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In this paper, we derive some Fourier transforms of confluent hypergeometric functions. We give generalizations of several well-known results involving Fourier transforms of gamma functions. In particular, the generalizations include some Ramanujan's remarkable formulas. [ABSTRACT FROM AUTHOR]

In this paper, we deduct some properties of the Fourier transform on arbitrary time scales. We define the generalized shifting problem and we prove the existence of solutions. We define a generalized convolution on anarbitrary time scale and we deduct and prove the generalized convolution theorem. [ABSTRACT FROM AUTHOR]

Quantitative Shapiro's dispersion uncertainty principle and umbrella theorem are proved for the multivariate continuous shearlet transform S H ψ introduced earlier in Dahlke et al. [The continuous shearlet transform in arbitrary space dimensions. Preprint Nr. 2008-7, Philipps-Universität Marburg; 2008; The continuous shearlet transform in arbitrary space dimensions. J Fourier Anal Appl. 2010;16:340–364]. Also, we extend local uncertainty principles for a set of finite measure to S H ψ . [ABSTRACT FROM AUTHOR]

In this paper, we introduce the definitions of the Fourier transform and spherical Fourier transform on quaternionic Heisenberg group, then we present some of their properties, in particular, we will determine the spherical functions and the heat kernel of quaternionic Heisenberg group. Finally, we prove a qualitative uncertainty principle 'Donoho–Stark's uncertainty principle'. [ABSTRACT FROM AUTHOR]

We prove a Heisenberg type uncertainty principle for the Gabor spherical mean transform, and we study its generalization. Next, we extend local uncertainty principle for sets of finite measure to the latter transform. [ABSTRACT FROM AUTHOR]

In this paper some new classes of two-variable orthogonal functions by using Fourier transforms of two-variable orthogonal polynomials are introduced. Orthogonality relations are obtained by using the Parseval identity. Recurrence relations for new families of orthogonal functions are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we study Boas transforms of wavelets and obtain a sufficient condition under which the Boas transform of a wavelet is the derivative of another wavelet. Also, a characterization of the Boas transform of a wavelet ψ ∈ B R − [ − 1 , 1 ] is given. A sufficient condition is given to obtain higher order vanishing moments of Boas transforms of wavelets. Further, we study the Boas transform of wavelets in L 2 (R 2). Finally, higher order vanishing moments of Boas transforms of wavelets have been used to approximate finite energy functions. [ABSTRACT FROM AUTHOR]

Index transforms with the product of the associated Legendre functions are introduced. Mapping properties are investigated in the Lebesgue spaces. Inversion formulas are proved. The results are applied to solve a boundary value problem in a wedge for a third order partial differential equation. [ABSTRACT FROM AUTHOR]

We define the windowed Fourier transform, called also the Gabor transform, associated with singular partial differential operators. We establish Plancherel theorem, orthogonality property and inversion formula for this transform. Next, we define the localization operators , called also Gabor multipliers, associated with two windows and with symbol σ. First, we study the boundedness and compactness of the operators. Last, we define the Schatten-von Neumann class and we prove that the localization operators belong to. [ABSTRACT FROM AUTHOR]

In this paper, using the Mellin transform of Wright function we derive an addition formula for the Wright function. In some special cases, addition formulas for the Hermite, Bessel and Mittag-Leffler functions are also given and the Green's function of two-dimensional time-fractional diffusion equation is presented in the whole plane. [ABSTRACT FROM AUTHOR]

New index transforms with Weber-type kernels, consisting of products of Bessel functions of the first and second kind, are investigated. Mapping properties and inversion formulas are established for these transforms in Lebesgue spaces. The results are applied to solve a boundary value problem on the wedge for a fourth-order partial differential equation. [ABSTRACT FROM PUBLISHER]

