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We demonstrate that the Fourier partial sums of a Lipschitz continuous function on the two-dimensional special unitary group converge uniformly to the function.

In this study, we present some formulae for the Fourier transform of O ( p , q ) -invariant temperate distributions. The formulae are used to calculate fundamental solutions of homogeneous and non-homogeneous ultra-hyperbolic operators.

By using the convolution theorem of the Fourier transform (Faltung theorem), the following integral involving the Macdonald function is calculated, ∫ − ∞ ∞ | t ′ | α + 2 n K α ( a | t ′ | ) | t − t ′ | β + 2 m K β ( a | t − t ′ | ) d t ′ . As a consistency test of the result obtained, setting the parameters α, β, m, n, t and a to particular values, some integrals reported in the literature are recovered. It turns out that the calculation method of integrals via the convolution theorem is useful for calculating other infinite integrals involving the Macdonald function.

New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. After various preceding works of the last 25 years where the behavior of the Fourier transform has been considered on specific subspaces of the space of functions of bounded variation, in this paper such problems are considered on the whole space of functions of bounded variation. The widest subspaces of the space of functions of bounded variation are studied for which the cosine and sine Fourier transforms are integrable. The main result of the paper is an asymptotic formula for the sine Fourier transform of an arbitrary locally absolutely continuous function of bounded variation. Interrelations of various function spaces are studied, in particular, the sharpness of Hardy's inequality is established and the inequality itself is strengthened in certain cases. A way to extend the obtained results to the radial case is shown.

We obtain some well-known expansions for the Lerch transcendent and the Hurwitz zeta function using elementary Fourier analytic methods. These Fourier series can be used to analytically continue the functions and prove the classical functional equations, which arise from the relations satisfied by the Fourier conjugate and flat Fourier series. In particular, the functional equation for the Riemann zeta function can be obtained in this way without contour integrals. The conjugate series for special values of the parameters yields analogous results for the Bernoulli and Apostol-Bernoulli polynomials. Finally, we give some consequences derived from the Fourier series. © 2015 Elsevier Inc.

The quaternion Fourier transform – a generalized form of the classical Fourier transform – has been shown to be a powerful analyzing tool in image and signal processing. This paper investigates Pitt's inequality and uncertainty principle associated with the two-sided quaternion Fourier transform. It is shown that by applying the symmetric form f = f 1 + i f 2 + f 3 j + i f 4 j of quaternion from Hitzer and the novel module or L p -norm of the quaternion Fourier transform f ˆ , then any nonzero quaternion signal and its quaternion Fourier transform cannot both be highly concentrated. Two part results are provided, one part is Heisenberg–Weyl's uncertainty principle associated with the quaternion Fourier transform. It is formulated by using logarithmic estimates which may be obtained from a sharp of Pitt's inequality; the other part is the uncertainty principle of Donoho and Stark associated with the quaternion Fourier transform.

In this paper we state new theorems of Hardy–Littlewood type for functions with general monotone Fourier coefficients. Sharpness of stated results is discusses.

The nonlinear Fourier basis has shown its advantages over the classical Fourier basis in the time-frequency analysis. The need of processing large amount of high dimensional data motivates the extension of the methods based upon the nonlinear Fourier basis to high dimensions. We consider the multi-dimensional nonlinear Fourier basis, which is the tensor product of univariate nonlinear Fourier basis. We investigate the convergence order in norm and also the almost everywhere convergence of the nonlinear Fourier expansions. In order to compute fast and efficiently the nonlinear Fourier expansions of d-dimensional functions, we introduce the sparse nonlinear Fourier expansion and develop a fast algorithm for evaluating it. We also prove that the fast sparse nonlinear Fourier expansions enjoy the optimal convergence order and reduce the computational costs to O ( n log 2 d − 1 n ) . Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed method.

In this paper we obtain new sufficient conditions for representation of a function as an absolutely convergent Fourier integral. Unlike those known earlier, these conditions are given in terms of belonging to weighted spaces. Adding weights allows one to extend the range of application of such results to Fourier multipliers with unbounded derivatives.

