Fourier Analysis and Approximation

0 Preliminaries.- 0 Preliminaries.- 0.1 Fundamentals on Lebesgue Integration.- 0.2 Convolutions on the Line Group.- 0.3 Further Sets of Functions and Sequences.- 0.4 Periodic Functions and Their Convolution.- 0.5 Functions of Bounded Variation on the Line Group.- 0.6 The Class BV2?.- 0.7 Normed Linear Spaces, Bounded Linear Operators.- 0.8 Bounded Linear Functional, Riesz Representation Theorems.- 0.9 References.- I Approximation by Singular Integrals.- 1 Singular Integrals of Periodic Functions.- 1.0 Introduction.- 1.1 Norm-Convergence and-Derivatives.- 1.1.1 Norm-Convergence.- 1.1.2 Derivatives.- 1.2 Summation of Fourier Series.- 1.2.1 Definitions.- 1.2.2 Dirichlet and Fejer Kernel.- 1.2.3 Weierstrass Approximation Theorem.- 1.2.4 Summability of Fourier Series.- 1.2.5 Row-Finite ?-Factors.- 1.2.6 Summability of Conjugate Series.- 1.2.7 Fourier-Stieltjes Series.- 1.3 Test Sets for Norm-Convergence.- 1.3.1 Norms of Some Convolution Operators.- 1.3.2 Some Applications of the Theorem of Banaeh-Steinhaus.- 1.3.3 Positive Kernels.- 1.4 Pointwise Convergence.- 1.5 Order of Approximation for Positive Singular Integrals.- 1.5.1 Modulus of Continuity and Lipschitz Classes.- 1.5.2 Direct Approximation Theorems.- 1.5.3 Method of Test Functions.- 1.5.4 Asymptotic Properties.- 1.6 Further Direct Approximation Theorems, Nikolski? Constants.- 1.6.1 Singular Integral of Fejer-Korovkin.- 1.6.2 Further Direct Approximation Theorems.- 1.6.3 Nikolski? Constants.- 1.7 Simple Inverse Approximation Theorems.- 1.8 Notes and Remarks.- 2 Theorems of Jackson and Bernstein for Polynomials of Best Approximation and for Singular Integrals.- 2.0 Introduction.- 2.1 Polynomials of Best Approximation.- 2.2 Theorems of Jackson.- 2.3 Theorems of Bernstein.- 2.4 Various Applications.- 2.5 1.- 4.2.1 The Case p = 2.- 4.2.2 The Case p ? 2.- 4.3 Finite Fourier-Stieltjes Transforms.- 4.3.1 Fundamental Properties.- 4.3.2 Inversion Theory.- 4.3.3 Fourier-Stieltjes Transforms of Derivatives.- 4.4 Notes and Remarks.- 5 Fourier Transforms Associated with the Line Group.- 5.0 Introduction.- 5.1 L1-Theory.- 5.1.1 Fundamental Properties.- 5.1.2 Inversion Theory.- 5.1.3 Fourier Transforms of Derivatives.- 5.1.4 Derivatives of Fourier Transforms, Moments of Positive Functions Peano and Riemann Derivatives.- 5.1.5 Poisson Summation Formula.- 5.2 Lp-Theory, 1 < p ? 2.- 5.2.1 The Case p = 2.- 5.2.2 The Case 1 2.- 5.2.3 Fundamental Properties.- 5.2.4 Summation of the Fourier Inversion Integral.- 5.2.5 Fourier Transforms of Derivatives.- 5.2.6 Theorem of Plancherel.- 5.3 Fourier-Stieltjes Transforms.- 5.3.1 Fundamental Properties.- 5.3.2 Inversion Theory.- 5.3.3 Fourier-Stieltjes Transforms of Derivatives.- 5.4 Notes and Remarks.- 6 Representation Theorems.- 6.0 Introduction.- 6.1 Necessary and Sufficient Conditions.- 6.1.1 Representation of Sequences as Finite Fourier or Fourier-Stieltjes Transforms.- 6.1.2 Representation of Functions as Fourier or Fourier-Stieltjes Transforms.- 6.2 Theorems of Bochner.- 6.3 Sufficient Conditions.- 6.3.1 Quasi-Convexity.- 6.3.2 Representation as L1/2? Transform.- 6.3.3 Representation as L1-Transform.- 6.3.4 A Reduction Theorem.- 6.4 Applications to Singular Integrals.- 6.4.1 General Singular Integral of Weierstrass.- 6.4.2 Typical Means.- 6.5 Multipliers.- 6.5.1 Multipliers of Classes of Periodic Functions.- 6.5.2 Multipliers on LP.- 6.6 Notes and Remarks.- 7 Fourier Transform Methods and Second-Order Partial Differential Equations.- 7.0 Introduction.- 7.1 Finite Fourier Transform Method.- 7.1.1 Solution of Heat Conduction Problems.- 7.1.2 Dirichlet's and Neumann's Problem for the Unit Disc.- 7.1.3 Vibrating String Problems.- 7.2 Fourier Transform Method in L1.- 7.2.1 Diffusion on an Infinite Rod.- 7.2.2 Dirichlet's Problem for the Half-Plane.- 7.2.3 Motion of an Infinite String.- 7.3 Notes and Remarks.- III Hilbert Transforms.- 8 Hilbert Transforms on the Real Line.