In this paper, we deal with a new class of non-local operators that we term integro-differential systems of mixed type. We study the behaviour of solutions of this system when the diffusion term involves higher order fractional powers of the Laplacian. Moreover, we prove that the solution of the system decays faster than a power with an exponent given by the smallest index of the fractional power of the Laplacian. [ABSTRACT FROM PUBLISHER]

The fractional Radon transform defined, based on the Fourier slice theorem and the fractional Fourier transform, has many potential applications in optics and the pattern-recognition field. Here we study many properties of the fractional Radon transform using existing theory of the regular Radon transform: the inversion formulas, stability estimates, uniqueness and reconstruction for a local data problem, and a range description. Also, we define the fractional exponential Radon transform and present its inversion. [ABSTRACT FROM PUBLISHER]

In this paper, we get three integral representations for the linear independent solutions of differential equationin terms of the quartic Lévy stable functions. We also present the products of these solutions with new integral representations including the Bessel functions. Finally, we obtain the Riesz fractional derivative of solutions. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce the concept of the-product which is a very useful tool to obtain theanalytic Fourier–Feynman transform. We then use this concept to obtain various integration formulas involving the-product, the convolution product and the first variation. [ABSTRACT FROM AUTHOR]

New index transforms, involving the squares of Bessel functions of the first kind as the kernel, are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution to the initial value problem for the third-order partial differential equation, involving the Laplacian, is obtained. [ABSTRACT FROM AUTHOR]

In this paper we examine necessary and sufficient conditions on the sequences of norm of the derivatives of functions insuch that their spectrum (the support of their Fourier transform) is contained in a given compact set in. [ABSTRACT FROM PUBLISHER]

Spaces of Boehmians having-growth () are constructed and studied. For, the Fourier transform is defined and properties are established. An inversion formula and an uncertainty principle are given for Boehmians. [ABSTRACT FROM PUBLISHER]

New index transforms, involving the real part of the modified Bessel function of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution of the initial value problem for the second-order partial differential equation, involving the Laplacian, is obtained. It is noted that the corresponding operators with the imaginary part of the modified Bessel function of the first kind lead to the familiar Kontorovich–Lebedev transform and its inverse. [ABSTRACT FROM AUTHOR]

New index transforms are investigated, which contain as the kernel products of the Bessel and modified Bessel functions. Mapping properties and invertibility in Lebesgue spaces are studied for these operators. Relationships with the Kontorovich–Lebedev and Fourier cosine transforms are established. Inversion theorems are proved. As an application, a solution of the initial value problem for the fourth order partial differential equation, involving the Laplacian is presented. [ABSTRACT FROM AUTHOR]

Starting from the classical summation formulas and basing on properties of the Mellin transform and Ramanujan's identities, which represent a ratio of products of Riemann's zeta functions of different arguments in terms of the Dirichlet series of arithmetic functions, we obtain a number of new summation formulas of the Poisson, Müntz, Möbius and Voronoi type. The corresponding analogues of the Müntz operators are investigated. Interesting and curious particular cases of summation formulas involving arithmetic functions are exhibited. Necessary and sufficient conditions for the validity of the Riemann hypothesis are derived. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the Fourier-type functionals introduced in [Chung HS, Tuan VK. Fourier-type functionals on Wiener space. Bull Korean Math Soc. 2012;49:609–619] and used in [Lee IY, Chung HS, Chang SJ. Series expansions of the analytic Feynman integral for the Fourier-type functional. J Korean Soc Math Educ Ser B. 2012;19:87–102]. We first establish the existence of the transform with respect to the Gaussian process for the Fourier-type functionals. We then obtain the various relationships for the transform with respect to the Gaussian process of the Fourier-type functionals which involved the ⋄-product and in addition, establish various integration formulas. [ABSTRACT FROM AUTHOR]

We use real Paley–Wiener theorems to give a characterization of the functions whose Fourier transform has support in antipodal points, and apply the results to study Roe's theorem. [ABSTRACT FROM AUTHOR]

