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

Abstract

We establish an analog of the Riesz interpolation formula owing to which it is possible to obtain a sharp estimate for the first order derivative of the spline of minimal defect with equidistant knots jπ/σ, j ∈ ℤ, in terms of the first order difference in the uniform metric. Based on the constructed identity, it is possible to improve the inequality by replacing the right-hand side with a linear combination of differences, including higher order differences, of the spline. In the case of the difference step π/σ, iterations of this identity lead to formulas analogous to the Riesz formula for higher order derivatives or differences, which makes it possible to obtain the corresponding Riesz and Bernstein type inequalities in strengthened form.