- 8.0 Introduction.- 8.1 Existence of the Transform.- 8.1.1 Existence Almost Everywhere.- 8.1.2 Existence in L2-Norm.- 8.1.3 Existence in Lp-Norm, 1 ?.- 8.2 Hilbert Formulae, Conjugates of Singular Integrals, Iterated Hilbert Transforms.- 8.2.1 Hilbert Formulae.- 8.2.2 Conjugates of Singular Integrals: 1 ?.- 8.2.3 Conjugates of Singular Integrals: p = 1.- 8.2.4 Iterated Hilbert Transforms.- 8.3 Fourier Transforms of Hilbert Transforms.- 8.3.1 Signum Rule.- 8.3.2 Summation of Allied Integrals.- 8.3.3 Fourier.- 8.3.4 Norm-Convergence of the Fourier Inversion Integral.- 8.4 Notes and Remarks.- 9 Hilbert Transforms of Periodic Functions.- 9.0 Introduction.- 9.1 Existence and Basic Properties.- 9.1.1 Existence.- 9.1.2 Hilbert Formulae.- 9.2 Conjugates of Singular Integrals.- 9.2.1 The Case 1 ?.- 9.2.2 Convergence in C2? and L1/2?.- 9.3 Fourier Transforms of Hilbert Transforms.- 9.3.1 Conjugate Fourier Series.- 9.3.2 Fourier Transforms of Derivatives of Conjugate Functions, the Classes (W~)xr2?'(V~)rx2?.- 9.3.3 Norm-Convergence of Fourier Series.- 9.4 Notes and Remarks.- IV Characterization of Certain Function Classes 355.- 10 Characterization in the Integral Case.- 10.0 Introduction.- 10.1 Generalized Derivatives, Characterization of the Classes Wrx2?.- 10.1.1 Riemann Derivatives in X2?-Norm.- 10.1.2 Strong Peano Derivatives.- 10.1.3 Strong and Weak Derivatives, Weak Generalized Derivatives.- 10.2 Characterization of the Classes Vr2?.- 10.3 Characterization of the Classes (V~)rx2?.- 10.4 Relative Completion.- 10.5 Generalized Derivatives in Lp-Norm and Characterizations for 1 ? p ?2.- 10.6 Generalized Derivatives in X(R)-Norm and Characterizations of the Classes Wrx(R) and Vrx(R).- 10.7 Notes and Remarks.- 11 Characterization in the Fractional Case.- 11.0 Introduction.- 11.1 Integrals of Fractional Order.- 11.1.1 Integral of Riemann-Liouville.- 11.1.2 Integral of M. Riesz.- 11.2 Characterizations of the Classes W[LP; |?|?], V[LP; |?|?], 1 ? p ? 2.- 11.2.1 Derivatives of Fractional Order.- 11.2.2 Strong Riesz Derivatives of Higher Order, the Classes V[LP; |?|? ].- 11.3 The Operators R?{?} on Lp 1 ? p ? 2.- 11.3.1 Characterizations.- 11.3.2 Theorems of Bernstein-Titchmarsh and H. Weyl.- 11.4 The Operators R?(?} on 2?.- 11.5 Integral Representations, Fractional Derivatives of Periodic Functions.- 11.6 Notes and Remarks.- V Saturation Theory.- 12 Saturation for Singular Integrals on X2? and Lp, 1 ? p ? 2 433.- 12.0 Introduction.- 12.1 Saturation for Periodic Singular Integrals, Inverse Theorems.- 12.2 Favard Classes.- 12.2.1 Positive Kernels.- 12.2.2 Uniformly Bounded Multipliers.- 12.2.3 Functional Equations.- 12.3 Saturation in Lp, 1 ? p ? 2.- 12.3.1 Saturation Property.- 12.3.2 Characterizations of Favard Classes: p = 1.- 12.3.3 Characterizations of Favard Classes: 1 < p? 2.- 12.4 Applications to Various Singular Integrals.- 12.4.1 Singular Integral of Fejer.- 12.4.2 Generalized Singular Integral of Picard.- 12.4.3 General Singular Integral of Weierstrass.- 12.4.4 Singular Integral of Bochner-Riesz.- 12.4.5 Riesz Means.- 12.5 Saturation of Higher Order.- 12.5.1 Singular Integrals on the Real Line.- 12.5.2 Periodic Singular Integrals.- 12.6 Notes and Remarks.- 13 Saturation on X(R).- 13.0 Introduction.- 13.1 Saturation of D?(f;x;t) in X(R), Dual Methods.- 13.2 Applications to Approximation in Lp, 2 ?.- 13.2.1 Differences.- 13.2.2 Singular Integrals Satisfying (12.3.5).- 13.2.3 Strong Riesz Derivatives.- 13.2.4 The Operators R?{?}.- 13.2.5 Riesz and Fejer Means.- 13.3 Comparison Theorems.- 13.3.1 Global Divisibility.- 13.3.2 Local Divisibility.- 13.3.3 Special Comparison Theorems with no Divisibility Hypothesis.- 13.3.4 Applications to Periodic Continuous Functions.- 13.4 Saturation on Banach Spaces.- 13.4.1 Strong Approximation Processes.- 13.4.2 Semi-Groups of Operators.- 13.5 Notes and Remarks.- List of Symbols.- Tables of Fourier and Hilbert Transforms.