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The aim of this paper is to establish several versions of Paley–Wiener-type theorems for the canonical Fourier–Bessel transform, which is a Bessel type of linear canonical transform indexed by a matrix parameter m ∈ SL(2, ℝ). [ABSTRACT FROM AUTHOR]

For the octonion Fourier transform, we establish the real Paley‐Wiener theorem. It relates the mean of derivatives of a function with the support of its octonion Fourier transform via limm→∞‖∂mαf‖L2(ℝ3,핆)1/m=(2π)α‖wα‖L∞(suppF핆{f},핆)for any octonion‐valued Sobolev function f∈H∞(ℝ3,핆). [ABSTRACT FROM AUTHOR]

Abstract. In this paper, we establish some analogues of the Paley-Wiener theorem for the Hankel type transform of half-integer order less than -1 in weighted L²-spaces. These Paley-Wiener-type theorems give a description of the class of even entire functions of exponential type σ ≤ 1 under this transformation in terms of solutions from the corresponding spaces of some differential equations. [ABSTRACT FROM AUTHOR]

The paper promotes a new sparse approximation for fractional Fourier transform, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the upper half-plane. Under this methodology, the local polynomial Fourier transform characterization of Hardy space is established, which is an analog of the Paley-Wiener theorem. Meanwhile, a sparse fractional Fourier series for chirp L 2 function is proposed, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the unit disk. Besides the establishment of the theoretical foundation, the proposed approximation provides a sparse solution for a forced Schr o ¨ dinger equations with a harmonic oscillator. [ABSTRACT FROM AUTHOR]

We develop a photonic description of short, one-dimensional electromagnetic pulses, specifically in the language of electrical transmission lines. Current practice in quantum technology, using arbitrary waveform generators, can readily produce very short, few- cycle pulses in a very-low-noise, low-temperature setting. We argue that these systems attain the limit of producing pure coherent quantum states, in which the vacuum has been displaced for a short time, and therefore over a short spatial extent. When the pulse is bipolar, that is, the integrated voltage of the pulse is zero, then the state can be described by the finite displacement of a single mode. Therefore there is a definite mean number of photons, but which have neither a well-defined frequency nor position. Due to the Paley--Wiener theorem, the two-component photon "wavefunction" of this mode, while somewhat localized, is not strictly bounded in space even if the vacuum displacement that defines it is bounded. When the pulse is unipolar, no photonic description is possible--the photon number can be considered to be divergent. We consider properties that photon counters and quantum non-demolition detectors must have to optimally convert and detect the photons in several example pulses. We develop a conceptual test system for implementing short-pulse quantum key distribution, building on the design of a recently achieved Bell's theorem test in a cryogenic microwave setup. [ABSTRACT FROM AUTHOR]

Let L² ((0; 1); x4 dx) be the weighted Lebesgue space of all measurable functions f : (0; 1) → C, satisfying ∫0¹ t4 |f(t)|² dt < +∞. Let J−5/2 be the Bessel function of the first kind of index −5/2 and (ρk)k∈N be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system {ρk² √xρk J−5/2(xρk) : k ∈ N} in the space L² ((0; 1); x4 dx) are found in terms of an entire function with the set of zeros coinciding with the sequence (ρk)k∈N. In this case, we study an integral representation of some class E4,+ of even entire functions of exponential type σ ≤ 1. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than −1, due to B. Vynnyts’kyi, V. Dilnyi, O. Shavala and the author. [ABSTRACT FROM AUTHOR]

We consider the problem of restoring a two‐dimensional convex source of gravity based on values of its potential function on a union of half‐planes. With that aim we prove a Paley–Wiener type theorem for functions of exponential type in a sector. In particular, we find the maximal set of analytic continuation whose complement is convex inside a sector. Our result serves as a sectorial analog of G. Polya's indicator theorem. It corrects, simplifies, and extends a result of M. Morimoto. It also improves a related result of M. Dzhrbashyan and A. Avetisyan. [ABSTRACT FROM AUTHOR]

