The construction of Dirichlet and de la Vallée-Poussin-Nikol'skiĭ kernels for -Bessel Fourier integrals

Author:
L. N. Lyakhov

Translated by:
E. Khukhro

Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom **76** (2015), vypusk 1.

Journal:
Trans. Moscow Math. Soc. **2015**, 55-69

MSC (2010):
Primary 33C10; Secondary 42A38

DOI:
https://doi.org/10.1090/mosc/242

Published electronically:
November 17, 2015

MathSciNet review:
3467260

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Abstract | References | Similar Articles | Additional Information

Abstract: We give elementary proofs of some properties of the generalized shift generated by a spherical symmetry. We construct B-kernels for Fourier integrals with respect to Bessel j-functions (Fourier-Bessel transforms). These are designed to play the same role as Dirichlet and de la Vallée-Poussin-Nikol'skiĭ kernels in the theory of trigonometric Fourier integrals and in the theory of function approximation.

**1.**B. M. Levitan,*Expansion in Fourier series and integrals with Bessel functions*, Uspehi Matem. Nauk (N.S.)**6**(1951), no. 2(42), 102–143 (Russian). MR**0049376****2.**I. A. Kipriyanov,*\cyr Singulyarnye èllipticheskie kraevye zadachi*, Fizmatlit “Nauka”, Moscow, 1997 (Russian, with English and Russian summaries). MR**1659097****3.**S. M. Nikol'skiĭ,*Approximations of functions of many variables and embedding theorems*, Nauka, Moscow, 1977. (Russian)**4.**Ya. I. Zhitomirskiĭ,*Cauchy's problem for systems of linear partial differential equations with differential operators of Bessel type*, Mat. Sb.**36**(1955), 299-310. (Russian)**5.**G. N. Watson,*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110****6.**L. N. Lyakhov and E. L. Sanina,*Schlömilch polynomials: the Riesz interpolation formula for the 𝐵-derivative and Bernstein’s inequality for Weyl-Marchaud fractional 𝐵-derivatives*, Dokl. Akad. Nauk**417**(2007), no. 5, 592–596 (Russian); English transl., Dokl. Math.**76**(2007), no. 3, 916–920. MR**2459424**, https://doi.org/10.1134/S1064562407060270**7.**I. A. Kiprijanov and V. V. Katrahov,*A class of one-dimensional singular pseudodifferential operators*, Mat. Sb. (N.S.)**104(146)**(1977), no. 1, 49–68, 175 (Russian). MR**0513018****8.**V. V. Katrakhov and L. N. Lyakhov,*The full Fourier-Bessel transform and the algebra of singular pseudodifferential operators*, Differ. Uravn.**47**(2011), no. 5, 681–695 (Russian, with Russian summary); English transl., Differ. Equ.**47**(2011), no. 5, 681–695. MR**2918284**, https://doi.org/10.1134/S0012266111050077**9.**E. L. Sanina,*Fractional Weyl B-derivatives of j-Bessel expansions and Bernshtein's inequality for B-derivatives of even Schlómilch polynomials*, Diss., Candidate Physics-Math. Sci., Voronezh Univ., Voronezh, 2008. (Russian)**10.**I. A. Kipriyanov and L. A. Ivanov,*The obtaining of fundamental solutions for homogeneous equations with singularities with respect to several variables*, Imbedding theorems and their applications to problems of mathematical physics, Trudy Sem. S. L. Soboleva, No. 1, vol. 1983, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1983, pp. 55–77 (Russian). MR**738988****11.**L. N. Lyakhov,*On Schlómilch's j-series*, Nauchn. Vedom. Belgorod Univ. Ser. Mat. Fiz. 2013, no. 12 (155), issue 31, 62-73. (Russian)**12.**A. A. Feoktistova,*B-Liouville operations and approximation of functions in weighted classes*, Diss., Candidate Physics-Math. Sci., Voronezh Univ., Voronezh, 2012. (Russian)**13.**Francesco G. Tricomi,*Lezioni sulle equazioni a derivate parziali*, Corso di analisi superiore, anno accademico 1953–1954, Editrice Gheroni, Torino, 1954 (Italian). MR**0067293****14.**N. N. Lebedev,*Special functions and their applications*, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR**0350075****15.**Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756****16.**L. N. Lyakhov, I. P. Polovinkin, and E. L. Shishkina,*On a Kipriyanov problem for a singular ultrahyperbolic equation*, Differ. Equ.**50**(2014), no. 4, 513–525. Translation of Differ. Uravn. 50 (2014), no. 4, 516–528. MR**3300061**, https://doi.org/10.1134/S0012266114040090**17.**I. A. Kipriyanov and M. I. Klyuchantsev,*Singular integrals that are generated by a generalized shift operator*. II, Sibirsk. Mat. Zh.**11**(1970), 1060-1083; English transl., Siber. Math. J.**11**(1971), 787-804.**18.**S. M. Nikol′skiĭ,*Constructive representation of the zero-classes of differentiable functions of several variables*, Dokl. Akad. Nauk SSSR**170**(1966), 512–515 (Russian). MR**0225152****19.**A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,*Higher transcendental functions. Vol.*II.*Based, in part, on notes left by Harry Bateman*, McGraw-Hill, New York-Toronto-London, 1953. MR**0698780 (84h:33001b)**

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Additional Information

**L. N. Lyakhov**

Affiliation:
Voronezh State University

Email:
levnlya@mail.ru

DOI:
https://doi.org/10.1090/mosc/242

Keywords:
Generalized shift,
Bessel function,
even and odd Bessel j-functions,
Hankel (Bessel) transform,
Fourier--Bessel transform,
Dirichlet kernel,
de la Vall\'ee-Poussin kernel.

Published electronically:
November 17, 2015

Article copyright:
© Copyright 2015
L. N. Lyakhov