GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 3, 1998, 213-232

ALLIED INTEGRALS, FUNCTIONS, AND SERIES FOR THE UNIT SPHERE *

O. DZAGNIDZE

Abstract. Among the functions defined on the two-dimensional unit sphere we distinguish functions generalizing the conjugate integral, the conjugate function, and the conjugate series which depend on one variable. We establish the properties of these functions whose structures essentially differ from those of integrals, functions, and series based on the theory of analytic functions of two complex variables.

Introduction

0.1. Let a function F (X, Y, Z) be defined on the two-dimensional unit sphere σ = {(X, Y, Z) : Z^{2} + Y^{2} + Z^{2} = 1} and F ∈ L(σ). Denote by U_{F} (z, y, z) the Poisson integral for the three-dimensional unit ball 𝔹 = {(x, y, z) : x^{2} + y^{2} + z^{2} < 1} with density F (X, Y, Z), (X, Y, Z) ∈ σ, (x, y, z) ∈ 𝔹, which has the form

U_{F} (x, y, z) =
= \frac{1}{4π} ∫_{σ} F (X, Y, Z) \frac{1 − (x^{2} + y^{2} + z^{2})}{[(X − x)^{2} + (Y − y)^{2} + (Z − z)^{2}]^{3/2}} dS. (0.1)

By introducing the spherical coordinates (r, θ, ϕ), x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ (0 ≤ r < 1, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π), the function F is transformed to a function f(θ, ϕ) defined on the rectangle R = {(θ, ϕ) : 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π} so that f(θ, ϕ) · sin θ ∈ L(R). Hence U_{F} (x, y, z)

1991 Mathematics Subject Classification. 33A45, 33C55, 42A50. Key words and phrases. Spherical Poisson integral, allied functions (integrals, kernels,

series).

* The results of this paper were announced in the author’s paper [1], where the terms used differ slightly from those of this paper.

213

1072-947X/98/0500-0213$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

takes the form (see, e.g., [2], p. 445)

U_{f}(r, θ, ϕ) = \frac{1}{4π} ∫_{0}^{π} ∫_{0}^{2π}f(θ′, ϕ′)P_{r}(θ, ϕ; θ,′ ϕ′) sin θ′ dθ′dϕ′,  (0.2)

which is called the spherical Poisson integral of f(θ, ϕ), (θ, ϕ) ∈ R.

The spherical Poisson kernel P_{r} can be represented as follows (see, e.g., [3], pp. 335 and 143):

P_{r}(θ, ϕ; θ′, ϕ′)= \frac{1 − r^{2}}{(1 − 2r[cos θ cos θ′+sin θ sin θ′ cos(ϕ−ϕ′)]+r^{2})^{3/2}} (0.3)

and

P_{r}(θ, ϕ; θ′, ϕ′) = 1 + ∑_{n=1}^{∞}(2n + 1)r^{n}P_{n}(cos θ)P_{n}(cos θ′) +

+2 ∑_{n=1}^{∞}(2n + 1)r^{n} ∑_{m=1}^{n} \frac{(n − m)!}{(n + m)!} P_{nm}(cos θ)P_{nm}(cos θ′) cos m(ϕ − ϕ′). (0.4)

First and second order derivatives with respect to θ and to ϕ of the spherical Poisson integral U_{f}(r, θ, ϕ) have been investigated, together with their boundary values, in [4]–[9].

0.2. The main objects for one-dimensional Fourier analysis are functions defined on or inside the circle. Among them functions which in the circle can be represented by the Poisson integrals, and their conjugate functions (conjugate Poisson integrals) play a special role, since the Poisson integral and the conjugate Poisson integral make up a pair which is an analytic function inside the circle. This analyticity enables one to represent the radial derivative of the Poisson integral in terms of the derivative of the conjugate Poisson integral with respect to a polar angle. If the density of the Poisson integral possesses the summable conjugate function with a finite derivative at some point ϕ_{0}, then by virtue of Smirnov’s theorem ([10], p. 263; [11], p. 583) and Fatou’s theorem ([10], p. 100) this derivative is an angular limit at the point (1, ϕ_{0}) for the radial derivative of the Poisson integral.

Analogous questions naturally arise for the spherical Poisson integral U_{f}(r, θ, ϕ). The representation of the radial derivative \frac{∂}{∂r} U_{f}(r, θ, ϕ) becomes a more difficult problem, in particular, because of the fact that for functions defined in the three-dimensional real ball there is no theory that would be analogous to the theory of analytic functions in the disk. The formula for \frac{∂}{∂r} U_{f}(r, θ, ϕ) considered in the author’s paper “The radial derivative with boundary values of the spherical Poisson integral” has turned out to be closely connected with functions that have not been considered previously.

This paper is dedicated to the investigation of such functions having an independent interest for Fourier analysis on the sphere.

Namely, the triple of harmonic functions U_{f}^{∗}(r, θ, ϕ), Ũ_{f}(r, θ, ϕ) and Ũ_{f}^{∗}(r, θ, ϕ) in the ball is connected with the spherical Poisson integral U_{f}(r, θ, ϕ). These functions are called allied with U_{f} harmonic functions with respect to θ, to ϕ, and to (θ, ϕ), respectively (see §1). The triple of harmonic functions P_{r}^{∗}(θ, ϕ; θ′, ϕ′), P̃_{r}(θ, ϕ; θ′, ϕ′) and P̃_{r}^{∗}(θ, ϕ; θ′, ϕ′) in the ball is connected with the spherical Poisson kernel P_{r}(θ, ϕ; θ′, ϕ′). These functions are called allied with P_{r} kernels with respect to θ, to ϕ, and to (θ, ϕ). These kernels can be regarded as generalizations of the conjugate Poisson kernel Q_{r}(t) = ∑_{n=1}^{∞} r^{n} sin nt (see §2). Moreover, the triple of allied harmonic functions admits representations in the form of integrals with allied kernels and are called by the author allied integrals with U_{f}. These integrals are generalizations of the conjugate Poisson integral (see §3). If f(θ, ϕ) ∈ L^{2}, then in L^{2} there exist functions f^{∗}(θ, ϕ), f̃(θ, ϕ), and f̃^{∗}(θ, ϕ) called allied with f(θ, ϕ) functions with respect to θ, ϕ, and (θ, ϕ). In that case the allied integrals are Poisson integrals for the allied functions (see §4). We have derived an estimate of the L^{2}-norm of the functions f^{∗}, f̃, and f̃^{∗} through the L^{2}-norms of the function f ∈ L^{2}. The obtained inequalities can be regarded as generalizations of M. Riesz’s inequality ([10], p. 253) for the ball when p = 2 (see §5). The application of the above arguments to the Fourier–Laplace series S[f], f ∈ L(R), enables us to obtain the allied Laplace series S^{∗}[f], S̃[f], and S̃^{∗}[f] with respect to θ, to ϕ, and to (θ, ϕ), respectively. For f ∈ L^{2} these Laplace series become Fourier–Laplace series S[f^{∗}], S[f̃], and S[f̃^{∗}], respectively. This fact generalizes, for the ball, Smirnov’s equality ([10], p. 263; [11], p. 583) and M. Riesz’s equality ([10], p. 253) when p = 2 (see §6).

