Equiconvergence of Spectral Decompositions for Sturm–Liouville Operators: Triples of Lebesgue Spaces

Autor: I. V. Sadovnichaya, Artem Markovich Savchuk
Rok vydání: 2021
Předmět:
Zdroj: Lobachevskii Journal of Mathematics. 42:1027-1052
ISSN: 1818-9962
1995-0802
DOI: 10.1134/s1995080221050164
Popis: The paper deals with the Sturm–Liouville operator generated on the finite interval $$[0,\pi]$$ by the differential expression $$-y^{\prime\prime}+q(x)y$$ , where $$q=u^{\prime}$$ , $$u\in L_{\varkappa}[0,\pi]$$ for some $$\varkappa\geq 2$$ , and arbitrary regular boundary conditions. Consider two such operators with different potentials but the same boundary conditions. We prove that the difference between spectral decompositions $$S_{m}^{1}(f)-S_{m}^{2}(f)$$ of this operators tends to zero as $$m\to\infty$$ for any $$f\in L_{\mu}[0,\pi]$$ in the norm of the space $$L_{\nu}[0,\pi]$$ if the indices satisfy the inequality $$1/\varkappa+1/\mu-1/\nu\leq 1$$ (except for the case $$\varkappa=\nu=\infty$$ , $$\mu=1$$ ). In particular, in the case of a square summable function $$u$$ the uniform equiconvergence on the whole interval $$[0,\pi]$$ is proved for an arbitrary function $$f\in L_{2}[0,\pi]$$
Databáze: OpenAIRE
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