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In this paper, estimates of the rate of convergence almost everywhere of the Fourier series of a continuous function and its conjugate are obtained. These estimates, expressed in terms of its best approximation functions and its modulus of continuity, cannot be strengthened in a number of cases.Bibliography: 9 items.

A procedure using Fourier transforms in Luttinger-Tisza calculations for minimizing the interaction energy of a magnetic system is described, which is equivalent to the recently introduced procedure using permutation groups.

In this paper we prove that for the piecewise-linear unit jump approximation the sum of the moduli of the Fourier coefficients with respect to an arbitrary complete orthonormal system, which is totally bounded, has, when averaged over sections, a lower bound of order log N, where N−1 is the approximation step.

There exists a continuous function f f on the real line, vanishing at infinity, such that, for every measure-preserving transformation h h , the composition f ∘ h f \circ h fails to be a Fourier transform. This fact is a consequence of a theorem about measurable functions which is obtained from the theory of idempotents.

A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering. The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator. A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed. The relationship of the scattering theory and Backlund transformations is brought out. In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems.

Carleson and Hunt proved that the space of functions with almost everywhere convergent Fourier series contains Lp (p > l) as a subspace. We shall give two kinds of subspaces of the spaces of functions with everywhere convergent or uniformly convergent Fourier series.

In this paper we consider spaces Ap, p ∈ [1, 2], and multipliers (Ap, Aq), p ∈ [1, 2], q ∈ [1, 2]. In 4.4 and 6.1 we identify (Ap, Aq) for p ∈ [1, 2], q ∈ [p, 2], and in 7.3 we identify (A2, A1). In 7.1 we give a sufficient condition, and in 7.5 a necessary condition, for membership of (Ap, Aq), p ∈ [1, 2], q ∈ [1, 2]. We give, in 7.2, a necessary condition for membership of (A2, Aq), q ∈[l,2). We include constructive proofs of some strict inclusion results for Ap, p ∈ [1,2], (3.1 and 3.2), and also, in 5.3, for (AP,AP), p∈[l,2].

We extend to L p , 1 ≦ p ≦ ∞ {L^p},1 \leqq p \leqq \infty , the L 2 {L^2} results of Bramble and Hilbert on convergence of discrete Fourier transforms and on approximation using smooth splines. The main tools are the estimates of [1] for linear functionals on Sobolev spaces and elementary results on Fourier multipliers.

If f ( x ) f(x) and g ( x ) g(x) satisfy the equations \[ g ( x ) = d d x ∫ 0 ∞ 1 t f ( t ) k 1 ( x t ) d t , f ( x ) = d d x ∫ 0 ∞ 1 t g ( t ) k 1 ( x t ) d t , g(x) = \frac {d}{{dx}}\int _0^\infty \frac {1}{t}f(t){k_1}(xt)dt,\quad f(x) = \frac {d}{{dx}}\int _0^\infty \frac {1}{t}g(t){k_1}(xt)dt, \] then we call f and g a pair of k 1 {k_1} -transforms, where \[ k 1 = 1 2 π i ∫ 1 / 2 − i ∞ 1 / 2 + i ∞ K ( s ) 1 − s x 1 − s d s . k_1 = \frac {1}{2\pi i} \int _{1/2 - i\infty }^{1/2 + i\infty } \frac {K(s)}{1 - s} x^{1-s}\,ds. \] In this paper alternative sets of conditions are established for f and g to be k 1 {k_1} -transform provided K ( s ) K(s) is decomposable in a special way. These conditions involve simpler functions, which replace the kernel k 1 ( x ) {k_1}(x) . Results are proved for the function spaces L 2 {L^2} . The necessary and sufficient conditions are established for the two functions to be self-reciprocal. Conditions are given for generating pairs of transforms for a given kernel. Two examples are given at the end to illustrate the methods and the advantage of the results.

The paper is devoted to the absolute convergence of Fourier series of absolutely continuous functions in arbitrary bounded complete orthonormal systems of functions. The results obtained are new even for the classical systems, namely the trigonometric system and the Walsh system. Bibliography: 8 items.

In this work there is constructed a complete orthonormal system with respect to which the Fourier series of piecewise continuous non-identically-zero functions diverge a.e. There is also constructed an orthonormal system with respect to which the Fourier series of continuous functions converge a.e., while the Fourier series of bounded functions having at least one discontinuity of the first kind diverge a. e. Bibliography: 10 items.

