TY - JOUR

T1 - Analog of the Riesz Identity and Sharp Inequalities for Derivatives and Differences of Splines in the Uniform Metric

AU - Vinogradov, O. L.

PY - 2019/6/7

Y1 - 2019/6/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85065442070&partnerID=8YFLogxK

U2 - 10.1007/s10958-019-04303-z

DO - 10.1007/s10958-019-04303-z

M3 - Article

AN - SCOPUS:85065442070

VL - 239

SP - 268

EP - 281

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 3

ER -