An analog of Bernstein's inequality for fractional $B$-derivatives of Schlemilch $j$-polynomials in weighted function classes
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.