A function f analytic in the upper half-plane Π + {\Pi ^ + } is said to be of class E p ( Π + ) ( 0 > p > ∞ ) {E_p}({\Pi ^ + })(0 > p > \infty ) if there exists a constant C such that ∫ − ∞ ∞ | f ( x + i y ) | p d x ≤ C > ∞ \smallint _{ - \infty }^\infty |f(x + iy){|^p}dx \leq C > \infty for all y > 0 y > 0 . These classes are an extension of the H p {H_p} spaces of the unit disc U. For f belonging to E p ( Π + ) ( 0 > p ≤ 2 ) {E_p}({\Pi ^ + })(0 > p \leq 2) , there exists a Fourier transform f with the property that f ( z ) = 2 ( π ) − 1 ∫ 0 ∞ f ^ ( t ) e i z t d t f(z) = 2{(\pi )^{ - 1}}\smallint _0^\infty \hat f(t){e^{izt}}dt . This makes it possible to give a definition for the multiplication of E p ( Π + ) ( 0 > p ≤ 2 ) {E_p}({\Pi ^ + })(0 > p \leq 2) into L q ( 0 , ∞ ) {L_q}(0,\infty ) that is analogous to the multiplication of H p ( U ) {H_p}(U) into l q {l_q} . In this paper, we consider the case 0 > p > 1 0 > p > 1 and p ≤ q p \leq q and derive a necessary and sufficient condition for multiplying E p ( Π + ) {E_p}({\Pi ^ + }) into L q ( 0 , ∞ ) {L_q}(0,\infty ) .

Some very fast numerical methods have been developed in recent years for the solution of elliptic differential equations which allow for separation of variables. In this paper, a Fourier-Toeplitz method is developed as an alternative to the well-known methods of Hockney and Buneman. It is based on the fast Fourier transform and Toeplitz factorizations. The use of Toeplitz factorizations combined with the Sherman-Morrison formula is also systematically explored for linear systems of algebraic equations with band matrices of Toeplitz, or almost Toeplitz form. Finally, results of numerical experiments are described.

The (A)-summability and mean-convergence of Fourier series in terms of certain biorthonormal systems have been investigated.

An on-line method to compute the spectra associated with the Fourier transform M of a data sequence is developed, whereM and N are finite positive integers. This method provides a simple means of generating time-frequency-amplitude plots of Fourier power and phase spectra. Such plots may be used to display Fourier spectra for pedagogical purposes, and in the general area of the classification of transient waveforms whose durations are unknown. An illustrative example is included.

The Fourier transform of a product of two functions onSL(2,C) is expressed as a convolution integral of the Fourier transforms of its factors. With the help of this convolution integral we present the Fourier transform of a polynomially bounded function as a finite linear combination of analytic delta functionals applied to a continuous function on the real line in an improper sense.

G G is a locally compact abelian group with dual Γ \Gamma . If p ( γ ) = ∑ 1 N a n ( x n , γ ) p(\gamma ) = \sum \nolimits _1^N {{a_n}({x_n},\gamma )} is a trigonometric polynomial, its capacity, by definition is Σ | a n | \Sigma |{a_n}| . The main theorem is: Let φ \varphi be a measurable function defined on the measurable subset Λ \Lambda of Γ \Gamma . If φ \varphi can be approximated on finite sets in Λ \Lambda by trigonometric polynomials of capacity at most C C (constant), then φ = μ ^ \varphi = \hat \mu , locally almost everywhere on Λ \Lambda , where μ \mu is a regular bounded measure on G G and | | μ | | ≦ C ||\mu || \leqq C .

The object of this introduction is to give the reader an idea of the various kinds of problems in which the Fourier series or integral is useful, and to discuss not only the essential properties which make it useful, but also to point out the limitations and restrictions imposed by these properties. Collaterally it is significant to recognize the flexibility that is inherent in the Fourier representation, if we consider its mathematical form from a nonconventional point of view. Thus it is a common belief, that only the integral representation applies to aperiodic functions, and that the Fourier series is restricted to the representation of periodic time functions. If we are interested in sampled time functions, this attitude turns out to be incorrect, and we find that the series rather than the integral is at once the pertinent tool. Inherent properties of the partial sums, such as the Gibbs phenomenon and the mean-square-error criterion are discussed, as well as the fact that in dealing with stable systems for which the response must be zero before the advent of the excitation, it turns out that the real or imaginary parts of the system function are separately sufficient to characterize the response: and one may dispense with the need of having to deal with complex system functions, or to distinguish between such things as minimum-phase and nonminimum-phase transfer functions.

The authors have defined the product of two summability methods and have applied it to Fourier series. The criterion are analogous to the known criterion Convergence of Fourier series.

Keywords: approximation and series expansion Reference CTN-ARTICLE-1966-001doi:10.1007/BF01111289 Record created on 2008-12-12, modified on 2016-08-08

The purpose of this note is to show that while I holds, when appropriately interpreted, for Fourier series on arbitrary compact groups, II does not hold in general if the group is noncommutative. The fact that a generalization of I holds for arbitrary groups follows easily from Billard's original proof and some results of Kahane on random series in Banach spaces [4, Theoreme 2] and [5, Chapter II, Theorem 1]. With the help of another result of Kahane [4, Theoreme 3] and [5, Chapter II, Theorem 5] one can easily show that II holds for Fourier series defined on an arbitrary commutative compact group. Our example in ?3 shows that II can fail at least for some noncommutative groups. One should notice that II is a stronger result than I, in fact an application of Fubini's theorem yields that if II is true, the conclusion also holds for random Fourier series of the type

A theorem by Varshney [8] on an interesting sequence related to Fourier series is generalized.