0.3. In what follows we shall use the representation of the spherical Poisson integral as a series (see, e.g., [2], p. 444)

U_{f}(r, θ, ϕ) = a_{00} + ∑_{n=1}^{∞} a_{n0}r^{n}P_{n}(cos θ) +

+ ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} (a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ),  (0.5)

coinciding with the Abel–Poisson mean values (A-mean values) of the following Fourier–Laplace series S[f] for the function f(θ, ϕ) ∈ L(R) (see, e.g., [2], p. 444):

S[f] = a_{00} + ∑_{n=1}^{∞} a_{n0}P_{n}(cos θ) +

+ ∑_{n=1}^{∞} ∑_{m=1}^{n} (a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ),  (0.6)

where the Fourier–Laplace coefficients a_{n0}, a_{nm}, b_{nm} of the function f are defined by the following equalities:

a_{n0} = \frac{2n + 1}{4π} ∫_{0}^{π} ∫_{0}^{2π}f(θ′, ϕ′)P_{n}(cos θ′) sin θ′ dθ′dϕ′,  (0.7)

a_{nm} = \frac{2n + 1}{2π} · \frac{(n − m)!}{(n + m)!} ∫_{0}^{π} ∫_{0}^{2π}f(θ′, ϕ′)P_{nm}(cos θ′) ×
× cos mϕ′ sin θ′ dθ′dϕ′,  (0.8)
b_{nm} = \frac{2n + 1}{2π} \frac{(n − m)!}{(n + m)!} ∫_{0}^{π} ∫_{0}^{2π}f(θ′, ϕ′)P_{nm}(cos θ′) ×
× sin mϕ′ sin θ′ dθ′dϕ′.  (0.9)

The Legendre polynomials P_{n}(x) and the associated Legendre functions P_{nm}(x) figuring in these equalities are defined on [−1, 1] by the following equalities:

P_{n}(x) = \frac{1}{n!2^{n}} · \frac{d^{n}}{dx^{n}} (x^{2} − 1)^{n}, n = 0, 1, 2, …,  (0.10)
P_{nm}(x) = (1 − x^{2})^{\frac{1}{2} m} \frac{d^{m}}{dx^{m}} P_{n}(x) =  (0.11)
= \frac{(1 − x^{2})^{\frac{1}{2} m}}{n!2^{n}} \frac{d^{n+m}}{dx^{n+m}} (x^{2} − 1)^{n},  (0.12)
1 ≤ m ≤ n, n = 1, 2, …,

where (1 − x^{2})^{\frac{1}{2}} is a non-negative value of the square root. Note that P_{n}^{0}(x) = P_{n}(x), and P_{nm}(x) = 0 for m > n.

Remark 0.1. In defining the associated Legendre functions, some authors use the multiplier (−1)^{m} (see, e.g., [3], p. 98 or 107; [12], p. 240, formula (7.12.7); [13], p. 67, 191), while others do not (see, e.g., [14], p. 481; [15], p. 246; [16], p. 384).

Remark 0.2. In the case where the associated Legendre functions are defined by equality (0.11) or (0.12), they are sometimes called Ferrers’ functions ([15], p. 246) and denoted by the symbol T_{n}^{m}(x) ([3], p. 93) or P_{n,m}(x) ([14], p. 481; [16], p. 384^{1}).

Remark 0.3. In what follows P_{nm}(x) will be assumed to be defined by equality (0.11) or (0.12). If P_{nm}(x) were defined by the multiplier (−1)^{m}, then the terms of series (0.4) would have (−1)^{m} · (−1)^{m} = 1 as multipliers, while in series (0.5) and (0.6) (−1)^{m} would arise the first time from the coefficients a_{nm} and b_{nm} and the second time from the Legendre functions. As a result, the products a_{nm}P_{nm}(x) and b_{nm}P_{nm}(x) would have (−1)^{m} · (−1)^{m} = 1 as multipliers. Thus, what will be proved for the Fourier series with functions P_{nm}(x) defined by equality (0.11) or (0.12) will also be true for the Fourier series with associated Legendre functions with multipliers (−1)^{m}.

§ 1. Allied Harmonic Functions

1.1. We introduce the following functions connected with representation (0.5) of the harmonic function U_{f}(r, θ, ϕ) in the ball 𝔹 :

U_{f}^{∗}(r, θ, ϕ) = ∑_{n=1}^{∞} a_{n0}λ_{n0}r^{n}P_{n}(cos θ) +

+ ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}(a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ)  (1.1)

and

Ũ_{f}^{∗}(r, θ, ϕ) = ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}(a_{nm} sin mϕ − b_{nm} cos mϕ)P_{nm}(cos θ), (1.2)

where the numbers λ_{nm} are defined by the equalities

λ_{nm} = \frac{1}{n + m} + \frac{1}{n − m + 1} for 0 ≤ m ≤ n, n = 1, 2, …. (1.3)

Besides the functions U_{f}^{∗} and Ũ_{f}^{∗}, we introduce one more function again connected with representation (0.5) of the function U_{f},

Ũ_{f}(r, θ, ϕ) = ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} (a_{nm} sin mϕ − b_{nm} cos mϕ)P_{nm}(cos θ). (1.4)

It will be proved below that the functions U_{f}^{∗}, Ũ_{f}^{∗} = (Ũ_{f})^{∗} ≡ Ũ_{f}^{∗} and Ũ_{f} are harmonic in the ball 𝔹 (see Theorem 1.1).

1On pages 420–423 of [16] one will also find the tables of surface spherical harmonics and solid spherical harmonics for 0 ≤ n ≤ 4, 0 ≤ m ≤ 4.