A generalization of Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series is investigated with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the Marcinkiewicz-θ-means of a tempered distribution is bounded from H p ( X d ) to L p ( X d ) for all d / ( d + α ) p ⩽ ∞ and, consequently, is of weak type ( 1 , 1 ) , where 0 α ⩽ 1 is depending only on θ and X = R or X = T . As a consequence we obtain a generalization of a summability result due to Marcinkiewicz and Zhizhiashvili for d-dimensional Fourier transforms and Fourier series, more exactly, the Marcinkiewicz-θ-means of a function f ∈ L 1 ( X d ) converge a.e. to f. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces H p ( X d ) and so they converge in norm ( d / ( d + α ) p ∞ ) . Similar results are shown for conjugate functions. Some special cases of the Marcinkiewicz-θ-summation are considered, such as the Fejer, Cesaro, Weierstrass, Picar, Bessel, de La Vallee–Poussin, Rogosinski and Riesz summations.

Firstly, we study the uniform convergence of cosine and sine Fourier transforms. Secondly, we obtain Pitt–Boas type results on L p -integrability of Fourier transforms with the power weights. The solutions of both problems are written as criteria in terms of general monotone functions.

In this paper, we develop some important Fourier analysis tools in the context of time scales. In particular, we present a generalized Fourier transform in this context as well as a critical inversion result. This leads directly to a convolution for signals on two (possibly distinct) time scales as well as several natural classes of time scales which arise in this setting: dilated, closed under addition, and additively idempotent. We explore the properties of these time scales and demonstrate the utility of these concepts in discrete convolution, Mellin convolution, and transformations of a random variable.

We obtain a sharp L 2 ( R n ) bound for the maximal directional Hilbert transform over an arbitrary set of directions.

In this paper we complete a distributional Fourier analysis developed by Howell in a serie of papers. We investigate convolution operators in the corresponding distribution spaces.

A uniqueness theorem is proven for the problem of the recovery of a complex valued compactly supported 2-D function from the modulus of its Fourier transform. An application to the phase problem in optics is discussed.

Complementary spaces for Fourier series were introduced by G. Goes and generalized by M. Tynnov. In this paper we investigate a notion of complementary space for double Fourier series of functions of bounded variation. Various applications are given.

For any continuous and 2π-periodic function f , the GW condition is known to be necessary and sufficient for the everywhere convergence of the Fourier series of f ∘ g for every homeomorphism g of [−π, π] with itself. It is shown that we may replace the continuous functions in this result by those which have right and left limits at each point.

We prove Abel-Tauber theorems for Fourier cosine transforms. They treat the boundary cases of the Abel-Tauber theorem of Pitman. and Soni and Soni. A similar result for Fourier cosine series is obtained as a corollary. The latter gives an answer to an open problem in Boas′ book on Fourier series. Applications to probability distributions and stationary processes are given.

In this paper, we prove a representation theorem for the usual distributional Fourier transform over the spaces S′k, k ∈ Z, k < 0. An inversion formula is also obtained, which enables us to prove that S′k is a commutative convolution algebra with unit element.

Basic general definitions for the multiplication and convolution of elements in the dual to the space of "exponentially decreasing" analytic test functions are developed and analyzed with regard to continuity and the Fourier-Laplace transform.

The dual to the space of analytic test functions is introduced along with operators generalizing translation, scaling, differentiation, and the Fourier-Laplace transform. Baste aspects and formulas concerning Fourier analysis on this dual is developed. It is shown that all elements of this dual space can be represented using Integration on C n. It is also shown that this dual contains all exponentially integrable functions. Results derived include formulas for the Fourier transforms of arbitrary exponential functions and step functions.

This is the second in a series of papers extending the theory supporting Fourier analysis. Bounds for basic mappings on the space of test functions, the convergence of certain sequences of lest functions, and the convolution of test functions are discussed.

It is shown that the Fourier coefficients satisfying Stanojevic's Tauberian condition can be represented in terms of O-regularly varying sequences.

It is well known that the Lp norm is an essential tool in the study of Fourier series. In letting p + co the Lp norm becomes the essential upper bound and Lp behavior formally becomes Lipschitz behavior. It is a natural way to generalize results on Fourier series by replacing the powerfunction by more general classes of functions. In this paper we shall focus on the application of a class of functions which has been used extensively in classical analysis by a number of authors [S, 9, 12, 133. Following the notation in [6] we denote this class by Y[a, b] (a < b), which is defined as the collection of all positive functions @ defined on (0, co) such that @(u)/u” is nondecreasing and @(u)/u” is nonincreasing. The class Y[a, b] is known to be equivalent to other class of functions of the “power type” such as those introduced by Marcinkiewicz [ 111, Koizumi [lo] , Woyczynski [ 151, and the class of R 0 varying functions [ 141. For the details the readers are referred to [4, 6, 71. The author [4] has proved the following integral relationship for functions in Y[a, h]

