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

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.