UDC 517.521.8

MATHEMATICS

Sh. A. ALIMOV

ON THE SUMMATION OF SERIES IN EIGENFUNCTIONS

(Presented by Academician A. N. Tikhonov, February 26, 1968)

Consider, in (N)-dimensional space, a normal domain (g) with boundary (\Gamma) and a complete orthonormal system ({u_n(x)}) of eigenfunctions of the Laplace operator for the first boundary-value problem

[
\Delta u+\lambda u=0,\quad x\in g,\qquad u\big|_{\Gamma}=0.
]

We define a continuous triangular summation method generated by a function (\varphi_\lambda(t)=0) for (t\geq \lambda), absolutely continuous for (t\geq 0), and satisfying the condition
[
\lim_{\lambda\to\infty}\varphi_\lambda(t)=1.
]

A series (\sum b_n) will be called summable by the method (\varphi) to the number (s) if there exists the limit

[
\lim_{\lambda\to\infty}\sum_{\lambda_n<\lambda}\varphi_\lambda(\lambda_n)b_n=s.
]

The present note is devoted to the study of the properties of the kernel of this summation method

[
\Phi(x,y,\lambda)=\sum_{\lambda_n<\lambda}\varphi_\lambda(\lambda_n)u_n(x)u_n(y).
]

For the Riesz method ((R,\lambda_n,\alpha)) exact results are known ((^{1-4})), which make it possible to sum the Fourier series of (f(x)\in L_2(g)). In the case of multiple Fourier series, E. Stein ((^5)) obtained results for (f(x)\in L_p(g)), (p\geq 1).

For any (\xi>0) and integer (k\geq 0), define

[
d_k(\xi)=\left(\frac{\lambda}{2\pi}\right)^{N/2}\frac{2^{-k}}{k!}
\int_0^1 \varphi_\lambda(\lambda t^2)
\frac{J_{(N-2)/2+k}(\xi t)}{\xi^{(N-2)/2+k}}
t^{N/2+k}\,dt.
]

For an arbitrary fixed (R>0), denote by (g_R) the set of points of the domain (g) whose distance from (\Gamma) is greater than (R), and by (D_s(r,\lambda)) the function which, for (r>R), is equal to zero and, for (r\leq R), is defined by the equality

[
D_s(r,\lambda)=\sum_{k=s}^{\infty} d_k(R\sqrt{\lambda})\lambda^k(R^2-r^2)^k,\qquad s=0,1,2,\ldots
]

Then from the mean-value formula (( (^{1}),\ \text{p. }230)) there follows the equality

[
\Phi(x,y,\lambda)=D_s(r_{xy},\lambda)+\sum \gamma_n u_n(x)u_n(y),
\tag{1}
]

where

[
\gamma_n=(2\lambda)^s s!\,(2\pi)^{N/2}
\int_R^{\infty} d_s(r\sqrt{\lambda})\,
\frac{J_\nu(r\sqrt{\lambda_n})}{\lambda_n^{\nu/2}}\,
r^{\nu+1}\,dr,\qquad
\nu=(N-2)/2+s.
]

Note that (D_0(r_{xy},\lambda)) for (r_{xy}\leq R) coincides with (\Phi^*(x,y,\lambda)), the kernel of summation by the method (\varphi) of the expansion into an ordinary (N)-fold Fourier integral.

Theorem 1. Let the summability function (\varphi_\lambda(t)) have a derivative of order (l>N/p,\ 1\le p\le 2), satisfying the condition

[
\left|\varphi_\lambda^{(l)}(t)\right|<c\lambda^{-l}.
\tag{2}
]

Then, uniformly with respect to (x\in g_R), the formula

[
\left|\Phi(x,y,\lambda)-\Phi^*(x,y,\lambda)\right|_{L_q(g)}=o(1)
]

holds. The norm is taken with respect to (y\in g,\ q=p/(p-1)).

To prove the theorem it is enough to estimate the right-hand side of (1), which is not difficult to do by using the properties of fractional-order kernels ((^6)). The indicated scheme corresponds to the method of Minakshisundaram ((^1,^2)).

Theorem 1 asserts the equisummability of expansions in series in eigenfunctions and in the ordinary Fourier integral. Following S. Bochner ((^7)), one can establish the summability of the expansion in the Fourier integral of an arbitrary function (f(x)\in L(g)) to the value of the function at every Lebesgue point in the case where (\varphi) satisfies condition (2) with (l>(N+1)/2).

Corollary. Let (p) belong to the interval (1\le p\le 2N/(N+1)), and let (2) hold for (\varphi_\lambda(t)) with (l>N/p).

Then the Fourier series of an arbitrary function (f(x)\in L_p) is summed by the method (\varphi) to the value of the function at every Lebesgue point.

For the Riesz method, from equality (1) one can obtain a more precise result.

Theorem 2. If (f(x)\in L_p(g),\ 1\le p\le 2), then its Fourier series in eigenfunctions is summed to it by the Riesz method ((R,\lambda_n,\alpha)) of order (\alpha>N/p-\tfrac12) at every Lebesgue point.

If (f(x)\in L_p(g),\ 1