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Mat. Zametki, 2019, Volume 106, Issue 4, Pages 549–564 (Mi mz12552)  

Norms of the Positive Powers of the Bessel Operator in the Spaces of Even Schlömilch j-Polynomials

L. N. Lyakhov, E. Sanina

Voronezh State University

Abstract: The definition of a $B$-derivative is based on the notion of generalized Poisson shift; this derivative coincides, up to a constant, with the singular Bessel differential operator. We introduce the fractional powers of a $B$-derivative by analogy with fractional Marchaud and Weyl derivatives. We prove statements on the coincidence of these derivatives for the classes of even smooth integrable functions. We obtain analogs of Bernstein's inequality for $B$-derivatives of integer and fractional order in the space of even Schlömilch j-polynomials with sup-norm and $L_p^\gamma$-norm (the Lebesgue norm with power weight $x^\gamma$, $\gamma>0$). The resulting estimates are sharp and define the norms of powers of the Bessel operator in the spaces of even Schlömilch j-polynomials.

Keywords: Bessel j-function, generalized Poisson shift, Liouville, Marchaud, and Weyl fractional derivatives, Schlömilch polynomial, Riesz interpolation formula, Bernstein's inequality, Bernstein–Zygmund inequality, operator norm.

DOI: https://doi.org/10.4213/mzm12552

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English version:
Mathematical Notes, 2019, 106:4, 577–590

Bibliographic databases:

UDC: 519.216
Received: 20.11.2018

Citation: L. N. Lyakhov, E. Sanina, “Norms of the Positive Powers of the Bessel Operator in the Spaces of Even Schlömilch j-Polynomials”, Mat. Zametki, 106:4 (2019), 549–564; Math. Notes, 106:4 (2019), 577–590

Citation in format AMSBIB
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