We describe the precise relationship between the local smoothness behavior of an integrable function and the asymptotic tail behavior of its Fourier transform. This has special relevance for probability and spectral density functions with a discontinuous m th derivative. Simply stated, if the function's Fourier transform behaves asymptotically as the product of the Fourier series for a discrete complex measure and a regularly varying function, then an m th derivative of the function behaves at its discontinuities like the density of a regularly varying function. With side conditions, the converse also holds.

In the present paper the authors prove a Parseval-Goldstein type theorem involving the classical Laplace transform, the Fourier sine transform, and Widder's potential transform. The theorem is then shown to yield a simple algorithm for evaluating infinite integrals. Some illustrative examples are also given.

Let μ be a signed measure with a density. If the components of p: R n → R d are linearly independent polynomials, an estimate for the Fourier transform of the signed measure μp−1 on R d is derived.

A Fourier transform for sufficiently bounded and regular functionals on C([0, 1], R) is introduced, which interchanges multiplication and differentiation. We define convolutions and prove a convolution theorem. Finally, we use the transform to uniquely solve the Cauchy problem for second-order parabolic equations in function-space, where the coefficients depend only on t.

If ƒ(z) is an entire function of order larger than one that satisfies certain additional conditions, characterizations of the growth of its Fourier transform are given, in terms of the growth of ƒ(z) . The results obtained are extensions of theorems of Paley and Wiener, and are a refinement of prior work of Gel'fand and Silov.

L1-convergence theorems for Fourier series with complex quasimonotone coefficients are generalized. The class of null sequences of bounded variation of order m is extended and the point-wise theorem for Fourier series with such coefficients is proved.

A space of pseudoquotients is introduced that is shown to be isomorphic to the space of tempered distributions on RN. The Fourier transform is defined as a map from the space of pseudoquotients to the space of tempered distributions and as a transformation on pseudoquotients.

After the analysis of the q-even translation and the q-cosine Fourier transform by A. Fitouhi and F. Bouzeffour in [q-cosine Fourier transform and q-heat equation, Ramanujan J., in press], it is natural to look for the q-analogue of some well-known classical theorems. In this paper, we purpose to give a q-version of the Paley–Wiener theorem related to this Fourier transform using standard methods of the q-calculus.

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy–Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type (1,1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that σTθf→f over a cone-like set a.e. for all f∈L1(Rd). Moreover, σTθf(x) converges to f(x) over a cone-like set at each Lebesgue point of f∈L1(Rd) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.

In this paper extensions of the classical Fourier, fractional Fourier and Radon transforms to superspace are studied. Previously, a Fourier transform in superspace was already studied, but with a different kernel. In this work, the fermionic part of the Fourier kernel has a natural symplectic structure, derived using a Clifford analysis approach. Several basic properties of these three transforms are studied. Using suitable generalizations of the Hermite polynomials to superspace (see [H. De Bie, F. Sommen, Hermite and Gegenbauer polynomials in superspace using Clifford analysis, J. Phys. A 40 (2007) 10441-10456]) an eigenfunction basis for the Fourier transform is constructed., Comment: 20 pages, accepted for publication in J. Math. Anal. Appl

INTRODUCTION The theory of pseudodifferential operators is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the Ž w x. so-called Fourier integral operators see 1, 10, 11 . The Fourier integral operators are represented by formulas of the type

In this paper we develop a characterization of the Fourier transform as a continuous operator from L 2 (R n ) to L 2 (R n ) behaving like the Fourier transform under rotations, dilation, and translations. This gives a new approach to a result of A. Hertle [6] based on the Hecke-Bochner identities.

In this paper, we prove a simple integrability condition for Fourier transforms of functions in L(p, q) and for functions in Lytil!,-,,, . With the help of this condition we prove a strong version of a conjecture of Boas [I] concerning integrability of Fourier transforms of monotone functions, and of functions whose Fourier transforms are monotone. Our condition has a natural analog for periodic functions, which in turn provides a new proof for a classical theorem of Hardy and Littlewood on functions whose Fourier coefficients are monotone. We will review in brief the definitions of the spaces used: If (X, Z, ,u) is a u-finite measure space, define