The functions U_{f}^{∗}, Ũ_{f}, and Ũ_{f}^{∗} will be called allied with U_{f} harmonic functions with respect to θ, to ϕ, and to (θ, ϕ), respectively.

Speaking in general, if there is a harmonic function in the ball 𝔹

U(r, θ, ϕ) = A_{00} + ∑_{n=1}^{∞} A_{n0}r^{n}P_{n}(cos θ) +

+ ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} (A_{nm} cos mϕ + B_{nm} sin mϕ)P_{nm}(cos θ),  (1.5)

then the function

Ũ(r, θ, ϕ) = ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} (A_{nm} sin mϕ − B_{nm} cos mϕ)P_{nm}(cos θ) (1.6)

will be called the allied function of U(r, θ, ϕ) with respect to ϕ.

Similarly, using the numbers λ_{nm}, one can define the functions U^{∗}(r, θ, ϕ) and Ũ^{∗} allied with the function U(r, θ, ϕ) with respect to θ and to (θ, ϕ).

1.2. Now we shall prove the theorem on the functions U_{f}^{∗}, Ũ_{f}^{∗}, and Ũ_{f} being harmonic for f ∈ L(R).

Theorem 1.1. For every function f(θ, ϕ), summable on the rectangle R, the functions U_{f}^{∗}(r, θ, ϕ), Ũ_{f}^{∗}(r, θ, ϕ), and Ũ_{f}(r, θ, ϕ) defined by equalities (1.1)–(1.4) are hamonic functions in the ball 𝔹.

Proof. If we introduce the functions

α_{n}(θ) = a_{n0}λ_{n0}P_{n}(cos θ),
β_{n}(θ, ϕ) = ∑_{m=1}^{n} λ_{nm}(a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ), (1.7)

then series (1.1) will take the form

U_{f}^{∗}(r, θ, ϕ) = ∑_{n=1}^{∞} r^{n}α_{n}(θ) + ∑_{n=1}^{∞} r^{n}β_{n}(θ, ϕ).  (1.8)

By virtue of equality (0.8) we shall have

a_{nm}P_{nm}(cos θ) = \frac{2n + 1}{2π} ∫_{0}^{π} ∫_{0}^{2π}f(θ′ϕ′) \frac{(n − m)!}{(n + m)!} ×
× P_{nm}(cos θ)P_{nm}(cos θ′) cos mϕ′ sin θ′ dθ′dϕ′.  (1.9)

But since for 0 ≤ θ ≤ π, 0 ≤ θ′ ≤ π, 1 ≤ m ≤ n, n = 1, 2, …, the inequality

\frac{(n − m)!}{(n + m)!} |P_{nm}(cos θ)P_{nm}(cos θ′)| < 1  (1.10)

is fulfilled (see [3], p. 418, or [17], p. 271), we obtain |a_{nm}P_{nm}(cos θ)| < \frac{1}{2} nI(f), where

I(f) = ∫_{0}^{π} ∫_{0}^{2π}|f(θ′ϕ′)| sin θ′ dθ′dϕ′.  (1.11)

A similar estimate holds for b_{nm}P_{nm}(cos θ) and a_{n0}P_{n}(cos θ). Thus we have the estimates

|a_{n0}P_{n}(cos θ)| < \frac{1}{4} nI(f), |a_{nm}P_{nm}(cos θ)| < \frac{1}{2} nI(f),

|b_{nm}P_{nm}(cos θ)| < \frac{1}{2} nI(f),

(1.12)

|(a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ)| < nI(f)  (1.13)

for the above-mentioned n, m, θ and all ϕ.

Since^{2}

|λ_{nm}| ≤ \frac{3}{2} , 0 ≤ m ≤ n, n = 1, 2, …,  (1.14)

from (1.7) we obtain, in view of (1.12) and (1.13), the equalities

|α_{n}(θ)| < \frac{1}{2} nI(f), |β_{n}(θ, ϕ)| < 2n^{2}I(f)  (1.15)

for all θ, ϕ and n = 1, 2, . . ..

Therefore the power series ∑_{n=1}^{∞} n^{2}r^{n} converging for r < 1 is, to within a constant multiplier, a majorant series for series (1.1). This implies that series (1.1) converges uniformly and absolutely in every closed ball 𝔹_{0} = {(x, y, z) : x^{2} + y^{2} + z^{2} ≤ r_{0}^{2}}, where 0 < r_{0} < 1. Thus the sum of series (1.1) exists and is continuous in the unit open ball. We denote this sum by U_{f}^{∗}(r, θ, ϕ).

λ_{n,m+1} − λ_{n,m} = 2m · \frac{2n + 1}{[(n + 1)^{2} − m^{2}](n^{2} − m^{2})} > 0

λ_{n1} < λ_{n2} < ⋯ < λ_{n,n−1} < λ_{nn} = \frac{1}{2n} + 1 ≤ \frac{3}{2} , n = 1, 2, ….

2By equality (1.3) we have λ_{n0} = λ_{n1} = \frac{1}{n} + \frac{1}{n+1} ≤ \frac{3}{2} for all n = 1, 2, . . .. A simple calculation shows that for 1 ≤ m ≤ n − 1 and n = 1, 2, . . .. Therefore

Now we shall show that the function U_{f}^{∗}(r, θ, ϕ) is harmonic in 𝔹. For this it is sufficient to prove that the function U_{f}^{∗}(r, θ, ϕ) in the ball 𝔹 satisfies the equation

\frac{∂}{∂r} (r^{2} \frac{∂ψ}{∂r} ) + \frac{1}{sin θ} \frac{∂}{∂θ} ( sin θ \frac{∂ψ}{∂θ} ) + \frac{1}{sin^{2} θ} \frac{∂^{2}ψ}{∂ϕ^{2}} = 0.  (1.16)

Using the function β_{n}(θ, ϕ) from equality (1.7), we introduce the function v(r, θ, ϕ) = r^{n}β_{n}(θ, ϕ) for which we have the equalities

\frac{∂v}{∂r} = nr^{n−1}β_{n}, r^{2} \frac{∂v}{∂r} = nr^{n+1}β_{n}, \frac{∂}{∂r} (r^{2} \frac{∂v}{∂r} ) = n(n + 1)r^{n}β_{n}; (1.17)

sin θ \frac{∂v}{∂θ} = r^{n} sin \frac{∂}{∂θ} β_{n},

\frac{1}{sin θ} \frac{∂}{∂θ} ( sin θ \frac{∂v}{∂θ} ) = r^{2}(\frac{∂^{2}}{∂θ^{2}} β_{n} + cot θ \frac{∂}{∂θ} β_{n});

