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 transยญ 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 convoluยญ 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
CONTENT
0 Preliminaries -- 0 Preliminaries -- I Approximation by Singular Integrals -- 1 Singular Integrals of Periodic Functions -- 2 Theorems of Jackson and Bernstein for Polynomials of Best Approximation and for Singular Integrals -- 3 Singular Integrals on the Line Group -- II Fourier Transforms -- 4 Finite Fourier Transforms -- 5 Fourier Transforms Associated with the Line Group -- 6 Representation Theorems -- 7 Fourier Transform Methods and Second-Order Partial Differential Equations -- III Hilbert Transforms -- 8 Hilbert Transforms on the Real Line -- 9 Hilbert Transforms of Periodic Functions -- IV Characterization of Certain Function Classes 355 -- 10 Characterization in the Integral Case -- 11 Characterization in the Fractional Case -- V Saturation Theory -- 12 Saturation for Singular Integrals on X2? and Lp, 1 ? p ? 2 433 -- 13 Saturation on X(R) -- List of Symbols -- Tables of Fourier and Hilbert Transforms
