Schlömilch polynomials: Riesz’s interpolation formula for B-derivatives and Bernstein’s inequality for Weyl-Marchaud fractional B-derivatives

Abstract

It is well known that trigonometric polynomials have subtle properties that cannot sometimes be entirely extended to polynomials in terms of arbitrary transcendental functions (a detailed presentation can be found in [1]). It is also well know that among the series in terms of Bessel functions (i.e., Neumann, Kapteyn, Fourier‐Bessel, and Dini series) the Schlomilch series are most similar to the trigonometric Fourier series since they are derived from the latter by applying Sonin or Schlomilch’s integral (see [2, 3]). We study the even component of the general Schlomilch polynomial. For Schlomilch polynomials over j -Bessel functions, it is found that the derivative in the Bessel singular differential operator (referred to hereafter as the B-derivative) can be expressed by an interpolation formula similar to Riesz’s one, which is well known in the theory of trigonometric polynomials. This result follows from Riesz’s formula for trigonometric polynomials and cannot be derived in a similar manner for other polynomials over Bessel functions. As consequences of this formula, we derive Bernstein’s inequalities for the Bderivative and the fractional B-derivative. The latter have the type of the Marchaud and Weyl fractional derivatives. It is also shown that the fractional B-derivatives have this type for all functions represented by series over j -Bessel functions. The generalized Schlomilch series are defined as