Bessel Functions and Sharp Bernstein-Type Inequalities

It is well known that the classical sharp Bernstein inequality for entire functions of exponential type, where the norm of the derivative of a function is estimated by the norm of the function itself, follows from Riesz’s interpolation identity. In this identity, the derivative is expanded in a series over equidistant shifts of the function itself. For fractional derivatives of order less than one, interpolation formulas with equidistant shifts do not lead to sharp inequalities. In the present paper, a method to obtain sharp inequalities from interpolation formulas is described. Then, we construct new interpolation formulas with nonequidistant shifts, related to the zeros of entire functions of exponential type and, among them, Bessel functions. The considered interpolation formulas lead to sharp Bernstein-type inequalities for a certain family of differential operators, which generalize the Riesz derivative of noninteger order.