Introduction to the Theory of Fourier Integrals

  title={Introduction to the Theory of Fourier Integrals},
  author={A. C. Offord},
SINCE the publication of Prof. Zygmund's “Trigonometric Series” in 1935, there has been considerable demand for another book dealing with trigonometric integrals. Prof. Titchmarsh's book meets this demand. He is already well known to students of mathematics by his text-book on the theory of functions, and his new book comes up to the high standard of the former.Introduction to the Theory of Fourier Integrals By Prof. E. C. Titchmarsh. Pp. x + 390. (Oxford: Clarendon Press; London: Oxford… 

Reciprocal Convergence Classes for Fourier Series and Integrals

  • A. Guinand
  • Mathematics
    Canadian Journal of Mathematics
  • 1961
The classical result of Plancherel for Fourier cosine transforms of functions f(x) of the class L 2(0, ∞) states that (see (7) for references) converges in mean square to a function g(x) which also

On the curious series related to the elliptic integrals

By using the theory of the elliptic integrals, a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise

Moduli of continuity and average decay of Fourier transforms: two-sided estimates

We study inequalities between general integral moduli of continuity of a function and the tail integral of its Fourier transform. We obtain, in particular, a refinement of a result due to D. B. H.

Harmonic analysis associated to the canonical Fourier Bessel transform

ABSTRACT The aim of this paper is to develop a new harmonic analysis related to a Bessel type operator on the real line: We define the canonical Fourier Bessel transform and study some of its

Arithmetical Aspects of Beurling's Real Variable Reformulation of the Riemann Hypothesis

The paper presents two arithmetical versions of the Nyman-Beurling equivalence with the Riemann hypothesis, proved by classical, quasi elementary, number-theoretic methods, based on an integrated

Voronoi–Nasim summation formulas and index transforms

Using L 2-theory of the Mellin and Fourier–Watson transformations, we relax Nasim’s conditions to prove the summation formula of Voronoi. It involves sums of the form , where d(n) is the number of

The Mellin transform of powers of the zeta-function

0 f(x)x s 1 dx with s = + it denote the Mellin transform of f(x). Mellin transforms play a fundamental role in Analytic Number Theory. They can be viewed, by a change of variable, as special cases of

Twisted second moments and explicit formulae of the Riemann zeta-function

Several aspects connecting analytic number theory and the Riemann zeta-function are studied and expanded. These include: 1. explicit formulae relating the Mobius function to the non-trivial zeros of

Multiplicator type operators and approximation of periodic functions of one variable by trigonometric polynomials

The norms of the images of multiplier type operators generated by an arbitrary generator are estimated in terms of the best approximations of univariate periodic functions by trigonometric

On the Integrability and Uniform Convergence of Multiplicative Fourier Transforms

Abstract Analogues of two Hardy–Littlewood theorems are proved for a multiplicative Fourier transform. A Szasz type condition for a multiplicative Fourier transform is given and its nonimprovability