Geodesic
An analog of Bernstein's inequality for fractional $B$-derivatives of Schlemilch $j$-polynomials in weighted function classes
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Physics, Tome 198 (2021), pp. 80-88

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The $B$-derivative defined by generalized displacements coincides with the singular Bessel differential operator up to a constant. Similarly to the Riemann–Liouville, Marchot, and Weil fractional derivatives, we introduce fractional powers of the $B$-derivative and prove that these derivatives coincide on the corresponding functional classes. Also, we prove Bernstein's inequalities for the $B$-derivative and fractional $B$-derivative of even Schlemilch $j$-polynomials in the classes of continuous and Lebesgue-measurable functions.
Keywords: $j$-Bessel functions; generalized Poisson distribution; fractional derivatives of Liouville, Weil; Riesz interpolation formula; Schlemilch polynomial; Stepanov space generated by a generalized shift; Bernstein–Zygmund inequality.
Mots-clés : Marchot 
@article{INTO_2021_198_a8,
     author = {L. N. Lyakhov and E. Sanina},
     title = {An analog of {Bernstein's} inequality for fractional $B$-derivatives of {Schlemilch} $j$-polynomials in weighted function classes},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {80--88},
     publisher = {mathdoc},
     volume = {198},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_198_a8/}
}
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L. N. Lyakhov; E. Sanina. An analog of Bernstein's inequality for fractional $B$-derivatives of Schlemilch $j$-polynomials in weighted function classes. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Physics, Tome 198 (2021), pp. 80-88. http://geodesic.mathdoc.fr/item/INTO_2021_198_a8/