly bounded continuous functions each of which can be developed into a series

0o

un(q) = Un,(,) +- CnkUk() n , n , 2 ..... (2) k=l

where the coefficients have the values

r27r

Cnk = / [U,(O) - Un(()]Uk(p)dp. (3) Jo

Suppose further that the three series

E E t , (4) n,k1l = k=l / k=l

converge and that the sum of the first is less than unity. Then there exists a set of continuous functions {Vn(sp)} such that {un(ip)}

and {v,n() } are biorthogonal sets:

r ()vk(o)d = O, n = k,

-o, n=k.

Furthermore, if f((p) is any function integrable and with an integrable square

(in the sense of Lebesgue) on the interval 0 ? <p e 27r, then the two series

f(s) E a,n(o) , a = f(<)vn(<)dp, n=l J

fo - r2r ,

f() ~ > bnn ,, b f(~o)Un(n<p)dnU, n=l JQ

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VoL. 13, 1927 MATHEMATICS: J. L. WALSH 177

have essentially the same convergence properties, in the sense that their term-

by-term difference converges uniformly and absolutely to the sum zero on the interval 0 ?< s < 27r.

Theorem 2 will be shown to yield the following result:4 THEOREM 3. Let the functions

pl(x', y'), p(x', y'), p(x', y'),...

be harmonic for p < 1 + C, e > 0, where (x', y') are rectangular coordinates in the plane and (p, sp) the corresponding polar coordinates, and suppose we have on and within the circle y': p = 1 -+ , the following inequalities:

pl(x, y')-- < , 427T

2n(x', y') - p COs nSp 2n , (5)

n =1,2,3,....

P2n+l(X ', ) - p sin ny ? C_2n+ t

c 2 1

whe converes a i e the series converges and its sum is ess than and where the n=1 2Ir

series E C,, converges. Then any function f'(x', y') which is continuous n=l

and of bounded variation on the circle y: p = 1, can be developed into a series

f'(x', y') = alp(x', y') + ap (x', y') + ....,

where the series converges uniformly for p < 1; the sum of the series is thus harmonic for p < 1 and continuous for p < 1. There exists a set of functions qn(x', y'), defined and continuous on y, such that the coefficients a' are given by the formulas

a = f'(x' yq( x' y')dp;

the functions q'(x', y') do not depend on the function f'(x', y'), but merely on the functions p(x', y').

Theorem 3 is proved from Theorem 2 by setting on y, 0 < s < 27r,

un(c) = P(x', y'), n = 1, 2, 3, ....,

1 1 - 1 u =(vo ) -= - cos np, U2n+l (O) = V- sin nyo, n= 1, 2, 3,....

The sets of functions {u } and {u } are obviously uniformly bounded, and the set {Un } can be developed into series (2) in terms of the functions

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178 MATHEMATICS: J. L. WALSH PROC. N. A. S.

u,n}, for the functions P',(x', y') are harmonic and hence analytic functions of the real variables x' and y' (and of the variable sp) for p _ 1 + E. It remains merely to consider the series (4).

In order to compute the coefficients cnk, we find it convenient to expand p'(x', y') into a Fourier Series on the circle y':5

pj(x', y') - -rn(9) =2 n P( x} y) -p Un() = --- + (Cn2 cos S+cn sin p) n even,

p2 = n-I

2 + - (c%4 cos 2s + Cn,5 sin 2() + ', ' 2 ' n odd,

(6)

pkCn2k = - [ptn(X, Y')-pmu n(Q) ]u2k( p)d, k 1, 2, 3, ....,

27r

p Cn2k+l = [pn(x , y) - p n(p)]2kl()d k = 0, 1, 2, ...

the development being valid for p = 1 + e and hence likewise for all values of p less than 1 + E, in particular valid for p = 1.

Bessel's Inequality as derived from (6) for p = 1 is

Ecnk -<- [n( -) U-n((p) ] 2dqo. k=l

This integral is not greater than 2Ir e-. Hence the first of the series (4) converges to a sum less than unity, and the second of those series converges.

From (6) for p = 1 + e and from the inequalities (5) we derive

I c1 < A/2T En, I 1 <

2IC I c I2k+ < ? +2V n2kj (1+)k)'

n,k = 1, 2,3,....