We introduce the translation operators and convolution product connected with some singular partial differential operators defined on the half plane [0,+∞[×ℝ. We investigate Calderon's reproducing formula associated with the convolution * involving finite Borel measures, leading to results onLp-norm for functions on half plane. [ABSTRACT FROM PUBLISHER]

In this paper we obtain solutions in closed form for some special cases of the Toeplitz plus Hankel integral equation on the whole real line with the help of a new Hartley–Fourier cosine generalized convolution. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the theory of variation diminishing kernels introduced by Hirschman, Jr and van de Wetering for a class of singular second-order differential operators introduced by Chébli and Trimèche. We give a necessary and sufficient condition such that a function Λ is a variation diminishing convolution kernel. [ABSTRACT FROM PUBLISHER]

In this paper, we construct and study a new polyconvolution for the Hartley, Fourier cosine and Fourier sine transforms. Here, it is the first time that four different integral transforms appear in the factorization equalities. Next, we get the norm inequalities for this polyconvolution operator in the function spacesL1(ℝ) andLp(ℝ). The applications in solving integral equations and systems of two integral equations are presented. [ABSTRACT FROM AUTHOR]

We define convolution of two distributions in the spaces, in two ways, which turn out to be equivalent, and show that the space is closed under convolution. We also define the Fourier transform of elements of, and prove the Parseval equality. [ABSTRACT FROM AUTHOR]

In this paper, we show that slight modifications of the Kontorovich–Lebedev (KL) transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found. [ABSTRACT FROM PUBLISHER]

In this paper, we obtain solutions in closed form for some special cases of the Toeplitz plus Hankel integral equation with the help of generalized convolutions. [ABSTRACT FROM AUTHOR]

On the one hand, the Fermi-Dirac and Bose-Einstein functions have been extended in such a way that they are closely related to the Riemann and other zeta functions. On the other hand, the Fourier transform representation of the gamma and generalized gamma functions proved useful in deriving various integral formulae for these functions. In this paper, we use the Fourier transform representation of the extended functions to evaluate integrals of products of these functions. In particular, we evaluate some integrals containing the Riemann and Hurwitz zeta functions, which had not been evaluated before. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce some basic concepts of microlocal analysis suitable to the notion of R-regularity within Colombeau algebra. Main result is an extension of the well-known Hormander's theorem on the wave front set of the product of two distributions. [ABSTRACT FROM AUTHOR]

Different approaches are used to introduce a direct product Kontorovich-Lebedev transformation, its essentially multidimensional analogue and certain modifications in terms of multiple integrals over n+. Mapping properties are investigated. Inversion formulas are proved. [ABSTRACT FROM AUTHOR]

We continue to investigate boundedness properties in a two-parametric family of Lebesgue spaces for convolutions related to the Fourier and Kontorovich-Lebedev transforms. Norm estimations in the weighted Lp-spaces are obtained and applications to the corresponding class of convolution integral equations are demonstrated. Necessary and sufficient conditions are found for the solvability of these equations in the weighted L2-spaces. [ABSTRACT FROM AUTHOR]

The theory of Fourier transform on spaces of Boehmians is well known. Several spaces of Boehmians are constructed, and the Fourier transform on these spaces are defined and investigated in the recent literature. Most recently, Dennis Nemzer introduced a new space of Boehmians [image omitted] called the space of integrable Boehmians, which is properly larger than the space [image omitted] of all integrable functions on [image omitted] (the real axis) and investigates the theory of Fourier transform on this space. In this article, we shall find a necessary and sufficient condition for a continuous function to be the Fourier transform of some integrable Boehmian. This answers a question raised by Nemzer. [ABSTRACT FROM AUTHOR]

We consider a class of polynomials related to the kernel Kiτ(x) of the Kontorovich-Lebedev transformation. Algebraic and differential properties are investigated and integral representations are derived. We draw a parallel and establish a relationship with the Bernoulli's and Euler's numbers and polynomials. Finally, as an application we invert a discrete transformation with the introduced polynomials as the kernel, basing it on a decomposition of Taylor's series in terms of the Kontorovich-Lebedev operator. [ABSTRACT FROM AUTHOR]