(1.18)

\frac{1}{sin^{2} θ} \frac{∂^{2}v}{∂ϕ^{2}} = r^{n} \frac{1}{sin^{2} θ} \frac{∂^{2}}{∂ϕ^{2}} β_{n}.  (1.19)

By inequality (1.15) we have the estimate

|\frac{∂}{∂r} (r^{2} \frac{∂v}{∂r} )| ≤ 4n^{4}r^{n}I(f)  (1.20)

while (1.18), (1.19) give

|\frac{1}{sin θ} \frac{∂}{∂θ} ( sin θ \frac{∂v}{∂θ} )| ≤ r^{n}(|\frac{∂^{2}}{∂θ^{2}} β_{n}| + \frac{1}{sin θ} |\frac{∂}{∂θ} β_{n}|),  (1.21)
|\frac{1}{sin^{2} θ} \frac{∂^{2}v}{∂ϕ^{2}} | ≤ \frac{r^{n}}{sin^{2} θ} |\frac{∂^{2}}{∂ϕ^{2}} β_{n}|.  (1.22)

It is necessary to estimate |\frac{∂}{∂θ} β_{n}|, |\frac{∂^{2}}{∂θ^{2}} β_{n}| and |\frac{∂^{2}}{∂ϕ^{2}} β_{n}|. To this end, we use S. Bernstein’s well-know inequality ([10], p.118; [11], p. 47): the inequality

|T′_{n}(x)| ≤ n · max_{0≤x≤2π} |T_{n}(x)|  (1.23)

is fulfilled for a trigonometric polynomial T_{n}(x) of order not higher than n.

Since P_{nm}(cos θ), 0 ≤ m ≤ n, is a trigonometric polynomial of order n, by (1.23) we conclude that

|a_{nm} \frac{d}{dθ} P_{nm}(cos θ)| ≤ n · max_{0≤θ≤π} |a_{nm}P_{nm}(cos θ)|,  (1.24)
|b_{nm} \frac{d}{dθ} P_{nm}(cos θ)| ≤ n · max_{0≤θ≤π} |b_{nm}P_{nm}(cos θ)|  (1.25)

hold for all θ, 0 ≤ θ ≤ π.

Now using inequality (1.12) we obtain

|a_{n0} \frac{d}{dθ} P_{n}(cos θ)| < \frac{1}{4} n^{2}I(f),  (1.26)
|a_{nm} \frac{d}{dθ} P_{nm}(cos θ)| ≤ \frac{1}{2} n^{2}I(f), |b_{nm} \frac{d}{dθ} P_{nm}(cos θ)| ≤ \frac{1}{2} n^{2}I(f) (1.27)

for all θ, 0 ≤ θ ≤ π, and n = 1, 2, . . ..

Inequalities (1.26) and (1.27) imply

max_{0≤θ≤π} |\frac{d}{dθ} (a_{n0}P_{n}(cos θ))| ≤ \frac{1}{4} n^{2}I(f), n = 1, 2, …,  (1.28)
max_{\begin{matrix}0≤θ≤π \\ 0≤ϕ≤2π \end{matrix}} |\frac{∂}{∂θ} (a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ)| ≤ n^{2}I(f) (1.29)

for 1 ≤ m ≤ n, n = 1, 2, . . ..

Using (1.7), (1.28), and (1.29) we obtain

|\frac{d}{dθ} α_{n}(θ)| ≤ \frac{1}{2} n^{2}I(f), n = 1, 2, …,  (1.30)
|\frac{∂}{∂θ} β_{n}(θ, ϕ)| ≤ 2 n^{3}I(f), n = 1, 2, …,  (1.31)

for all (θ, ϕ), 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π. By virtue of inequalities (1.23) and (1.31) we obtain the estimate

|\frac{∂^{2}}{∂θ^{2}} β_{n}(θ, ϕ)| ≤ 2 n^{4}I(f), n = 1, 2, ….  (1.32)

Further, since

\frac{∂^{2}}{∂ϕ^{2}} β_{n}(θ, ϕ) =

= − ∑_{m=1}^{n} m^{2}λ_{nm}(a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ), (1.33)

by (1.13) and (1.14) we have

max_{\begin{matrix}0≤θ≤π \\ 0≤ϕ≤2π \end{matrix}} |\frac{∂^{2}}{∂ϕ^{2}} β_{n}(θ, ϕ)| ≤ 2n^{4}I(f), n = 1, 2, ….  (1.34)

On account of estimates (1.31), (1.32), and (1.33) inequalities (1.21) and (1.22) imply that

|\frac{1}{sin θ} \frac{∂}{∂θ} ( sin θ \frac{∂v}{∂θ} )| ≤ \frac{4}{sin θ} n^{4}r^{n}I(f), n = 1, 2, …,  (1.35)

|\frac{1}{sin^{2} θ} \frac{∂^{2}v}{∂ϕ^{2}} | ≤ \frac{2}{sin^{2} θ} n^{4}r^{n}I(f), n = 1, 2, …,  (1.36)

for all (θ, ϕ), 0 < θ < π, 0 ≤ ϕ ≤ 2π.

From (1.20), (1.35), and (1.36) it follows that the power series ∑_{n=1}^{∞} n^{4}r^{n}, which converges for r < 1, is, to within a constant multiplier, the majorant series for the series^{3} ∑_{n=1}^{∞} \frac{∂}{∂r} (r^{2} \frac{∂}{∂r})r^{n}β_{n}(θ, ϕ), ∑_{n=1}^{∞} \frac{1}{sin θ} \frac{∂}{∂θ} (sin θ \frac{∂}{∂θ}) r^{n}β_{n}(θ, ϕ), ∑_{n=1}^{∞} \frac{1}{sin^{2} θ} \frac{∂^{2}}{∂ϕ^{2}} r^{n}β_{n}(θ, ϕ). Therefore these series uniformly and absolutely converge in the ball 𝔹_{0} of radius r_{0} < 1. Hence in these series the operation of differentiation can be put before the summation sign, which enables us to conclude that the function U_{f}^{∗}(r, θ, ϕ) satisfies equation (1.16) so that it is harmonic in 𝔹.

