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UDC 517.994+517.43
MATHEMATICS
Sh. A. Alimov
ON THE EXPANSION OF FUNCTIONS OF THE CLASS \(B^\alpha_{p,\theta}\) IN A FOURIER SERIES WITH RESPECT TO AN ARBITRARY FUNDAMENTAL SYSTEM OF FUNCTIONS OF THE LAPLACE OPERATOR
(Presented by Academician A. N. Tikhonov on February 24, 1970)
Consider an arbitrary \(N\)-dimensional domain \(g\), star-shaped with respect to some ball, and an arbitrary fundamental system of functions (f.s.f.) \(\{u_n(x)\}\) of the Laplace operator in this domain (i.e., a complete orthonormal system of eigenfunctions of an arbitrary nonnegative self-adjoint extension of the Laplace operator in the domain \(g\), with spectrum consisting of eigenvalues \(\lambda_n \geq 0\)).
The present work is devoted to the study of expansions of functions from the class \(B^\alpha_{p,\theta}\) in a Fourier series with respect to the system \(\{u_n(x)\}\).* In what follows, by \(\overset{\circ}{B}{}^\alpha_{p,\theta}(g)\) we shall denote the set of functions \(f(x)\) belonging in the whole space to the Besov class \(B^\alpha_{p,\theta}(E_N)\) and equal to zero outside the domain \(g\).
For an arbitrary function \(f(x)\in L_2(g)\), with Fourier coefficients \(f_n\) with respect to the system \(\{u_n(x)\}\), and for any \(a>0\), define the quantity
\[ \rho_a(f)=\left(\sum_{a<\sqrt{\lambda_n}\leq 2a} f_n^2\right)^{1/2}. \tag{1} \]
The following theorem on the Fourier coefficients of functions \(f(x)\in \overset{\circ}{B}{}^\alpha_{2,\theta}(g)\) is valid.
Theorem 1. Let \(\alpha>0\) and let \(\theta\) satisfy the condition \(1\geq \theta\leq \infty\). Then, for an arbitrary function \(f(x)\in \overset{\circ}{B}{}^\alpha_{2,\theta}(g)\), the inequality
\[ \left[\sum_{m=0}^{\infty}\left(2^{m\alpha}\rho_{2^m}(f)\right)^\theta\right]^{1/\theta} \leq c\|f\|_{B^\alpha_{2,\theta}} \tag{2} \]
holds.
Let us briefly discuss the scheme of the proof of the theorem. For any vector \(u\) and any number \(m\), introduce the notation
\[ \Delta_u^m f(x)=\sum_{k=0}^{m}(-1)^{m+k} C_m^k f(x+ku). \tag{3} \]
For \(h>0\), define the modulus of continuity of order \(m\) by
\[ \omega_m(f,h)=\sup_{|u|\leq h}\|\Delta_u^m f(x)\|_{L_2(E_N)}. \]
Choose a spherical coordinate system \((r,\theta)\) with origin at the center of the ball with respect to which the domain \(g\) is star-shaped, and introduce the notation
\[ \widetilde{\Delta}_h^m f(x)=\sum_{k=0}^{m}(-1)^{m+k} C_m^k f(r+kh,\theta). \]
* For the definition of the Besov classes \(B^\alpha_{p,\theta}\), as well as the Nikol’skii classes \(H^\alpha_p\) and Liouville classes \(L^\alpha_p\), see (1).
Define the spherical modulus of continuity of order \(m\)
\[ \widetilde{\omega}_m(f,h)=\sup_{0<t<h}\left\|\widetilde{\Delta}_t^{\,m} f(x)\right\|_{L_2(E_N)} . \]
For any number \(m\) and \(t\ge 0\), introduce the function
\[ \psi_m(t)=2^{N/2-1}\Gamma\left(\frac{N}{2}\right) \sum_{k=0}^{m}(-1)^{m+k} C_m^k(kt)^{1-N/2}J_{N/2-1}(kt). \]
Applying the mean-value formula to the function (3) and using Parseval’s equality, we obtain the principal estimate
\[ \left[\sum_{n=1}^{\infty}\varphi_m^2\left(h\sqrt{\lambda_n}\right)f_n^2\right]^{1/2} \le c\left[\omega_m(f,h)+\widetilde{\omega}_m(t,h)\right]. \tag{4} \]
From the membership of the function \(f(x)\) in the class \(B_{2,\theta}^{\alpha}\), for \(\omega_m(f,h)\) there follows the estimate \((m>\alpha)\)
\[ \left(\int_0^\infty \left(h^{-\alpha}\omega_m(f,h)\right)^\theta \frac{dh}{h}\right)^{1/\theta} \le c\|f\|_{B_{2,\theta}^{\alpha}} . \tag{5} \]
Representing \(f(x)\in B_{2,\theta}^{\alpha}\) in the form of a series whose terms are functions of exponential type, it is not difficult to verify the validity of estimate (5) also for \(\widetilde{\omega}_m(f,h)\) (see \((^1)\), p. 260). Applying this estimate to inequality (4), we arrive at estimate (2).
Let us note that, for a function \(f(x)\) finite in an arbitrary strictly interior subdomain \(g'\) of the domain \(g\), when \(\theta\ge 2\), by means of the same arguments one can verify the validity of the reverse inequality
\[ \|f\|_{B_{2,\theta}^{\alpha}} \le c(g')\left[\sum_{m=1}^{\infty}\left(2^{m\alpha}\rho_{2^m}(f)\right)^\theta\right]^{1/\theta}. \]
In the case when the domain \(g\) is an \(N\)-dimensional cube with side \(2\pi\), and \(\{u_n(x)\}\) is the multiple trigonometric system, this result, as well as estimate (2), is known (see \((^1)\), Theorem 8.10.1).
