Fourier Analysis and Approximation: One Dimensional Theory by P.l. ButzerFourier Analysis and Approximation: One Dimensional Theory by P.l. Butzer

Fourier Analysis and Approximation: One Dimensional Theory

byP.l. Butzer, Nessel, Trebels

Paperback | May 3, 2012

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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.
Title:Fourier Analysis and Approximation: One Dimensional TheoryFormat:PaperbackDimensions:554 pagesPublished:May 3, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034874502

ISBN - 13:9783034874502


Table of Contents

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 <_29_rx23f_.- 9.3.3="" norm-convergence="" of="" fourier="" series.-="" 9.4="" notes="" and="" remarks.-="" iv="" characterization="" certain="" function="" classes="" 355.-="" 10="" in="" the="" integral="" case.-="" 10.0="" introduction.-="" 10.1="" generalized="" _derivatives2c_="" _wrx23f_.-="" 10.1.1="" riemann="" derivatives="" _x23f_-norm.-="" 10.1.2="" strong="" peano="" derivatives.-="" 10.1.3="" weak="" 10.2="" _vr23f_.-="" 10.3=""><_29_rx23f_.- 10.4="" relative="" completion.-="" 10.5="" generalized="" derivatives="" in="" lp-norm="" and="" characterizations="" for="" 1="" p="1.-" .-="" 10.6="" _x28_r29_-norm="" of="" the="" classes="" _wrx28_r29_="" _vrx28_r29_.-="" 10.7="" notes="" remarks.-="" 11="" characterization="" fractional="" case.-="" 11.0="" introduction.-="" 11.1="" integrals="" order.-="" 11.1.1="" integral="" riemann-liouville.-="" 11.1.2="" m.="" riesz.-="" 11.2="" _w5b_lp3b_="" _7c_3f_7c_3f_5d_2c_="" _v5b_lp3b_="" 2.-="" 11.2.1="" 11.2.2="" strong="" riesz="" higher="" _order2c_="" _7c_3f_7c_3f_="" _5d_.-="" 11.3="" operators="" _r3f_7b_3f_7d_="" on="" lp="" 11.3.1="" characterizations.-="" 11.3.2="" theorems="" bernstein-titchmarsh="" h.="" weyl.-="" 11.4="" _r3f_28_3f_7d_="" _23f_.-="" 11.5="" _representations2c_="" periodic="" functions.-="" 11.6="" v="" saturation="" theory.-="" 12="" singular="" _x23f_="" _lp2c_="" 2="" 433.-="" 12.0="" 12.1="" _integrals2c_="" inverse="" theorems.-="" 12.2="" favard="" classes.-="" 12.2.1="" positive="" kernels.-="" 12.2.2="" uniformly="" bounded="" multipliers.-="" 12.2.3="" functional="" equations.-="" 12.3="" 12.3.1="" property.-="" 12.3.2="" _classes3a_="" 12.3.3=""><_20_3f_.-20_13.2.120_differences.-20_13.2.220_singular20_integrals20_satisfying20_28_12.3.529_.-20_13.2.320_strong20_riesz20_derivatives.-20_13.2.420_the20_operators20_r3f_7b_3f_7d_.-20_13.2.520_riesz20_and20_fejc3a9_r20_means.-20_13.320_comparison20_theorems.-20_13.3.120_global20_divisibility.-20_13.3.220_local20_divisibility.-20_13.3.320_special20_comparison20_theorems20_with20_no20_divisibility20_hypothesis.-20_13.3.420_applications20_to20_periodic20_continuous20_functions.-20_13.420_saturation20_on20_banach20_spaces.-20_13.4.120_strong20_approximation20_processes.-20_13.4.220_semi-groups20_of20_operators.-20_13.520_notes20_and20_remarks.-20_list20_of20_symbols.-20_tables20_of20_fourier20_and20_hilbert20_transforms. -="" 13.2.1="" differences.-="" 13.2.2="" singular="" integrals="" satisfying="" _28_12.3.529_.-="" 13.2.3="" strong="" riesz="" derivatives.-="" 13.2.4="" the="" operators="" _r3f_7b_3f_7d_.-="" 13.2.5="" and="" _fejc3a9_r="" means.-="" 13.3="" comparison="" theorems.-="" 13.3.1="" global="" divisibility.-="" 13.3.2="" local="" 13.3.3="" special="" theorems="" with="" no="" divisibility="" hypothesis.-="" 13.3.4="" applications="" to="" periodic="" continuous="" functions.-="" 13.4="" saturation="" on="" banach="" spaces.-="" 13.4.1="" approximation="" processes.-="" 13.4.2="" semi-groups="" of="" operators.-="" 13.5="" notes="" remarks.-="" list="" symbols.-="" tables="" fourier="" hilbert="">