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On the convergence of eigenfunction expansions. (English) Zbl 0807.35103

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ n\) and \(\{u_ j\}\) a complete orthonormal system of functions in \(L^ 2 (\Omega)\) satisfying the equations \(-\Delta u_ j = \lambda_ j u_ j\), where \(\Delta\) is the Laplace operator and \(\lambda_ j \in \mathbb{C}\). Set \(S_ \mu (f,x) = \sum_{\rho_ j < \mu} f_ j u_ j (x)\) \((\mu>0)\), where \(f_ j\) are Fourier coefficients, and \(\rho_ j = \text{Re} \lambda_ j^{1/2} \geq 0\). The author proves three theorems. (1) Let \(f \in L_ p^ \alpha (\mathbb{R}^ 3)\), \(\alpha \geq 1\), \(\alpha p>3\), \(p \geq 1\), and \(\text{supp} f \subset \Omega\). Then \(S_ \mu (f,x) \to f\) uniformly on any compact subset of \(\Omega\) as \(\mu \to \infty\). (2) Let \(f \in L^ \alpha_ 2 (\mathbb{R}^ 3)\), \(\alpha \geq 1\), and \(f=0\) in a subdomain \(\Omega_ 0 \subset \Omega\). Then \(S_ \mu (f,x) \to 0\) uniformly in every compact subset of \(\Omega_ 0\) as \(\mu \to \infty\). (3) Let \(f \in L^ \alpha_ p (\mathbb{R}^ 3)\), \(\alpha > 3/2\), \(\alpha p > 3\), \(p \geq 1\), and \(\text{supp} f \subset \Omega\). Then the Fourier series of \(f\) with respect to \(\{u_ j\}\) converges absolutely and uniformly on every compact subset of \(\Omega\).

MSC:

35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
47F05 General theory of partial differential operators
42A20 Convergence and absolute convergence of Fourier and trigonometric series
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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