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V. A. Il’in
ON THE UNIFORM CONVERGENCE OF EXPANSIONS IN EIGENFUNCTIONS WHEN SUMMED IN THE ORDER OF INCREASING EIGENVALUES
(Presented by Academician S. L. Sobolev, 13 XII 1956)
In this paper we study the question of the uniform convergence of expansions in eigenfunctions of the equation \(\Delta u+\lambda u=0\) in an arbitrary domain \(g\) of any even number \(N\) of dimensions, with a homogeneous boundary condition of any of the three kinds, under the condition that the summation is carried out in the natural order of increasing eigenvalues.
The usual conditions for the expandability of a function \(f\), ensuring the absolute and uniform convergence of the Fourier series of this function, consist of the following two requirements:
1) the function \(f\) and its derivatives up to order* \([N/2]\) are continuous in the closed domain \(g\), while the derivatives of order \([N/2]+1\) have an integrable square in the domain \(g\);
2) the function \(f\) itself, its Laplacian \(\Delta f\), and its iterated Laplacians up to order \([N/4]\) in the case of the first boundary-value problem and up to order \([(N-2)/4]\) in the case of the second or third boundary-value problem satisfy the corresponding homogeneous boundary condition.
Remark 1. Requirements 1) and 2) may be generalized as follows: \(1'\)) instead of requirement 1), it is sufficient to subject the function \(f\) to the condition \(f\in W_2^{[(N/2)+1]}(g)\); \(2'\)) moreover, it is sufficient that the function \(f\) and its iterated Laplacians indicated in requirement 2) satisfy the corresponding boundary condition in the generalized sense (i.e. “on the average”).
It is natural to expect that the indicated conditions of expandability can be weakened if one abandons the requirement of absolute convergence and studies the uniform convergence of Fourier series when summed in the order of increasing eigenvalues. This expectation is justified.
For the special case when the domain \(g\) is an \(N\)-dimensional rectangular parallelepiped, i.e. for a multiple trigonometric Fourier series, and also for the Fourier integral, B. M. Levitan \((^1)\) succeeded in weakening the above usual conditions of expandability by one derivative.
In the present paper we have succeeded not only in obtaining the corresponding result for an arbitrary domain of any even number of dimensions, but also in proving a stronger assertion, which we state below.
Main theorem. Let \(g\) be an arbitrary domain of any even number \(N\) of dimensions, admitting the application of Green’s formulas to eigenfunctions, and let \(f\) be an arbitrary function defined in this domain and satisfying the following two requirements: 1) \(f\in W_p^{(N/2)}(g)\) \((p>2)\); 2) the function \(f\) itself, its Laplacian \(\Delta f\), and its iterated Laplacians up to order \([(N-2)/4]\) in the case of the first boundary-value problem and up to order \([(N-4)/4]\)
* The square brackets here and below mean that the integer part of the number enclosed in them is taken.
for the case of the 2nd or 3rd boundary-value problem, satisfy “on the average” the corresponding homogeneous boundary condition.
Then, when summing in the order of increasing eigenvalues, one may assert uniform convergence in any strictly interior subdomain \(g'\) not only of the Fourier series of the function \(f\), but also of a series of the more general form:
\[ \sum_{i=1}^{\infty} f_i u_i(P)\lambda_i^{\alpha}, \tag{1} \]
where \(\alpha\) is any number satisfying the requirement
\[
\alpha < N\frac{p-2}{4p}
\]
for \(p\) lying in the range
\[
2 < p < \frac{2N}{N-1},
\]
and
\[
\alpha < \frac14
\]
for
\[
p \geq \frac{2N}{N-1}.
\]
We emphasize that, for the uniform convergence of the Fourier series itself, it is sufficient that the number \(p\) in requirement 1) be greater than 2.
From the main theorem there follows the following estimate for the \(k\)-th remainder of the Fourier series of the function \(f\), valid in any strictly interior subdomain \(g'\):
\[ r_k(P)=\sum_{i=k}^{\infty} f_i u_i(P)=O\left(\frac{1}{\lambda_k^{\alpha}}\right) \tag{2} \]
(here \(\alpha\) is any fixed number satisfying the requirement indicated in the statement of the main theorem).
Remark 2. If \(f\in W_p^{(N/2)}\) (\(p>2\)), then from the known embedding theorems\({}^{(2)}\) it follows that \(f\) is continuous in the closed domain \(g\); \(f\) and its iterated Laplacians up to the order indicated in requirement 2) (and, in the case of the 2nd or 3rd boundary-value problem, also the normal derivatives of these Laplacians) are square-summable on any \((N-1)\)-dimensional piecewise smooth manifold.
Thus, satisfaction of the boundary conditions does not require that the function \(f\) satisfy additional smoothness conditions.