\(1^\circ\). We give some consequences of Theorem 1. Let us first formulate it for the cases \(\theta=\infty\) and \(\theta=2\).
Corollary 1. Let \(f(x)\in \dot H_2^\alpha(g)\), \(\alpha>0\). Then, for every \(\mu>0\), the inequality
\[ \sum_{\mu<\sqrt{\lambda_n}\le 2\mu} f_n^2\lambda_n^\alpha \le c\|f\|_{\dot H_2^\alpha}^{\,2} \tag{6} \]
is valid.
Corollary 2. For any function \(f(x)\in L_2^\alpha(g)\), with \(\alpha\ge 0\), the inequality
\[ \sum_{n=1}^{\infty} f_n^2\lambda_n^\alpha \le c\|f\|_{L_2^\alpha}^{\,2} \]
is valid.
For \(\theta=1\), the following assertion follows from Theorem 1.
Theorem 2. Let \(p\) satisfy the condition \(1\le p\le 2\). Then the Fourier series of an arbitrary function \(f(x)\in \dot B_{p,1}^{N/p}\) converges uniformly and absolutely in any strictly interior subdomain \(g'\) of the domain \(g\).
For the proof we use the Cauchy–Bunyakovsky inequality and definition (1):
\[ \sum_{n=1}^{\infty}|f_n u_n(x)| \le \sum_{m=0}^{\infty}\rho_{2^m}(f) \left[\sum_{\sqrt{\lambda_n}<2^m}u_n^2(x)\right]^{1/2}. \tag{7} \]
Let us further note that, uniformly in \(x\) belonging to \(g'\), the estimate
\[ \sum_{\sqrt{\lambda_n}<\mu} u_n^2(x) \le c\mu^N . \]
It remains to substitute this estimate into inequality (7) and apply estimate (2) with \(\theta=1\).
Corollary. The Fourier series of an arbitrary function \(f(x)\in \dot W_1^N(g)\) converges uniformly and absolutely in any strictly interior subdomain \(g'\) of the domain \(g\).
Let us note that Theorem 2 is final. For an arbitrary F.S.F. of the Laplace operator in any domain \(g\) and any interior point \(x_0\in g\) there exists a finite function \(f(x)\), belonging to the class \(B_{p,\theta}^{N/p}\) for any \(p\ge 1\) and any \(\theta>1\), whose Fourier series diverges at the point \(x_0\). Moreover, under the same conditions one can indicate a finite function \(f(x)\), for any \(p>2\) belonging to the class \(B_{p,1}^{N/p}\), whose Fourier series diverges absolutely at the point \(x_0\).
\(2^\circ\). Let us consider in more detail the case \(N=2\), when \(g\) is a two-dimensional domain star-shaped with respect to some circle. From the results of [2] it follows that, for localization of the partial sums of the Fourier series of an arbitrary function \(f(x)\in L_2(g)\), it is sufficient that the quantity standing on the left-hand side of inequality (6), with \(\alpha=1/2\), be uniformly bounded with respect to \(\mu\). Thus we obtain that a sufficient condition for localization is that the function \(f(x)\) belong to the class \(\dot H_2^{1/2}(g)\).
We shall say that a function \(f(x)\) is piecewise continuous in the domain \(g\) if this domain can be divided, by means of rectifiable curves, into a finite number of domains \(g_k\), in each of which \(f(x)\) is uniformly continuous. If, in addition, \(f(x)\) has uniformly continuous first-order derivatives in each of the domains \(g_k\), then the function \(f(x)\) will be called piecewise smooth. It is not difficult to show that every piecewise smooth function belongs to the class \(\dot H_2^{1/2}(g)\). Thus, for localization of the partial sums of the Fourier series in the case \(N=2\), there is no need to require that the function \(f(x)\) satisfy any boundary conditions. Let us note that in the case \(N>2\), even for the function identically equal to one in the \(N\)-dimensional ball, the Fourier series with respect to the eigenfunctions of the first boundary-value problem diverges at the center of the ball.
It can also be shown that any function belonging, in a two-dimensional domain \(g\) star-shaped with respect to some circle, to the space \(W_2^1(g)\), also belongs to the class \(\dot H_2^{1/2}(g)\). In connection with this, let us consider in the domain \(g\) with boundary \(\Gamma\) the Dirichlet problem
\[ \Delta u=0,\qquad x\in g,\qquad u|_\Gamma=\varphi \tag{8} \]
with an arbitrary admissible function \(\varphi\).
Since the generalized solution \(u(x)\in W_2^1(g)\), it follows that \(u(x)\) belongs to the class \(\dot H_2^{1/2}(g)\), and the following assertion is valid.
Theorem 3. Let \(g\) be an arbitrary two-dimensional domain, star-shaped with respect to some circle. Then for any function \(u(x)\) that is a generalized solution of the Dirichlet problem in the domain \(g\), the Fourier series of \(u(x)\) with respect to an arbitrary F.S.F. of the Laplace operator converges uniformly in any strictly interior subdomain \(g'\) of the domain \(g\).
The author expresses his deep gratitude to Prof. V. A. Il’in for the attention he has given to this work.
Moscow State University
named after M. V. Lomonosov
Received
17 II 1970
References
- S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems, Moscow, 1969.
- V. A. Il’in, Sh. A. Alimov, Dokl. Akad. Nauk SSSR, 193, no. 2 (1970).