Representation and boundedness properties of linear, right-shift invariant operators on half-line Bessel potential spaces (also known as fractional-order Sobolev spaces) as operator-valued multiplication operators in terms of the Laplace transform are considered. These objects are closely related to the input–output operators of linear, time-invariant control systems. Characterisations of when such operators map continuously between certain interpolation spaces and/or Bessel potential spaces are provided, including characterisations in terms of boundedness and integrability properties of the symbol, also known as the transfer function in this setting. The paper considers the Hilbert space case, and the theory is illustrated by a range of examples. [ABSTRACT FROM AUTHOR]

In this paper we prove several versions of real Paley–Wiener theorems for the Fourier–Bessel transform associated with the corresponding Laplace–Bessel operator which has many applications in pure and applied mathematics. We characterize functions in some Lebesgue space L γ p (R n) , 1 ≤ p ≤ ∞ , with their Fourier–Bessel transforms vanishing outside some polynomial domains and in a neighborhood of the origin respectively. Finally, a stronger version of real Paley–Wiener theorem for the Fourier–Bessel transform in L γ p (R n) is proved. [ABSTRACT FROM AUTHOR]

We study the phase retrieval problem for the short-time Fourier transform on the groups Z , Z d and R d . As is well-known, phase retrieval is possible once the windows’s ambiguity function vanishes nowhere. However, there are only few results for windows that fail to meet this condition. The goal of this paper is to establish new and complete characterizations for phase retrieval with more general windows and compare them to existing results. For a fixed window, our uniqueness conditions usually only depend on the signal’s support and are therefore easily comprehensible. In the discrete settings, we also provide examples which show that a non-vanishing ambiguity function is not necessary for a window to do phase retrieval. [ABSTRACT FROM AUTHOR]

It is well known that the kinetic and potential energies of a system governed by a linear wave equation are equipartitioned in a finite time. This result has been extended by previous researchers to several physical settings: elastodynamics, electrodynamics, and micropolar elasticity. In the case of a helix, there is a direct coupling of longitudinal and torsional wave motions due to the structure of constitutive relations. Providing that the initial conditions functions have compact support, we demonstrate that the equipartition phenomenon occurs after a finite time in this case as well—i.e. the potential energy and the kinetic energy of a helix become equal and remain constant after a finite time. [ABSTRACT FROM AUTHOR]

We open a new perspective on the sup-norm problem and propose a version for non-spherical Maaß forms when the maximal compact K is non-abelian and the dimension of the K -type gets large. We solve this problem for an arithmetic quotient of G = SL 2 (C) with K = SU 2 (C). Our results cover the case of vector-valued Maaß forms as well as all the individual scalar-valued Maaß forms of the Wigner basis, reaching sub-Weyl exponents in some cases. On the way, we develop analytic theory of independent interest, including uniform strong localization estimates for generalized spherical functions of high K -type and a Paley–Wiener theorem for the corresponding spherical transform acting on the space of rapidly decreasing functions. The new analytic properties of the generalized spherical functions lead to novel counting problems of matrices close to various manifolds that we solve optimally. [ABSTRACT FROM AUTHOR]

In the Clifford algebra setting, the present study develops efficient approximations by linear combinations of the parameterized kernel functions in monogenic reproducing kernel Hilbert spaces of Paley–Wiener type, which include the Paley–Wiener space, the Hardy space on strips, and the Bergman space on strips. [ABSTRACT FROM AUTHOR]

In the Clifford algebra setting the present study develops three reproducing kernel Hilbert spaces of the Paley–Wiener type, namely the Paley–Wiener spaces, the Hardy spaces on strips, and the Bergman spaces on strips. In particular, we give spectrum characterizations and representation formulas of the functions in those spaces and estimation of their respective reproducing kernels. [ABSTRACT FROM AUTHOR]

In this paper, we study an integral representation of one class of entire functions. Conditions for the existence of this representation in terms of certain solutions of some differential equations are found. We obtain asymptotic estimates of entire functions from the considered class of functions. We also give examples of entire functions from this class. [ABSTRACT FROM AUTHOR]

Abstract In (Rahman and Schmeisser in Trans. Amer. Math. Soc. 320: 91–103, 1990), the authors prove that the classical Bernstein inequality also holds for 0