In this paper we prove Hardy's theorem and a version of Cowling-Price theorem for all step two nilpotent Lie groups using Beurling's theorem on Euclidean space. Also a version of Beurling's theorem is proved for all step two nilpotent Lie groups. [ABSTRACT FROM AUTHOR]

The Stockwell transform of a signal, i.e., a function in L2(), is defined as a hybrid of the Gabor transform and the wavelet transform. The continuous and discrete reconstruction formulas for a signal from its Stockwell transform are established. [ABSTRACT FROM AUTHOR]

This article deals with a non-local problem for the harmonic function in a disc, when the value of the function on one part of its semi-circle and the relation of unknown function values on the semi-circles of concentrated circles on the other part of the circle are given. This problem is reduced to the integral equation with a periodical kernel. To solve it, the analogue of the Wiener–Hopf method for a discrete case is developed. Its use allows one to obtain the problem equivalent to the factorization problem which has a unique solution. An effective solution is constructed. [ABSTRACT FROM AUTHOR]

In this paper, the Fourier transform for some sequences of functions is defined. Also obtained is a generalization for the contructions of a ring of fractions, which is used to extend the class of sequences of functions on which the Fourier transform can be defined. [ABSTRACT FROM AUTHOR]

In this article, we deal with summation formulas for the series (3), referring partly to some results from our article [Stanković, M.S., Vidanović, M.V. and Tric˘ković, S.B., 2000, On the summation of series in involving Bessel or Struve functions. Journal of Mathematics and Analytical Applications , 247 , 15–26]. We show how these formulas arise from different representations of Bessel functions. In other words, we first apply Gegenbauer's integral, then in the sequel we define a function by means of the power series representation of Bessel functions and make use of Poisson's formula. Also, closed-form cases as well as those when it is necessary to take limit have been thoroughly analyzed. [ABSTRACT FROM AUTHOR]

Generalizing the band-limited signals appearing in Shannon's sampling theorem, we consider the signals of entire functions satisfying the growth condition for some positive constants C , L and p and prove that such a signal can be approximated by the interpolating trigonometric polynomials with nodes at the points , which does not exceed . As an application, we obtain a uniqueness theorem for some solutions of heat equation. [ABSTRACT FROM AUTHOR]

The C L classes including analytic and Gevrey ones are the basis of a microlocal analysis developed by Hörmander and refining the C ∞ one. To extend this analysis to the framework of 𝒢, the sheaf of the simplified Colombeau algebra, we begin to embed C L into 𝒢 L constructed as a subpresheaf of 𝒢 ∞ , the subsheaf of ‘regular sections’ of 𝒢. Then, by means of a special estimate for the Fourier transform of some sequence of generalized functions, we can give a microlocal description of the singularities related to 𝒢 L . [ABSTRACT FROM AUTHOR]

Cauchy problems that are not uniformly well posed, abstract ones, and those for differential systems are under consideration. Distribution, semigroup, and Laplace and Fourier transform techniques are used for constructing regularized (in a broad sense) solutions to the problems. The relationship between existence of generalized solutions and existence of convoluted semigroups is established. [ABSTRACT FROM AUTHOR]

We establish the Hausdorff-Young type theorem in Lp(R+), 1 ⩽ p ⩽ 2, which gives the norm estimates and an inversion formula for the Lebedev integral with respect to an index of the Macdonald function. The case p = 2 is known as the Plancherel theorem for the modified Kontorovich-Lebedev transform. [ABSTRACT FROM AUTHOR]

In this article, we prove an inversion formula for the distributional Fourier transform on the space , k ∈ ℤ, k < 0. This result is applied to obtain a representation on &Sscr;′ for any distribution of , as a limit of a sequence of ordinary functions. [ABSTRACT FROM AUTHOR]