Applying similar arguments, it can be proved that the functions Ũ_{f}^{∗}(r, θ, ϕ) and Ũ_{f}(r, θ, ϕ) are harmonic in the ball 𝔹.

1.3. The proof of Theorem 1.1 enables one to state that the harmonicity preserves after the transformation of the coefficients.

Theorem 1.2. Let the function Φ(θ, ϕ) be summable on R. Denote its Fourier–Laplace coefficients by A_{n0}, A_{nm}, B_{nm} and consider the harmonic in the ball 𝔹 function which is the spherical Poisson integral for Φ(θ, ϕ)

U_{Φ}(r, θ, ϕ) = A_{00} + ∑_{n=1}^{∞} [A_{n0}r^{n}P_{n}(cos θ) +

+ r^{n} ∑_{m=1}^{n}(A_{nm} cos mϕ + B_{nm} sin mϕ)P_{nm}(cos θ)].  (1.37)

Assume that a sequence of real numbers μ_{nm} is given such that the relations

|μ_{nm}| ≤ cn^{p} for 1 ≤ m ≤ n, n = 1, 2, …,  (1.38)

are fulfilled for some fixed constants c > 0 and p > 0.

Then the three series

∑_{n=1}^{∞} \frac{∂}{∂r} (r^{2} \frac{∂}{∂r} )[μ_{n0}A_{n0}r^{n}P_{n}(cos θ) +

+r^{n} ∑_{m=1}^{n} μ_{nm}(A_{nm} cos mϕ + B_{nm} sin mϕ)P_{nm}(cos θ)],  (1.39)

∑_{n=1}^{∞} \frac{1}{sin θ} \frac{∂}{∂θ} ( sin θ \frac{∂}{∂θ} )[μ_{n0}A_{n0}r^{n} +

3It is a well-known fact that the values θ = 0 and θ = π do not prevent one from making the same conclusions for the whole ball 𝔹 ([2], p. 273).

+r^{n} ∑_{m=1}^{n} μ_{nm}(A_{nm} cos mϕ + B_{nm} sin mϕ)P_{nm}(cos θ)],  (1.40)

∑_{n=1}^{∞} \frac{1}{sin^{2} θ} \frac{∂^{2}}{∂ϕ^{2}} [μ_{n0}A_{n0}r^{n} +

+r^{n} ∑_{m=1}^{n} μ_{nm}(A_{nm} cos mϕ + B_{nm} sin mϕ)P_{nm}(cos θ)]  (1.41)

converge absolutely and uniformly in each closed ball 𝔹_{0} of radius r_{0} < 1 and the sum of the series

∑_{n=1}^{∞} [μ_{n0}A_{n0}r^{n}P_{n}(cos θ) +

+r^{n} ∑_{m=1}^{n} μ_{nm}(A_{nm} cos mϕ + B_{nm} sin mϕ)P_{nm}(cos θ)]  (1.42)

is a harmonic function in the ball 𝔹.

§ 2. Allied Kernels

We introduce the functions

P_{r}^{∗}(θ, ϕ; θ′, ϕ′) = ∑_{n=1}^{∞}(2n + 1)λ_{n0}r^{n}P_{n}(cos θ)P_{n}(cos θ′) +

+ 2 ∑_{n=1}^{∞}(2n + 1)r^{n} ∑_{m=1}^{n} λ_{nm} · \frac{(n − m)!}{(n + m)!} ×

× P_{nm}(cos θ)P_{nm}(cos θ′) cos m(ϕ − ϕ′),  (2.1)
P̃_{r}^{∗}(θ, ϕ; θ′, ϕ′) = 2 ∑_{n=1}^{∞}(2n + 1)r^{n} ∑_{m=1}^{n} λ_{nm} · \frac{(n − m)!}{(n + m)!} ×
× P_{nm}(cos θ)P_{nm}(cos θ′) sin m(ϕ − ϕ′),  (2.2)

where λ_{nm} are defined by equality (1.3). Besides, we introduce one more function

P̃_{r}(θ, ϕ; θ′, ϕ′) = 2 ∑_{n=1}^{∞}(2n + 1)r^{n} ∑_{m=1}^{n} \frac{(n − m)!}{(n + m)!} ×
× P_{nm}(cos θ)P_{nm}(cos θ′) sin m(ϕ − ϕ′).  (2.3)

The functions P_{r}^{∗}, P̃_{r}, and P̃_{r}^{∗} = (P̃_{r})^{∗} ≡ P̃_{r}^{∗} will be called allied kernels with the Poisson kernel P_{r} with respect to θ, to ϕ, and to (θ, ϕ), respectively.

Let us establish some properties of the functions P_{r}^{∗}, P̃_{r}^{∗}, and P̃_{r}.

2.1. First, we shall prove that the following equalities are valid:

∫_{0}^{2π}P̃_{r}^{∗}(θ, ϕ; θ′, ϕ′) dϕ′ = 0,  (2.4)

∫_{0}^{2π}P̃_{r}(θ, ϕ; θ′, ϕ′) dϕ′ = 0,  (2.5)

∫_{0}^{π} ∫_{0}^{2π}P_{r}^{∗}(θ, ϕ; θ′, ϕ′) sin θ′ dθ′dϕ′ = 0,  (2.6)

∫_{0}^{π} ∫_{0}^{2π}P̃_{r}^{∗}(θ, ϕ; θ′, ϕ′)P_{k}(cos θ′) sin θ′ dθ′dϕ′ = 0, k = 1, 2, …,  (2.7)

∫_{0}^{π} ∫_{0}^{2π}P̃_{r}(θ, ϕ; θ′, ϕ′)P_{k}(cos θ′) sin θ′ dθ′dϕ′ = 0, k = 1, 2, ….  (2.8)

Equalities (2.4) and (2.5) are immediately obtained from equalities (2.2) and (2.3) by taking into account that the termwise integration is correct on account of inequalities (1.10) and (1.14).

To prove (2.6) note that equality (2.1) implies

∫_{0}^{2π}P_{r}^{∗}(θ, ϕ; θ′, ϕ′) dϕ′ = 2π ∑_{n=1}^{∞}(2n + 1)λ_{n0}r^{n}P_{n}(cos θ)P_{n}(cos θ′). (2.9)

Hence

∫_{0}^{π} ∫_{0}^{2π}P_{r}^{∗} sin θ′ dθ′dϕ′ =

= 2π ∑_{n=1}^{∞} λ_{n0}r^{n}P_{n}(cos θ) ∫_{0}^{π}(2n + 1)P_{n}(cos θ′) sin θ′ dθ′ =

= 2π ∑_{n=1}^{∞} λ_{n0}r^{n}P_{n}(cos θ) ∫_{−1}^{1}(2n + 1)P_{n}(x) dx.

