George Lorentz and inequalities in approximation. (English. Russian original) Zbl 1203.41001
St. Petersbg. Math. J. 21, No. 3, 365-405 (2010); translation from Algebra Anal. 21, No. 3, 1-57 (2009).
This is a survey paper on inequalities in approximation. The themes of the survey are:
(i) Bernstein-type inequalities for exponential sums; (ii) Remez-type inequalities for exponential sums; (iii) Lorentz degree of polynomials; (iv) Markov- and Bernstein-type inequalities for polynomials with constraints; (v) Müntz-type theorems; (vi) Remez-type inequalities and Newman’s product problem; (vii) Multivariate approximation; (viii) Newman’s inequality; (ix) Littlewood polynomials; (x) Inequalities for generalized polynomials; (xi) Markov- and Bernstein-type inequalities for rational functions; (xii) Nikolskiĭ-type inequalities for shift-invariant function spaces; (xiii) Inverse Markov- and Bernstein-type inequalities; (xiv) Ultraflat sequences of unimodular polynomials; (xv) Zeros of polynomials with restricted coefficients.
A list of 138 references is given at the end.
(i) Bernstein-type inequalities for exponential sums; (ii) Remez-type inequalities for exponential sums; (iii) Lorentz degree of polynomials; (iv) Markov- and Bernstein-type inequalities for polynomials with constraints; (v) Müntz-type theorems; (vi) Remez-type inequalities and Newman’s product problem; (vii) Multivariate approximation; (viii) Newman’s inequality; (ix) Littlewood polynomials; (x) Inequalities for generalized polynomials; (xi) Markov- and Bernstein-type inequalities for rational functions; (xii) Nikolskiĭ-type inequalities for shift-invariant function spaces; (xiii) Inverse Markov- and Bernstein-type inequalities; (xiv) Ultraflat sequences of unimodular polynomials; (xv) Zeros of polynomials with restricted coefficients.
A list of 138 references is given at the end.
Reviewer: T. D. Narang (Amritsar)
MSC:
| 41-02 | Research exposition (monographs, survey articles) pertaining to approximations and expansions |
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
| 41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
| 41A63 | Multidimensional problems |
| 42A05 | Trigonometric polynomials, inequalities, extremal problems |
Keywords:
Bernstein-type inequality; Remez-type inequality; Littlewood polynomials; generalized polynomials; multivariate approximationReferences:
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