Некоторые точные инт егральные оценки про изводных рациональных и алгеб раических функций. Пр иложения

Abstract

Exact estimates are obtained for integrals of absolute values of derivatives and gradients, for integral moduli of continuity and for major variations of piecewise algebraic functions (in particular, for polynomials, rational functions, splines, etc.). These results are applied to the problems of approximation theory and to the estimates of Laurent and Fourier coefficients. Typical results: 1. IfE is a measurable subset of the circle or of a line in thez-plane andR(z) is a rational function of degree ≦n, ¦R(z)¦≦ (z∈E), then ∝E ¦R′(z)¦dz¦≦ 2πn; the latter estimate is exact forn=0, 1, ... and everyE with positive measure; 2. Iff(x1,x2, ...,xm) is a real valued piecewise algebraic function of order (n, k) on the unit ballD⊂Rm (in particular, a real valued rational function of order ≦n), and ¦f¦≦1 onD, then ∝D¦gradf¦dx≦2πm/2n/Π(m/2); herem≧1, n≧0, 1≦k<∞; 3. LetE=Π={z∶¦z¦=1}, and letcm(R) be the mth Laurent coefficient ofR onΠ,Cm(n)=sup{¦cm(R)¦}, where sup is taken over allR from 1), then 1/7 min {n/¦m¦, 1} ≦Cm(n) ≦ min {n/¦m¦, 1}.