If we now use the equality (see, for instance, [3], p. 33, equality (34), or [12], p. 228, equality (7.8.2))

(2n + 1)P_{n}(x) = P′_{n+1}(x) − P′_{n−1}(x), P_{−1}(x) = 0  (2.10)

and the equalities P_{n}(1) = 1, P_{n}(−1) = (−1)^{n}, then we shall obtain the relations

∫_{−1}^{1} P_{n}(x) dx = 0 = ∫_{0}^{π} P_{n}(cos θ′) sin θ′ dθ′, n = 1, 2, …. (2.11)

We have thus proved equality (2.6).

Equalities (2.7), (2.8) are obtained from (2.2), (2.3) by virtue of the fact that on the rectangle R each of the systems (P_{nm}(cos θ′) · cos mϕ) and (P_{nm}(cos θ′) · sin mϕ) is orthogonal to the system (P_{k}(cos θ′)) with respect to the measure sin θ′ dθ′dϕ′, i.e.,

∫_{0}^{π} ∫_{0}^{2π}P_{k}(cos θ′)P_{nm}(cos θ′) cos mϕ′ sin θ′ dθ′dϕ′ = 0,  (2.12)
1 ≤ m ≤ n, k = 1, 2, …, n = 1, 2, …,
∫_{0}^{π} ∫_{0}^{2π}P_{k}(cos θ′)P_{nm}(cos θ′) sin mϕ′ sin θ′ dθ′dϕ′ = 0,  (2.13)
1 ≤ m ≤ n, k = 1, 2, …, n = 1, 2, ….

Equalities (2.7), (2.8) also hold for k = 0 but in that case they are weaker than equalities (2.4), (2.5).

2.2. Let us now prove

Theorem 2.1. The functions P_{r}^{∗}, P̃_{r}^{∗}, and P̃_{r} defined by equalities (2.1)– (2.3) are harmonic functions in the unit open ball 𝔹.

Proof. In the first place, inequalities (1.10) and (1.14) imply that the power series ∑_{n=1}^{∞} n^{2}r^{n}, converging for r < 1, is the majorant one for series (2.1)– (2.3) so that the functions P_{r}^{∗}, P̃_{r}^{∗}, and P̃_{r} are continuous in 𝔹.

To show that these functions are harmonic in 𝔹 we shall use the same arguments as in proving Theorem 1.1 with the only difference that there the presence of the function f ∈ L(R) was used to obtain adequate estimates. By analogy, we introduce the function

γ_{n}(θ, ϕ; θ′, ϕ′) = ∑_{m=1}^{n} 2(2n + 1)λ_{nm} \frac{(n − m)!}{(n + m)!} ×
× P_{nm}(cos θ)P_{nm}(cos θ′) cos m(ϕ − ϕ′),  (2.14)

for which by virtue of eastimates (1.10) and (1.14) we have the estimate

|γ_{n}(θ, ϕ′; θ′, ϕ′)| ≤ 9n^{2},  (2.15)

which repeats estimate (1.15) to within a constant multiplier.

If we introduce the function w(r, θ, ϕ; θ′, ϕ′) = r^{n}γ_{n}(θ, ϕ′; θ′, ϕ′), for w we shall obtain inequalities similar to (1.20)–(1.22).

Further, by inequality (1.23) we obtain the estimate

|\frac{d}{dθ} P_{nm}(cos θ)| ≤ n · max_{0≤θ≤π} |P_{nm}(cos θ)|,  (2.16)

which with (1.10) taken into account give the inequality

|\frac{∂}{∂θ} (\frac{(n − m)!}{(n + m)!} P_{nm}(cos θ)P_{nm}(cos θ′) cos m(ϕ − ϕ′))| ≤ n. (2.17)

Therefore

|\frac{∂}{∂θ} γ_{n}(θ, ϕ; θ′, ϕ′)| ≤ 9n^{3},  (2.18)

which is similar to estimate (1.31).

It is now clear how to complete the proof of the theorem.

§ 3. Allied Integrals

Theorem 3.1. For every function f(θ, ϕ)∈L(R) the functions U_{f}^{∗}(r, θ, ϕ), Ũ_{f}^{∗}(r, θ, ϕ), and Ũ_{f}(r, θ, ϕ) defined by equalities (1.1)–(1.4) admit the following integral representations:

U_{f}^{∗}(r, θ, ϕ) = \frac{1}{4π} ∫_{0}^{π} ∫_{0}^{2π}f(θ′, ϕ′)P_{r}^{∗}(θ, ϕ; θ′, ϕ′) sin θ′ dθ′dϕ′,  (3.1)

Ũ_{f}^{∗}(r, θ, ϕ) = \frac{1}{4π} ∫_{0}^{π} ∫_{0}^{2π}f(θ′, ϕ′)P̃_{r}^{∗}(θ, ϕ; θ′, ϕ′) sin θ′ dθ′dϕ′,  (3.2)

Ũ_{f}(r, θ, ϕ) = \frac{1}{4π} ∫_{0}^{π} ∫_{0}^{2π}f(θ′, ϕ′)P̃_{r}(θ, ϕ; θ′, ϕ′) sin θ′ dθ′dϕ′.  (3.3)

Proof. Taking into account equalities (0.7)–(0.8), from equality (2.1) we obtain

\frac{1}{4π} ∫_{0}^{π} ∫_{0}^{2π}f(θ′, ϕ′)P_{r}^{∗}(θ, ϕ; θ′, ϕ′) sin θ′ dθ′dϕ′ =

= ∑_{n=1}^{∞} λ_{n0}r^{n}P_{n}(cos θ) · \frac{2n + 1}{4π} ∫_{0}^{π} ∫_{0}^{2π}f(θ′, ϕ′)P_{n}(cos θ′) sin θ′ dθ′dϕ′ +

+ ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}P_{nm}(cos θ)[ cos mϕ · \frac{2n + 1}{2π} · \frac{(n − m)!}{(n + m)!} ×

× ∫_{0}^{π} ∫_{0}^{2π}f(θ′, ϕ′)P_{nm}(cos θ′) cos mϕ′ sin θ′ dθ′dϕ′ +

+ sin mϕ · \frac{2n + 1}{2π} · \frac{(n − m)!}{(n + m)!} ×

× ∫_{0}^{π} ∫_{0}^{2π}f(θ′, ϕ′)P_{nm}(cos θ′) sin mϕ′ sin θ′ dθ′dϕ′] = U_{f}^{∗}(r, θ, ϕ).