The series E c2 2k and E c 2k+l are both dominated by the series n=l n=l

3 47r 6 n=l ( + )2k

47 whose sum is less than Hence the series

(1 + )2k'

Es(sE ^2 )2 k=l \nl

converges and Theorem 3 is completely established. We are now in a position to prove Theorem 1. The interior of C can

be mapped one-to-one and conformally on the interior of y, and the map- ping will be conformal and smooth in larger regions containing C and y,

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VOL. 13, 1927 MATHMAATICS: J. L. WALSH 179

respectively, in their interiors.6 We choose a circle 7' within the latter

region but exterior to y and concentric with it; the transform of 7y' is a

simple analytic closed curve C'. The transforms of the functions

__1 _ 1 1 /=,- /r p cos P, - p sin , - p2 cos 2C , 2 p2 in 29,.. (7)

are harmonic in (x, y) in and on C', hence in the corresponding closed

region can be uniformly approximated as closely as desired by harmonic polynomials in (x, y).7 These approximating polynomials are defined to be the polynomials

PI(x,y), p(,y), p3(X, y), ....

respectively, and approximation is to be so close that for a suitable (that is, satisfying the requirements of theorem 3) set {cE}, inequalities (5) are satisfied in the (x', y')-plane; the transforms of the polynomials p,(x, y) are, of course, to be identified with the harmonic functions p'(x',y') of Theorem 3.

The functions q'(x, y) are to be defined on C in terms of the functions

q' (x', y') of Theorem 3 by the equation

d(p I I qn(x, y) = ds qn(x, y),

ds

where the equation refers to points of C and y which correspond under the conformal map already used, and where s is arc length on C.

If f(x, y) is continuous and of bounded variation on C, its transform in the (x',y')-plane is likewise continuous and of bounded variation. Any continuous function of bounded variation can be uniformly expanded on

7 in terms of the Fourier functions (7), hence on y in terms of the functions p'(x', y'), that is, on C in terms of the polynomials p,(x, y). The series of polynomials converges uniformly on C, hence uniformly in the closed region consisting of C and its interior. Series (1) thus furnishes a solution of the Dirichlet Problem for the boundary values f(x, y) in the region bounded by C, if f(x, y) is continuous and of bounded variation on C. If f(x, y) is merely known to be continuous on C, the sequence of first Cesaro means for (1) converges uniformly in and on C, and thus furnishes a solution of the Dirichlet Problem for these boundary values.

The writer hopes to study further properties of the expansions (1) and of analogous expansions of functions in harmonic polynomials, including orthogonal harmonic polynomials.

1 The present writer has studied the question from the standpoint of uniform ap- proximation (or of uniformly convergent developments), deriving extensions of Weier- strass's theorems on approximation of a continuous or harmonic function, rather than extensions of Fourier's Series.

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180 ASTRONOMY: B. P. GERASIMOVIC PROC. N. A. S.

See a forthcoming paper in Crelle's Journal. Compare also Bergmann, S., Math. Annalen, 86 (1922), 238-271, who considers, in

general, developments of harmonic functions which can be obtained from developments of analytic functions.

2 It is likewise sufficient if f(x, y) satisfies a Lipschitz Condition,

If(xl, Y) -f(x2, y2) I < M/(xi - X2)2 + (y - y2)2,

where M is constant and (xi, yi) and (X2, yz) are arbitrary points of C. 3 Walsh, J. L., Trans. Amer. Math. Soc., 22 (1921), 230-239, Theorem 1. 4 Theorem 3 is analogous to a theorem proved by means of Theorem 2, for analytic

instead of harmonic functions. See Walsh, Trans. Amer. Math. Soc., 26 (1924), 155-170. That theorem for analytic functions is essentially a modification of a result due to Birk- hoff, Paris, Comptes Rendus, 164 (1917), 942-945.

The same development is, of course, obtained by expanding pt (x', y') on -y instead of on 7y; the coefficients Cnk may be obtained by integration on either y or r'.

6 See, e.g., Picard, Traite d'Analyse, t. II (Paris, 1893), pp. 272, 276; Bieberbach, Einfiihrung in die konforme Abbildung (Sammlung G6schen, Berlin u. Leipzig, 1915), p. 120.

