On a generalization of Bessel's inequality and the Riesz-Fischer theorem to the case of expansions of functions in \(L_p\) with \(p\neq 2\) in eigenfunctions of the Laplace operator on an arbitrary \(N\)-dimensional domain (Q483644)

The author proves the following generalization of his results in [Proc. Steklov Inst. Math. 219, 205--214 (1997); translation from Tr. Mat. Inst. Steklova 219, 211--219 (1997; Zbl 0918.42022)]. Let \(f\in L_p(G)\) \((1<p<2)\), where \(\text{supp\,}G'\subset G\) with fixed compact subdomain \(G'\) of the \(N\)-dimensional domain \(G\). Assume that \(f\) has Fourier coefficients \(C_k\) in the system of eigenfunctions \(u_k\) of the Laplace operator in \(G\) with eigenvalues \(\lambda_k\). Then, \[ \sum^\infty_{k=1} C^2_k \lambda_k^{N({1\over 2}-{1\over p})}\leq A(N,p,G')\| f\|^2_{L_p(G)}. \] Furthermore, it is shown that if for some numbers the series \[ \sum^\infty_{k=1} C^2_k\lambda_k^{N({1\over 2}-{1\over q})} \] converges for some \(q\) satisfying \(2<q<\infty\), then there exists \(f\in L_2(G)\) such that \(f\in L_q(G')\) for any strictly interior subdomain \(G'\) of \(G\) and \(C_k\) are the Fourier coefficients in the system \(u_k\), such that \[ \| f\|^2_{L_q(G')}\leq B(N,q,G') \sum^\infty_{k=1} C^2_k\lambda_k^{N({1\over 2}-{1\over q})}. \]