Abstract:
Approximation problems for functions on the half-line [0,+∞) in a weighted Lp-metric are studied with the use of Bessel generalized translation. A direct theorem of Jackson type is proven for the modulus of smoothness of arbitrary order which is constructed on the basis of Bessel generalized translation. Equivalence is stated between the modulus of smoothness and the K-functional constructed by the Sobolev space corresponding to the Bessel differential operator. A particular class of entire functions of exponential type is used for approximation. The problems under consideration are studied mostly by means of Fourier–Bessel harmonic analysis.
Keywords:
approximation of functions, Jackson theorems, K-functional, Bessel generalized translation, moduli of smoothness, Bessel transforms, entire function of exponential type.
Citation:
S. S. Platonov, “Bessel generalized translations and some problems of approximation theory for functions on the half-line”, Sibirsk. Mat. Zh., 50:1 (2009), 154–174; Siberian Math. J., 50:1 (2009), 123–140
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\paper Bessel generalized translations and some problems of approximation theory for functions on the half-line
\jour Sibirsk. Mat. Zh.
\yr 2009
\vol 50
\issue 1
\pages 154--174
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\jour Siberian Math. J.
\yr 2009
\vol 50
\issue 1
\pages 123--140
\crossref{https://doi.org/10.1007/s11202-009-0015-6}
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Linking options:
https://www.mathnet.ru/eng/smj1946
https://www.mathnet.ru/eng/smj/v50/i1/p154
This publication is cited in the following 22 articles:
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Othman Tyr, Radouan Daher, “Jackson's inequalities in Mellin's analysis”, Ann Univ Ferrara, 70:1 (2024), 141
V. I. Ivanov, “Generalized one-dimensional Dunkl transform in direct problems of approximation theory”, Math. Notes, 116:2 (2024), 265–278
V. I. Ivanov, “Obobschennoe preobrazovanie Danklya na pryamoi v obratnykh zadachakh teorii priblizhenii”, Chebyshevskii sb., 25:2 (2024), 67–81
Othman Tyr, Radouan Daher, “On the Jackson–Stechkin Theorems for the Best Approximations of Functions in Clifford Algebras”, Adv. Appl. Clifford Algebras, 33:1 (2023)
S. S. Platonov, “Limits of some integrals connected with Bessel analysis”, Integral Transforms and Special Functions, 34:7 (2023), 495
Othman Tyr, Radouan Daher, “Jackson's inequalities in Laguerre hypergroup”, J. Pseudo-Differ. Oper. Appl., 13:4 (2022)
Platonov S.S., “on the Hankel Transform of Functions From Nikol'Skii Type Classes”, Integral Transform. Spec. Funct., 32:10 (2021), 823–838
Dzhabrailov A., Luchko Yu., Shishkina E., “Two Forms of An Inverse Operator to the Generalized Bessel Potential”, Axioms, 10:3 (2021), 232
Negzaoui S., Oukili S., “Modulus of Continuity and Modulus of Smoothness Related to the Deformed Hankel Transform”, Results Math., 76:3 (2021), 164
E. L. Shishkina, Trends in Mathematics, Operator Theory and Differential Equations, 2021, 229
Gorbachev V D., Ivanov V.I., Tikhonov S.Yu., “Sharp Approximation Theorems and Fourier Inequalities in the Dunkl Setting”, J. Approx. Theory, 258 (2020), 105462
Lyubov Britvina, Trends in Mathematics, Transmutation Operators and Applications, 2020, 49
Elina Shishkina, Sergei Sitnik, Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics, 2020, 433
Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics, 2020, 527
E. L. Shishkina, Trends in Mathematics, Transmutation Operators and Applications, 2020, 237
E. L. Shishkina, “Obschee uravnenie Eilera—Puassona—Darbu i giperbolicheskie $B$-potentsialy”, Uravneniya v chastnykh proizvodnykh, SMFN, 65, no. 2, Rossiiskii universitet druzhby narodov, M., 2019, 157–338
Gorbachev D.V., Ivanov V.I., Tikhonov S.Yu., “Positive l-P-Bounded Dunkl-Type Generalized Translation Operator and Its Applications”, Constr. Approx., 49:3 (2019), 555–605
Bayrakci S., “On the Boundedness of Square Function Generated By the Bessel Differential Operator in Weighted Lebesque Lp, Alpha Spaces”, Open Math., 16 (2018), 730–739
Mohamed El Hamma, Radouan Daher, Salah El Ouadih, “Some Results for the Bessel transform”, Malaya J. Mat., 3:02 (2015), 202
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