Here termwise integration is correct because of the uniform convergence of series (1.1) with respect to (θ, ϕ) ∈ R for r < 1 (see the proof of Theorem 1.1).

Equalities (3.2) and (3.3) are proved similarly.

Integrals (3.1), (3.3), and (3.2) will be called allied integrals with the integral U_{f}(r, θ, ϕ) with respect to θ, to ϕ, and to (θ, ϕ), respectively.

§ 4. Allied Functions

By Theorem 1.1, to each function f ∈ L(R) there corresponds with the aid of the Poisson integral U_{f} a triple of the harmonic functions U_{f}^{∗}, Ũ_{f}^{∗}, and Ũ_{f} in the ball 𝔹, which admits representations (1.1), (1.2), and (1.4) in the form of allied series, and representations (3.1)–(3.3) in the form of allied integrals.

We pose the natural question under what conditions these series will be Fourier–Laplace series and the integrals will be spherical Poisson integrals.

For the integrals (for the series see §6) the answer for the space L^{2}(R) is stated as

Theorem 4.1. For every function f(θ, ϕ) ∈ L^{2}(R) there exist functions f^{∗}(θ, ϕ) ∈ L^{2}(R), f̃^{∗}(θ, ϕ) ∈ L^{2}(R), and f̃(θ, ϕ) ∈ L^{2}(R) such that the equalities

U_{f}^{∗}(r, θ, ϕ) = U_{f^{∗}}(r, θ, ϕ),  (4.1)
Ũ_{f}^{∗}(r, θ, ϕ) = U_{f^{∗}}(r, θ, ϕ),  (4.2)
Ũ_{f}(r, θ, ϕ) = U_{f}(r, θ, ϕ)  (4.3)

hold in the ball 𝔹, i.e. (see equalities (3.1)–(3.3)),

\frac{1}{4π} ∫_{0}^{π} ∫_{0}^{2π}fP_{r}^{∗} sin θ′ dθ′dϕ′ = \frac{1}{4π} ∫_{0}^{π} ∫_{0}^{2π}f^{∗}P_{r} sin θ′ dθ′dϕ′,  (4.4)

\frac{1}{4π} ∫_{0}^{π} ∫_{0}^{2π}fP̃_{r}^{∗} sin θ′ dθ′dϕ′ = \frac{1}{4π} ∫_{0}^{π} ∫_{0}^{2π}f̃^{∗}P_{r} sin θ′ dθ′dϕ′,  (4.5)

\frac{1}{4π} ∫_{0}^{π} ∫_{0}^{2π}fP̃_{r} sin θ′ dθ′dϕ′ = \frac{1}{4π} ∫_{0}^{π} ∫_{0}^{2π}f̃P_{r} sin θ′ dθ′dϕ′.  (4.6)

Proof. For f ∈ L^{2}(R) we have Parseval’s equality (see, for instance, [16], p. 349)

∫_{0}^{π} ∫_{0}^{2π}f^{2}(θ′, ϕ′) sin θ′ dθ′dϕ′ = 4πa_{00}^{2} + 4π ∑_{n=1}^{∞} \frac{1}{2n + 1} a_{n0}^{2} +

+2π ∑_{n=1}^{∞} \frac{1}{2n + 1} ∑_{m=1}^{n} \frac{(n + m)!}{(n − m)!} (a_{nm}^{2} + b_{nm}^{2}).  (4.7)

Series (4.7) is therefore convergent. In that case the series obtained from (4.7) will converge if instead of (a_{nm}, b_{nm}) we successively put

(λ_{nm}a_{nm}, λ_{nm}b_{nm}), (−λ_{nm}b_{nm}, λ_{nm}a_{nm}), (−b_{nm}, a_{nm}). (4.8)

Hence by the Riesz–Fischer theorem we obtain the existence of the functions f^{∗} ∈ L^{2}(R), f̃^{∗} ∈ L^{2}(R) and f̃ ∈ L^{2}(R) with the Fourier–Laplace coefficients (4.8). Series (1.1), (1.2), and (1.4) are therefore A-mean values of the Fourier–Laplace series S[f^{∗}], S[f̃^{∗}] and S[f̃]. This in turn means that the harmonic functions U_{f}^{∗}(r, θ, ϕ), Ũ_{f}^{∗}(r, θ, ϕ), and Ũ_{f}(r, θ, ϕ) are the spherical Poisson integrals for f^{∗}, f̃^{∗}, and f̃, respectively. This is equivalent to equalities (4.1)–(4.3), which completes the proof of the theorem.

The functions f^{∗}, f̃, and f̃^{∗} from Theorem 4.1 will be called allied with f functions with respect to θ, to ϕ, and to (θ, ϕ).

§ 5. L^{2}-Inequalities for f^{∗}, f̃^{∗}, and f̃

For the operators transforming f ∈ L^{2}(R) into f^{∗} ∈ L^{2}(R), f̃^{∗} ∈ L^{2}(R), and f̃ ∈ L^{2}(R) the following statement holds.