Remark 3. We emphasize that the smoothness requirement established in the main theorem cannot be weakened: in the condition \(f\in W_p^{(N/2)}\), where \(p>2\), one cannot decrease either of the exponents (neither \(p\), nor \(N/2\)), since in that case the function \(f\), generally speaking, is no longer continuous.
For an arbitrary two-dimensional domain, the main theorem implies the expandability of a function without assuming the existence of its second derivatives. It is sufficient to require that the function \(f\) be summable, possess generalized first derivatives summable to a power greater than 2, and, for the case of the 1st boundary-value problem, satisfy on the average the homogeneous boundary condition of the 1st kind (for the case of the 2nd and 3rd boundary-value problems, satisfaction of the boundary condition is not required). If, however, the generalized first derivatives are summable to a power greater than 4, then for an arbitrary two-dimensional domain one may assert uniform convergence not only of the Fourier series, but also of a series of the more general form (1), where \(\alpha<1/4\); moreover, for the \(k\)-th remainder of the Fourier series the estimate (2) is valid, where also \(\alpha<1/4\).
Remark 4. In accordance with Remark 2, the requirements imposed on the function \(f\) ensure the continuity of this function. However, in the case of an arbitrary two-dimensional domain, by invoking the main result of the paper\({}^{(3)}\), we may dispense also with the requirement of strict continuity of the function \(f\) itself, replacing it by piecewise continuity in the sense indicated in paper\({}^{(3)}\). In this case the Fourier series for the function \(f\) will converge uniformly in any strictly interior subdomain \(g'\) from which have been removed arbitrarily small neighborhoods of those contours on which the function \(f\) itself has discontinuities of continuity.
We shall outline the scheme of the proof of the main theorem.
- The principal tool in the proof is the following asymptotic formula:
\[ \sum_{\sqrt{\lambda_i}\le \mu} u_i(P) f_i \lambda_i^{[(N+2)/4]} = O\left(\mu^{2[(N+2)/4]-N(p-2)/2p}\right) \qquad \left(2<p\le \frac{2N}{N-1}\right) \tag{3} \]
(The estimate of the \(O\)-terms is uniform with respect to \(P\), provided that \(P\) belongs to an arbitrary strictly interior subdomain \(g'\).)
Unlike the usual asymptotic formulas established by various methods by V. A. Il’in \((^3)\), B. M. Levitan \((^4)\), and K. I. Babenko,* formula (3) estimates the Laplacian of order \([(N+2)/4]\) of a finite sum of the Fourier series of an arbitrary function \(f\) satisfying certain smoothness conditions. To derive the basic asymptotic formula (3), one uses a preliminary asymptotic formula of the form
\[ \sum_{|\sqrt{\lambda_i}-\mu|\le \rho} \left|u_i(P)f_i\right|\lambda_i^{[(N+2)/4]} = \sqrt{\rho}\,O\left(\mu^{2[(N+2)/4]-1/2}\right) \qquad (1\le \rho\le \mu) \tag{4} \]
and the absolute and uniform convergence in any strictly interior subdomain \(g'\) of the series
\[ \sum_{i=1}^{\infty} u_i(P) f_i \lambda_i^{-\delta/2} \qquad (\delta>0). \tag{5} \]
Both of the last results follow almost directly from the convergence of the numerical series
\[ \sum_{i=1}^{\infty} f_i^{\,2}\lambda_i^{N/2} \]
and from the preliminary asymptotic formula
\[ \sum_{|\sqrt{\lambda_i}-\mu|\le \rho} u_i^2(P) = \rho O\left(\mu^{N-1}\right) \qquad (1\le \rho\le \mu), \tag{6} \]
established by the method indicated in \((^3)\).
For definiteness, let the number \(N/2\) be even, i.e. \([(N+2)/4]=N/4\). Consider the following specific function, which is a somewhat smoothed principal term of the usual asymptotic formula:
\[ v(r_{PQ})= \begin{cases} \left(\dfrac{\mu}{2\pi}\right)^{N/2} \left[ \dfrac{J_{N/2}(\mu r)}{r^{N/2}} - \dfrac{J_{N/2}(\mu R)}{R^{N/2}} \right], & \text{for } r\le R,\\[1.2em] 0, & \text{for } r\ge R, \end{cases} \tag{7} \]
(where it is assumed that the minimum distance of the point \(P\) from the boundary of the domain exceeds \(R\)). Computing the Fourier coefficient of the function (7) in exactly the same way as is done in \((^3)\), we shall have:
\[ v_i = \delta_i u_i(P) - u_i(P)\left(\mu/\sqrt{\lambda_i}\right)^{N/2} J_{N/2}(\mu R)J_{N/2}(R\sqrt{\lambda_i}) - \]
\[ - u_i(P)\, \frac{\mu^{N/2}}{\lambda_i^{(N-2)/4}} \int_R^{\infty} J_{N/2}(\mu r)J_{N/2-1}(r\sqrt{\lambda_i})\,dr, \tag{8} \]
where \(\delta_i=1\) for \(\sqrt{\lambda_i}<\mu\); \(\delta_i=0\) for \(\sqrt{\lambda_i}>\mu\).