7 This result may be easily proved from Runge's classical theorem on the expansion of an analytic function in terms of polynomials. A proof is given in the paper referred to under 1.

ON THE CORRECTION TO SAHtA'S FORMULA FOR SMALL DE- VIATIONS FROM TTIERMODYNAMIC EQUILIBRIUM

BY B. P. GERASIMOVIC

HARVARD COLLEGI OBSERVATORY, CAMBRIDGE, MASSACHUSETTS

Communicated March 4, 1927

All applications of the ionization theory to solar and stellar physics are

based on an implicit supposition that the layers of the star where spectral lines arise are in a state of thermodynamic equilibrium, which is deter-

mined by the effective temperature of a celestial body. In fact, the use

of Planck's law of radiation and the "principle of detailed equilibrium" of quantum processes alone allow us to deduce Saha's formula-the basis

of the modern theory of stellar spectra. This hypothesis is, of course,

quite justifiable if we have in view only a first approximation-some rough

explanation of the variation of spectra with temperature. But as soon as

we want to deepen the theory there arises the necessity for some revision

of the above-mentioned fundamental hypothesis. It is theoretically quite clear that the upper photospheric layer and the

adjacent thin reversing layer cannot be exactly in thermodynamic equi- librium, because their temperature is lower than that of the radiation

penetrating them from below by a factor of V2, according to Schwarz-

180 ASTRONOMY: B. P. GERASIMOVIC PROC. N. A. S.

See a forthcoming paper in Crelle's Journal. Compare also Bergmann, S., Math. Annalen, 86 (1922), 238-271, who considers, in

general, developments of harmonic functions which can be obtained from developments of analytic functions.

2 It is likewise sufficient if f(x, y) satisfies a Lipschitz Condition,

If(xl, Y) -f(x2, y2) I < M/(xi - X2)2 + (y - y2)2,

where M is constant and (xi, yi) and (X2, yz) are arbitrary points of C. 3 Walsh, J. L., Trans. Amer. Math. Soc., 22 (1921), 230-239, Theorem 1. 4 Theorem 3 is analogous to a theorem proved by means of Theorem 2, for analytic

instead of harmonic functions. See Walsh, Trans. Amer. Math. Soc., 26 (1924), 155-170. That theorem for analytic functions is essentially a modification of a result due to Birk- hoff, Paris, Comptes Rendus, 164 (1917), 942-945.

The same development is, of course, obtained by expanding pt (x', y') on -y instead of on 7y; the coefficients Cnk may be obtained by integration on either y or r'.

6 See, e.g., Picard, Traite d'Analyse, t. II (Paris, 1893), pp. 272, 276; Bieberbach, Einfiihrung in die konforme Abbildung (Sammlung G6schen, Berlin u. Leipzig, 1915), p. 120.

7 This result may be easily proved from Runge's classical theorem on the expansion of an analytic function in terms of polynomials. A proof is given in the paper referred to under 1.

ON THE CORRECTION TO SAHtA'S FORMULA FOR SMALL DE- VIATIONS FROM TTIERMODYNAMIC EQUILIBRIUM

BY B. P. GERASIMOVIC

HARVARD COLLEGI OBSERVATORY, CAMBRIDGE, MASSACHUSETTS

Communicated March 4, 1927

All applications of the ionization theory to solar and stellar physics are

based on an implicit supposition that the layers of the star where spectral lines arise are in a state of thermodynamic equilibrium, which is deter-

mined by the effective temperature of a celestial body. In fact, the use

of Planck's law of radiation and the "principle of detailed equilibrium" of quantum processes alone allow us to deduce Saha's formula-the basis

of the modern theory of stellar spectra. This hypothesis is, of course,

quite justifiable if we have in view only a first approximation-some rough

explanation of the variation of spectra with temperature. But as soon as

we want to deepen the theory there arises the necessity for some revision

of the above-mentioned fundamental hypothesis. It is theoretically quite clear that the upper photospheric layer and the

adjacent thin reversing layer cannot be exactly in thermodynamic equi- librium, because their temperature is lower than that of the radiation

penetrating them from below by a factor of V2, according to Schwarz-

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