Fourier Analysis and Approximation

At the international conference on 'Harmonic Analysis and Integral Transforms', conducted by one of the authors at the Mathematical Research Institute in Oberwolfach (Black Forest) in August 1965, it was felt that there was a real need for a book on Fourier analysis stressing (i) parallel treatment of Fourier series and Fourier trans­ forms from a transform point of view, (ii) treatment of Fourier transforms in LP(lRn)_ space not only for p = 1 and p = 2, (iii) classical solution of partial differential equations with completely rigorous proofs, (iv) theory of singular integrals of convolu­ tion type, (v) applications to approximation theory including saturation theory, (vi) multiplier theory, (vii) Hilbert transforms, Riesz fractional integrals, Bessel potentials, (viii) Fourier transform methods on locally compact groups. This study aims to consider these aspects, presenting a systematic treatment of Fourier analysis on the circle as well as on the infinite line, and of those areas of approximation theory which are in some way or other related thereto. A second volume is in preparation which goes beyond the one-dimensional theory presented here to cover the subject for functions of several variables. Approximately a half of this first volume deals with the theories of Fourier series and of Fourier integrals from a transform point of view.

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 Fejér-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 < p < 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 L½? 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 < p < ?.- 8.2 Hilbert Formulae, Conjugates of Singular Integrals, Iterated Hilbert Transforms.- 8.2.1 Hilbert Formulae.- 8.2.2 Conjugates of Singular Integrals: 1 < p < ?.- 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 < p < ?.- 9.2.2 Convergence in C2? and L½?.- 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 < p < ?.- 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 Fejér 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.