Multiplying both sides of (8) by \(f_i\lambda_i^{N/4}\), we obtain the formula:
\[ v_i f_i\lambda_i^{N/4} = \delta_i u_i(P) f_i\lambda_i^{N/4} - u_i(P) f_i\mu^{N/2}J_{N/2}(\mu R)J_{N/2}(R\sqrt{\lambda_i}) - \]
\[ - u_i(P) f_i\mu^{N/2}\sqrt{\lambda_i} \int_R^{\infty} J_{N/2}(\mu r)J_{N/2-1}(r\sqrt{\lambda_i})\,dr, \tag{9} \]
\[ \text{* Report at the Third All-Union Mathematical Congress.} \]
by means of which the coefficients with indices satisfying the condition \(\sqrt{\lambda_i}<\mu/2\) are estimated. Computing the integral standing on the right-hand side of (9) once by parts, we obtain another formula:
\[ v_i f_i \lambda_i^{N/4} = \delta_i u_i(P) f_i \lambda_i^{N/4} - u_i(P) f_i \mu^{N/2+1} \int_R^\infty J_{N/2+1}(\mu r) J_{N/2}(r\sqrt{\lambda_i})\,dr, \tag{10} \]
which may be used to estimate the coefficients for \(\sqrt{\lambda_i}\geq \mu/2\). The right-hand sides of formulas (9) and (10) are estimated in exactly the same way as in paper \((^3)\). For the estimate one uses the asymptotics of Bessel functions, the convergence of series (5), and the preliminary formula (4). Summing the left- and right-hand sides of formulas (9) and (10) over all indices, and taking into account that on the left-hand side there will then stand the quantity
\[ \sum_{i=1}^{\infty} v_i f_i \lambda_i^{N/4} = (-1)^{N/4}\sum_{i=1}^{\infty} v_i(\Delta^{N/4}f)_i = \]
\[ = (-1)^{N/4}\int_g v(P,Q)\,\Delta^{N/4}f(Q)\,dQ = O\!\left(\mu^{N/2-N(p-2)/2p}\right), \]
we obtain the desired asymptotic formula (3).
- Relying on the asymptotic formula (3), one can prove the convergence of series (1) by the method indicated in paper \((^5)\). This method is based on Courant’s asymptotic formula for eigenvalues and therefore assumes that the boundary of the domain satisfies certain (though not very high) smoothness requirements. Here we shall indicate another method of establishing convergence by means of an asymptotic formula free of any assumptions concerning the boundary of the domain.
We shall proceed from the following identity, obtained with the aid of the mean-value theorem:
\[ (2\pi)^{N/2}J_{N/2}(1) \sum_{i=1}^{n}\frac{u_i(P)f_i}{\lambda_i^{N/2-s}} = \sum_{k=1}^{n-1}\int_{C_k}\left(\sum_{i=1}^{k}u_i(Q)f_i\lambda_i^s\right)dQ + \int_{K_n}\left(\sum_{i=1}^{n}u_i(Q)f_i\lambda_i^s\right)dQ. \tag{11} \]
Here \(K_i\) is the \(N\)-dimensional ball of radius \(1/\sqrt{\lambda_i}\) with center at the point \(P\); \(C_i=K_i-K_{i+1}\); \(n\) is any index; \(s\) is any real number. If \(P\) belongs to an arbitrary interior subdomain \(g'\), then, for identity (11) to hold, it may be necessary to discard a finite number of the smallest eigenvalues (which does not affect convergence). In doing so, for simplicity we shall keep the numbering from \(i=1\). Putting in (11)
\[ s=\frac{N}{2}+\alpha=\frac{N}{2}+N\frac{p-2}{4p}-\delta \quad(\delta>0) \]
and using estimate (3) and Abel’s transformation, we easily prove the boundedness of the partial sums of the series
\[ \sum_{i=1}^{\infty} f_i u_i(P)\lambda_i^{\alpha+\delta/2}, \]
from which, by Abel’s test, the uniform convergence of series (1) follows.
The author thanks A. N. Tikhonov and B. M. Budak for their attention to this work, and S. L. Sobolev for a number of valuable comments.
Moscow State University
named after M. V. Lomonosov
Received
4 XII 1956
REFERENCES
\(^1\) B. M. Levitan, DAN, 102, No. 6 (1955).
\(^2\) S. L. Sobolev, Some applications of functional analysis in mathematical physics, L., 1950, p. 78.
\(^3\) V. A. Il’in, DAN, 109, No. 3 (1956).
\(^4\) B. M. Levitan, Matem. sborn., 35 (77), No. 2 (1954).
\(^5\) V. A. Il’in, DAN, 109, No. 1 (1956).