Analog of the Riesz Identity and Sharp Inequalities for Derivatives and Differences of Splines in the Uniform Metric

Abstract

An analog of the Riesz interpolation formula is established. It allows us to obtain a sharp estimate for the first order derivative of a spline of minimal defect with equidistant knots $$ \frac{j\pi}{\sigma },j\in \mathrm{\mathbb{Z}}, $$ in terms of the first order difference in the integral metric. Moreover, the constructed identity makes it possible to strengthen the inequality by replacing its right-hand side with a linear combination of differences, including higher order differences, of the spline. In the case of the difference step $$ \frac{\pi }{\sigma } $$ , iterations of this identity lead to formulas analogous to the Riesz formula for higher order derivatives and differences; this allows us to obtain Riesz and Bernstein type inequalities for them, also in a stronger form.