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Analog of Dirichlet multidimensional discontinuous factor for Riesz means in complex domain. (English. Russian original) Zbl 0623.47062

Math. Notes 40, 784-793 (1986); translation from Mat. Zametki 40, No. 4, 492-510 (1986).
Let \(\{\Lambda_ k\}\) be a sequence of complex numbers, Re \(\Lambda\) \({}_ k\geq 0\), \(| Im \Lambda_ k| \leq C_ 1=const\), and let \(\nu\geq -\), \(\alpha >-1\), \(\lambda\geq 1\). The author investigates an analog of the multidimensional discontinuous Dirichlet factor for Riesz means \[ \int^{R}_{0}J_{\nu +1+\alpha}(r\Lambda)J_{\nu}(r\Lambda_ k)r^{-\alpha}dr \] which arises under the expanding of a non selfadjoint extension of the Laplace operator into eigen and adjoined functions.
Reviewer: N.Vasilevski

MSC:

47Gxx Integral, integro-differential, and pseudodifferential operators
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
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[1] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second Order Differential Equations, Oxford Univ. Press (1962). · Zbl 0099.05201
[2] V. A. Il’in, ?On summability of Fourier series with respect to eigenfunctions by Riesz, Cesaro, and Poisson-Abel means,? Differents. Uravn.,2, No. 6, 816-827 (1966).
[3] V. A. Il’in, ?Necessary and sufficient conditions for spectral expansions to be basic and equiconvergent with trigonometric series,? II, Differents. Uravn.,16, No. 6, 980-1009 (1980).
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