Constructive approximation: advanced problems
This and the earlier book by R.A. DeVore and G.G. Lorentz (Vol. 303 of the same series), cover the whole field of approximation of functions of one real variable. The main subject of this volume is approximation by polynomials, rational functions, splines and operators. There are excursions into the related fields: interpolation, complex variable approximation, wavelets, widths, and functional analysis. Emphasis is on basic results, illustrative examples, rather than on generality or special problems. A graduate student can learn the subject from different chapters of the books; for a researcher they can serve as an introduction; for applied researchers a selection of tools for their endeavours.
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Problems of Polynomial Approximation
Distribution of Alternation Points of Polynomials of Best Approximation
3 Distribution of Zeros of Polynomials of Best Approximation
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analytic functions Appendix arbitrary assume Banach space belongs best approximation Blaschke product Borel measure bounded Chapter Chebyshev Chebyshev polynomial compact set compact subset constant contains continuous functions contradiction convergence convex Corollary define denote derive disjoint disk dominating set En(f entropy equivalent example exists extreme points finite fixed follows formula function f given Hardy spaces harmonic hence implies inequality integral coefficients interpolation interval knots Kolmogorov Lebesgue measure Lemma limsup linear Lorentz lower estimate measure metric monic polynomial non-negative obtain operator orthogonal orthonormal pn(f Pn(x polynomial of degree positive proof of Theorem Proposition prove rational approximation rational function replace Saff satisfies sequence spline subspace Theorem 1.1 theory trigonometric polynomials uniform norm uniformly unique unit ball vanishes vectors Weierstrass weight widths yields zeros