Methods of Approximation Theory

Stepanets A.I.

his work for advanced students and researchers explains methods for solving, within the framework of a common approach, traditional problems of approximation theory for large collections of functions, including the well-known Weyl-Nagy and Sobolev classes as particular cases, as well as classes of functions defined by convolutions with arbitrary summable kernels. Chapters cover areas including regularity of linear methods of summation of Fourier series, saturation of linear methods, classes of periodic functions, best approximations in the spaces C and L, interpolation, and approximation of Cauchy- type integrals

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Table of contents :
PREFACE……Page 12
1. Introduction……Page 20
2. Nikol’skii and Nagy Theorems……Page 25
3. Lebesgue Constants of Classical Linear Methods……Page 34
4. Lower Bounds for Lebesgue Constants……Page 40
5. Linear Methods Determined by Rectangular Matrices……Page 42
6. Estimates for Integrals of Moduli of Functions Defined by Cosine and Sine Series……Page 47
7. Asymptotic Equality for Integrals of Moduli of Functions Defined by Trigonometric Series. Telyakovskii Theorem……Page 62
8. Corollaries of Theorem 7.1. Regularity of Linear Methods of Summation of Fourier Series……Page 85
1. Statement of the Problem……Page 98
2. Sufficient Conditions for Saturation……Page 100
3. Saturation Classes……Page 103
4. Criterion for Uniform Boundedness of Multipliers……Page 109
5. Saturation of Classical Linear Methods……Page 117
1. Sets of Summable Functions. Moduli of Continuity……Page 120
2. Classes Hω[a, b] and Hω……Page 127
3. Moduli of Continuity in Spaces Lp. Classes Hωp……Page 129
4. Classes of Differentiable Functions……Page 131
5. Conjugate Functions and Their Classes……Page 135
6. Weyl–Nagy Classes……Page 138
7. Classes LβψN……Page 139
8. Classes CβψN……Page 145
9. Classes Lβ_ψN……Page 149
10. Order Relation for (ψβ_)-Derivatives……Page 152
11. ψ_-Integrals of Periodic Functions……Page 156
12. Sets M0,M∞, and MC……Page 166
13. Set F……Page 172
14. Two Counterexamples……Page 175
15. Function ηa(t) and Sets Defined by It……Page 179
16. Sets B and M0……Page 181
1. First Integral Representation……Page 184
2. Second Integral Representation……Page 186
3. Representation of Deviations of Fourier Sums on Sets Cψ_M and Lψ_……Page 192
5. APPROXIMATION BY FOURIER SUMS IN SPACES C AND L1……Page 206
1. Simplest Extremal Problems in Space C……Page 208
2. Simplest Extremal Problems in Space L1……Page 217
3. Approximations of Functions of Small Smoothness by Fourier Sums……Page 222
4. Auxiliary Statements……Page 226
5. Proofs of Theorems 3.1–3.3’……Page 244
6. Approximation by Fourier Sums on Classes Hω……Page 254
7. Approximation by Fourier Sums on Classes Η~ω……Page 258
8. Analogs of Theorems 3.1–3.3′ in Integral Metric……Page 262
9. Analogs of Theorems 6.1 and 7.1 in Integral Metric……Page 271
10. Approximations of Functions of High Smoothness by Fourier Sums in Uniform Metric……Page 272
11. Auxiliary Statements……Page 278
12. Proofs of Theorems 10.1–10.3’……Page 290
13. Analogs of Theorems 10.1–10.3′ in Integral Metric……Page 297
14. Remarks on the Solution of Kolmogorov–Nikol’skii Problem……Page 298
15. Approximation of ψ_-Integrals That Generate Entire Functions by Fourier Sums……Page 303
16. Approximation of Poisson Integrals by Fourier Sums……Page 313
17. Corollaries of Telyakovskii Theorem……Page 322
18. Solution of Kolmogorov–Nikol’skii Problem for Poisson Integrals of Continuous Functions……Page 329
19. Lebesgue Inequalities for Poisson Integrals……Page 357
20. Approximation by Fourier Sums on Classes of Analytic Functions……Page 364
21. Convergence Rate of Group of Deviations……Page 382
22. Corollaries of Theorems 21.1 and 21.2. Orders of Best Approximations……Page 393
23. Analogs of Theorems 21.1 and 21.2 and Best Approximations in Integral Metric……Page 397
24. Strong Summability of Fourier Series……Page 402
BIBLIOGRAPHICAL NOTES (Part I)……Page 412
REFERENCES (Part I)……Page 418
0. Introduction……Page 448
1. Approximations in the Space L2……Page 451
2. Direct and Inverse Theorems in the Space L2……Page 456
3. Extension to the Case of Complete Orthonormal Systems……Page 458
4. Jackson Inequalities in the Space L2……Page 463
5. Marcinkiewicz, Riesz, and Hardy–Littlewood Theorems……Page 467
6. Imbedding Theorems for the Sets Lψ_Lp……Page 471
7. Approximations of Functions from the Sets Lψ_Lp by Fourier Sums……Page 474
8. Best Approximations of Infinitely Differentiable Functions……Page 485
9. Jackson Inequalities in the Spaces C and Lp……Page 500
7. BEST APPROXIMATIONS IN THE SPACES C AND L……Page 508
1. Chebyshev and de la Vallée Poussin Theorems……Page 509
2. Polynomial of the Best Approximation in the Space L……Page 511
3. General Facts on the Approximations of Classes of Convolutions……Page 514
4. Orders of the Best Approximations……Page 524
5. Exact Values of the Upper Bounds of Best Approximations……Page 529
6. Dzyadyk–Stechkin–Xiung Yungshen Theorem. Korneichuk Theorem……Page 541
7. Serdyuk Theorem……Page 544
8. Bernstein Inequalities for Polynomials……Page 558
9. Inverse Theorems……Page 564
1. Interpolation Trigonometric Polynomials……Page 572
2. Lebesgue Constants and Nikol’skii Theorems……Page 576
3. Approximation by Interpolation Polynomials in the Classes of Infinitely Differentiable Functions……Page 579
4. Approximation by Interpolation Polynomials on the Classes of Analytic Functions……Page 591
5. Summable Analog of the Favard Method……Page 605
9. APPROXIMATIONS IN THE SPACES OF LOCALLY SUMMABLE FUNCTIONS……Page 616
1. Spaces L^p……Page 617
2. Order Relation for (ψ, β)-Derivatives……Page 620
3. Approximating Functions……Page 626
4. General Estimates……Page 634
5. On the Functions ψ(·) Specifying the Sets L^βψ……Page 643
6. Estimates of the Quantities ||r^σc(t, β)||1 for c = σ – h and h > 0……Page 645
7. Estimates of the Quantities ||r^σc(t, β)||1 for c = θσ, 0