ALLIED FUNCTIONS 229

Theorem 5.1. For every function f ∈ L^{2}(R) we have the inequalities

∫_{0}^{π} ∫_{0}^{2π}(f̃^{∗})^{2} sin θ′ dθ′dϕ′ ≤ ∫_{0}^{π} ∫_{0}^{2π}(f^{∗})^{2} sin θ′ dθ′dϕ′,  (5.1)
∫_{0}^{π} ∫_{0}^{2π}(f^{∗})^{2} sin θ′ dθ′dϕ′ ≤ \frac{9}{4} ∫_{0}^{π} ∫_{0}^{2π}f^{2} sin θ′ dθ′dϕ′,  (5.2)
∫_{0}^{π} ∫_{0}^{2π}(f̃)^{2} sin θ′ dθ′dϕ′ ≤ ∫_{0}^{π} ∫_{0}^{2π}f^{2} sin θ′ dθ′dϕ′  (5.3)

and the equalities

∫_{0}^{π} ∫_{0}^{2π}(f̃^{∗})^{2} sin θ′ dθ′dϕ′ =

= ∫_{0}^{π} ∫_{0}^{2π}(f^{∗})^{2} sin θ′ dθ′dϕ′ − 4π ∑_{n=1}^{∞} \frac{2n + 1}{n^{2}(n + 1)^{2}} a_{n0}^{2},  (5.4)
∫_{0}^{π}∫_{0}^{2π}(f̃)^{2} sin θ′ dθ′dϕ′ =2π ∑_{n=1}^{∞}\frac{1}{2n + 1} ∑_{m=1}^{n} \frac{(n + m)!}{(n − m)!} (a_{nm}^{2}+b_{nm}^{2}), (5.5)

∫_{0}^{π} ∫_{0}^{2π}(f̃)^{2} sin θ′ dθ′dϕ′ =

= ∫_{0}^{π} ∫_{0}^{2π}f^{2} sin θ′ dθ′dϕ′ − 4πa_{00}^{2} − 4π ∑_{n=1}^{∞} \frac{1}{2n + 1} a_{n0}^{2}.  (5.6)

Proof. Using the same reasoning as for the proof of Theorem 4.1, we obtain the equalities

∫_{0}^{π} ∫_{0}^{2π}(f̃^{∗})^{2} sin θ′ dθ′dϕ′ =

= 2π ∑_{n=1}^{∞} \frac{1}{2n + 1} ∑_{m=1}^{n} \frac{(n + m)!}{(n − m)!} λ_{nm}^{2}(a_{nm}^{2} + b_{nm}^{2}),  (5.7)

∫_{0}^{π} ∫_{0}^{2π}(f^{∗})^{2} sin θ′ dθ′dϕ′ = 4π ∑_{n=1}^{∞} \frac{1}{2n + 1} λ_{n0}^{2}a_{n0}^{2} +

+2π ∑_{n=1}^{∞} \frac{1}{2n + 1} ∑_{m=1}^{n} \frac{(n + m)!}{(n − m)!} λ_{nm}^{2}(a_{nm}^{2} + b_{nm}^{2}),  (5.8)

∫_{0}^{π} ∫_{0}^{2π}(f̃^{∗})^{2} sin θ′ dθ′dϕ′ =

= ∫_{0}^{π} ∫_{0}^{2π}(f^{∗})^{2} sin θ′ dθ′dϕ′ − 4π ∑_{n=1}^{∞} \frac{1}{2n + 1} λ_{n0}^{2}a_{n0}^{2}.  (5.9)

Equalities (5.7) and (5.8) imply (5.1) and (5.2) with (1.14) and (4.7) taken into account. Equality (5.4) is obtained from (5.9) by virtue of the fact that λ_{n0} = \frac{2n+1}{n(n+1)}. Furthermore, since (−b_{nm}, a_{nm}) are the Fourier–

Laplace coefficients for the function f̃(θ, ϕ), we obtain (5.5) and (5.6) from Parseval’s equality. Equality (5.3) follows in turn from (5.6).

§ 6. Allied Series

The technique used to make series (1.1), (1.2), and (1.4) correspond to series (0.5) enables one to write the allied series S^{∗}, S̃^{∗}, and S̃ for any Laplace series S obtained formally from series (1.5) by substituting r = 1. In particular, the series

S^{∗}[f] = ∑_{n=1}^{∞} (\frac{1}{n} + \frac{1}{n + 1})a_{n0}P_{n}(cos θ) +

+ ∑_{n=1}^{∞} ∑_{m=1}^{n} (\frac{1}{n + m} + \frac{1}{n − m + 1}) ×

× (a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ),  (6.1)
S̃^{∗}[f] = ∑_{n=1}^{∞} ∑_{m=1}^{n} (\frac{1}{n + m} + \frac{1}{n − m + 1}) ×
× (a_{nm} sin mϕ − b_{nm} cos mϕ)P_{nm}(cos θ),  (6.2)
S̃[f] = ∑_{n=1}^{∞} ∑_{m=1}^{n} (a_{nm} sin mϕ − b_{nm} cos mϕ)P_{nm}(cos θ)  (6.3)

are the allied Laplace series to the Fourier–Laplace series (0.6) with respect to θ, to (θ, ϕ), and to ϕ, respectively.

The proof of Theorem 4.1 gives rise to

Theorem 6.1. For every function f ∈ L^{2}(R) there exist functions f^{∗} ∈ L^{2}(R), f̃^{∗} ∈L^{2}(R), and f̃ ∈ L^{2}(R) for which the series S^{∗}[f], S̃^{∗}[f] and S̃[f] are the Fourier–Laplace series, i.e., S^{∗}[f]=S[f^{∗}], S̃^{∗}[f]=S[f̃^{∗}], S̃[f]=S[f̃].

Remark 6.1. Using the numbers λ_{nm} defined by equality (1.3), we introduce the polynomials

P_{n}^{∗}(x) = (\frac{1}{n} + \frac{1}{n + 1})P_{n}(x) (n = 1, 2, …)  (6.4)

and

P_{nm}^{∗} = (\frac{1}{n + m} + \frac{1}{n − m + 1})P_{nm}(x)  (6.5)
(1 ≤ m ≤ n, n = 1, 2, …),

which will be called allied Legendre polynomials and allied Legendre functions, respectively.

Then equalities (1.1), (1.2), (6.1) and (6.2) can be rewritten as

U_{f}^{∗}(r, θ, ϕ) = ∑_{n=1}^{∞} a_{n0}r^{n}P_{n}^{∗}(cos θ) +

+ ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} (a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}^{∗}(cos θ),  (6.6)

Ũ_{f}^{∗}(r, θ, ϕ) = ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} (a_{nm} sin mϕ − b_{nm} cos mϕ)P_{nm}^{∗}(cos θ), (6.7)

S^{∗}[f] = ∑_{n=1}^{∞} a_{n0}P_{n}^{∗}(cos θ) +

+ ∑_{n=1}^{∞} ∑_{m=1}^{n} (a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}^{∗}(cos θ),  (6.8)

S̃^{∗}[f] = ∑_{n=1}^{∞} ∑_{m=1}^{n} (a_{nm} sin mϕ − b_{nm} cos mϕ)P_{nm}^{∗}(cos θ).  (6.9)

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(Received 18.12.1995)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia