Chapter
8. Corollaries of Theorem 7.1. Regularity of Linear Methods of Summation of Fourier Series
pp.：
61 – 84
2. SATURATION OF LINEAR METHODS
pp.：
84 – 97
2. Sufficient Conditions for Saturation
pp.：
97 – 99
1. Statement of the Problem
pp.：
97 – 97
3. Saturation Classes
pp.：
99 – 102
4. Criterion for Uniform Boundedness of Multipliers
pp.：
102 – 108
5. Saturation of Classical Linear Methods
pp.：
108 – 116
3. CLASSES OF PERIODIC FUNCTIONS
pp.：
116 – 119
2. Classes Hω[a,b] and Hω
pp.：
119 – 126
1. Sets of Summable Functions. Moduli of Continuity
pp.：
119 – 119
3. Moduli of Continuity in Spaces Lp. Classes Hωp
pp.：
126 – 128
4. Classes of Differentiable Functions
pp.：
128 – 130
5. Conjugate Functions and Their Classes
pp.：
130 – 134
6. Weyl-Nagy Classes
pp.：
134 – 137
7. Classes LψϐN
pp.：
137 – 138
8. Classes CψϐN
pp.：
138 – 144
9. Classes LψϐN
pp.：
144 – 148
10. Order Relation for (ψ,ϐ) -Derivatives
pp.：
148 – 151
11. ψ-Integrals of Periodic Functions
pp.：
151 – 155
12. Sets M0, M∞, and Mc
pp.：
155 – 165
14. Two Counterexamples
pp.：
171 – 174
15. Function ηa(t) and Sets Defined by It
pp.：
174 – 178
16. Sets B and M0
pp.：
178 – 180
4. INTEGRAL REPRESENTATIONS OF DEVIATIONS OF POLYNOMIALS GENERATED BY LINEAR PROCESSES OF SUMMATION OF FOURIER SERIES
pp.：
180 – 183
2. Second Integral Representation
pp.：
183 – 185
1. First Integral Representation
pp.：
183 – 183
3. Representation of Deviations of Fourier Sums on Sets CψM and Lψ
pp.：
185 – 191
5. APPROXIMATION BY FOURIER SUMS IN SPACES C AND L1
pp.：
191 – 205
1. Simplest Extremal Problems in Space C
pp.：
205 – 207
2. Simplest Extremal Problems in Space L1
pp.：
207 – 216
3. Approximations of Functions of Small Smoothness by Fourier Sums
pp.：
216 – 221
4. Auxiliary Statements
pp.：
221 – 225
5. Proofs of Theorems 3.1-3.3'
pp.：
225 – 243
6. Approximation by Fourier Sums on Classes Hω
pp.：
243 – 253
7. Approximation by Fourier Sums on Classes Hω
pp.：
253 – 257
8. Analogs of Theorems 3.1-3.3' in Integral Metric
pp.：
257 – 261
9. Analogs of Theorems 6.1 and 7.1 in Integral Metric
pp.：
261 – 270
10. Approximations of Functions of High Smoothness by Fourier Sums in Uniform Metric
pp.：
270 – 271
11. Auxiliary Statements
pp.：
271 – 277
12. Proofs of Theorems 10.1-10.3'
pp.：
277 – 289
13. Analogs of Theorems 10.1-10.3' in Integral Metric
pp.：
289 – 296
14. Remarks on the Solution of Kolmogorov-Nikol’skii Problem
pp.：
296 – 297
15. Approximation of ψ-Integrals That Generate Entire Functions by Fourier Sums
pp.：
297 – 302
16. Approximation of Poisson Integrals by Fourier Sums
pp.：
302 – 312
17. Corollaries of Telyakovskii Theorem
pp.：
312 – 321
18. Solution of Kolmogorov-Nikol’skii Problem for Poisson Integrals of Continuous Functions
pp.：
321 – 328
19. Lebesgue Inequalities for Poisson Integrals
pp.：
328 – 356
20. Approximation by Fourier Sums on Classes of Analytic Functions
pp.：
356 – 363
21. Convergence Rate of Group of Deviations
pp.：
363 – 381
22. Corollaries of Theorems 21.1 and 21.2. Orders of Best Approximations
pp.：
381 – 392
23. Analogs of Theorems 21.1 and 21.2 and Best Approximations in Integral Metric
pp.：
392 – 396
24. Strong Summability of Fourier Series
pp.：
396 – 401
BIBLIOGRAPHICAL NOTES (Part I)
pp.：
401 – 411
REFERENCES (Part I)
pp.：
411 – 417
0. Introduction
pp.：
447 – 447
6. CONVERGENCE RATE OF FOURIER SERIES AND THE BEST APPROXIMATIONS IN THE SPACES Lp
pp.：
447 – 447
1. Approximations in the Space L2
pp.：
447 – 450
2. Direct and Inverse Theorems in the Space L2
pp.：
450 – 455
3. Extension to the Case of Complete Orthonormal Systems
pp.：
455 – 457
4. Jackson Inequalities in the Space L2
pp.：
457 – 462
5. Marcinkiewicz, Riesz, and Hardy-Littlewood Theorems
pp.：
462 – 466
6. Imbedding Theorems for the Sets LψLP
pp.：
466 – 470
7. Approximations of Functions from the Sets LψLp by Fourier Sums
pp.：
470 – 473
8. Best Approximations of Infinitely Differentiable Functions
pp.：
473 – 484
9. Jackson Inequalities in the Spaces C and Lp
pp.：
484 – 499
7. BEST APPROXIMATIONS IN THE SPACES C AND L
pp.：
499 – 507
1. Chebyshev and de la Vallée Poussin Theorems
pp.：
507 – 508
2. Polynomial of the Best Approximation in the Space L
pp.：
508 – 510
3. General Facts on the Approximations of Classes of Convolutions
pp.：
510 – 513
4. Orders of the Best Approximations
pp.：
513 – 523
5. Exact Values of the Upper Bounds of Best Approximations
pp.：
523 – 528
6. Dzyadyk-Stechkin-Xiung Yungshen Theorem. Korneichuk Theorem
pp.：
528 – 540
7. Serdyuk Theorem
pp.：
540 – 543
8. Bernstein Inequalities for Polynomials
pp.：
543 – 557
9. Inverse Theorems
pp.：
557 – 563
8. INTERPOLATION
pp.：
563 – 571
2. Lebesgue Constants and Nikol’skii Theorems
pp.：
571 – 575
1. Interpolation Trigonometric Polynomials
pp.：
571 – 571
3. Approximation by Interpolation Polynomials in the Classes of Infinitely Differentiable Functions
pp.：
575 – 578
4. Approximation by Interpolation Polynomials on the Classes of Analytic Functions
pp.：
578 – 590
5. Summable Analog of the Favard Method
pp.：
590 – 604
9. APPROXIMATIONS IN THE SPACES OF LOCALLY SUMMABLE FUNCTIONS
pp.：
604 – 615
1. Spaces Lp
pp.：
615 – 616
2. Order Relation for (ψ, ß)-Derivatives
pp.：
616 – 619
3. Approximating Functions
pp.：
619 – 625
4. General Estimates
pp.：
625 – 633
5. On the Functions ψ(•) Specifying the Sets Lψß
pp.：
633 – 642
6. Estimates of the Quantities ║ȓcσ(t, ß)║1 for c = σ - h and h > 0
pp.：
642 – 644
7. Estimates of the Quantities ║ȓcσ(t, ß)║1 for c = θσ, 0 ≤θ≤ 1, and ψ ∈ Uc
pp.：
644 – 650
8. Estimates of the Quantities ║ȓcσ(t, ß)║1 for c = 2σ - η(σ) and ψ ∈ U∞
pp.：
650 – 652
9. Estimates of the Quantities ║ȓcσ(t, 0)║1 for c = θσ, 0 ≤ θ ≤ 1, and ψ ∈ U0
pp.：
652 – 653
10. Estimates of the Quantities ║δσ,c(t,ß)║1
pp.：
653 – 654
11. Basic Results
pp.：
654 – 657
12. Upper Bounds of the Deviations ρσ(f;•) in the Classes Ĉψß,∞ and ĈψßHω
pp.：
657 – 666
13. Some Remarks on the Approximation of Functions of High Smoothness
pp.：
666 – 684
14. Strong Means of Deviations of the Operators Fσ(f;x)
pp.：
684 – 686
10. APPROXIMATION OF CAUCHY-TYPE INTEGRALS
pp.：
686 – 697
1. Definitions and Auxiliary Statements
pp.：
697 – 698
2. Sets of ψ-Integrals
pp.：
698 – 714
3. Approximation of Functions from the Classes Cψ(T)+
pp.：
714 – 720
4. Landau Constants
pp.：
720 – 731
5. Asymptotic Equalities
pp.：
731 – 734
6. Lebesgue-Landau Inequalities
pp.：
734 – 741
7. Approximation of Cauchy-Type Integrals
pp.：
741 – 745
11. APPROXIMATIONS IN THE SPACES SP
pp.：
745 – 759
2. ψ-Integrals and Characteristic Sequences
pp.：
759 – 763
3. Best Approximations and Widths of p-Ellipsoids
pp.：
763 – 765
4. Approximations of Individual Elements from the Sets
pp.：
765 – 768
5. Best n-Term Approximations
pp.：
768 – 773
6. Best n-Term Approximations (q>p)
pp.：
773 – 789
7. Proof of Lemma 6.1
pp.：
789 – 795
8. Best Approximations by q-Ellipsoids in the Spaces Spφ
pp.：
795 – 826
9. Application of Obtained Results to Problems of Approximation of Periodic Functions of Many Variables
pp.：
826 – 829
10. Remarks
pp.：
829 – 835
11. Theorems of Jackson and Bernstein in the Spaces Sp
pp.：
835 – 840
12. APPROXIMATIONS BY ZYGMUND AND DE LA VALLÉE POUSSIN SUMS
pp.：
840 – 865
1. Fejér Sums: Survey of Known Results
pp.：
865 – 866
2. Riesz Sums: A Survey of Available Results
pp.：
866 – 878
3. Zygmund Sums: A Survey of Available Results
pp.：
878 – 881
4. Zygmund Sums on the Classes Cψß,∞
pp.：
881 – 884
5. De la Vallée Poussin Sums on the Classes Wrß and WrßHw
pp.：
884 – 889
6. De la Vallée Poussin Sums on the Classes CψßN and CψN
pp.：
889 – 895
BIBLIOGRAPHICAL NOTES (Part II)
pp.：
895 – 899
REFERENCES (Part II)
pp.：
899 – 903
Methods of Approximation Theory
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