New representations for an integral kernel of the transmutation operator and for a regular solution of the perturbed Bessel equation of the form −u″+ℓ(ℓ+1)x2+q(x)u=ω2u are obtained. The integral kernel is represented as a Fourier‐Jacobi series. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to ω. For the coefficients of the series convenient for numerical computation recurrent integration formulas are obtained. The new representation improves the ones previously obtained by the authors for large values of ω and ℓ and for noninteger values of ℓ. The results are based on application of several ideas from the classical transmutation (transformation) operator theory, asymptotic formulas for the solution, results connecting the decay rate of the Fourier transform with the smoothness of a function, the Paley‐Wiener theorem, and some results from constructive approximation theory. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy. [ABSTRACT FROM AUTHOR]

We comprehensively review several general methods and analytical tools used for causality evaluation of photonic materials. Our objective is to call to mind and then formulate, on a mathematically rigorous basis, a set of theorems which can answer the question whether a considered material model is causal or not. For this purpose, a set of various distributional theorems presented in literature is collected as the distributional version of the Titchmarsh theorem, allowing for evaluation of causality in complicated electromagnetic systems. Furthermore, we correct the existing material models with the use of distribution theory in order to obtain their causal formulations. In addition to the well-known Kramers–Krönig (K–K) relations, we overview four further methods which can be used to assess causality of given dispersion relations, when calculations of integrals involved in the K–K relations are challenging or even impossible. Depending on the given problem, optimal approaches allowing us to prove either the causality or lack thereof are pointed out. These methodologies should be useful for scientists and engineers analyzing causality problems in electrodynamics and optics, particularly with regard to photonic materials, when the involved mathematical distributions have to be invoked.

A classical result due to Paley and Wiener characterizes the existence of a nonzero function in $L^{2}(\mathbb{R})$ , supported on a half-line, in terms of the decay of its Fourier transform. In this paper, we prove an analogue of this result for Damek–Ricci spaces. [ABSTRACT FROM AUTHOR]

In this paper, for an arbitrary fixed compact set K, we find necessary and sufficient conditions on the Taylor expansion coefficients of entire functions of exponential type so that these functions are the Fourier image of distributions supported in K. In other words, we state the Paley–Wiener theorem in the language of Taylor expansion coefficients. [ABSTRACT FROM AUTHOR]

In this paper we introduce appropriate associated function to the sequence M p = p τ p σ , p ∈ N , τ > 0 , σ > 1 , and derive its sharp asymptotic estimates in terms of the Lambert W function. These estimates are used to prove a Paley–Wiener type theorem for compactly supported functions from extended Gevrey classes. As an application, we discuss properties of the corresponding wave front sets. [ABSTRACT FROM AUTHOR]

In the framework of Clifford analysis, we consider the Paley‐Wiener type theorems for a generalized Clifford‐Fourier transform. This Clifford‐Fourier transform is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra. [ABSTRACT FROM AUTHOR]

The main aim of this work is to obtain Paley-Wiener and Wiener's Tauberian results associated with an oscillatory integral operator, which depends on cosine and sine kernels, as well as to introduce a consequent new convolution. Additionally, a new Young-type inequality for the obtained convolution is proven, and a new Wiener-type algebra is also associated with this convolution. [ABSTRACT FROM AUTHOR]

Abstract: Paley–Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundles on a compact Riemannian symmetric space U / K. It characterizes f with small support in terms of holomorphic extendability and exponential growth of their χ‐spherical Fourier transforms, where χ is a character of K. An important tool in our proof is a generalization of Opdam's estimate for the hypergeometric functions associated to multiplicity functions that are not necessarily positive. At the same time the radius of the domain where this estimate is valid is increased. This is done in an appendix. [ABSTRACT FROM AUTHOR]

In this article, we discuss Benedicks' theorem for the Weyl transform associated with the Heisenberg group. Suppose be a function with compact support and if the Weyl transform is of finite rank, then we show that σ is zero almost everywhere on . [ABSTRACT FROM AUTHOR]

Here we give a new approach to the Paley-Wiener theorem in a Mellin analysis setting which avoids the use of the Riemann surface of the logarithm and analytical branches and is based on new concepts of polar-analytic function in the Mellin setting and Mellin-Bernstein spaces. A notion of Hardy spaces in the Mellin setting is also given along with applications to exponential sampling formulas of optical physics. [ABSTRACT FROM AUTHOR]

We describe the relationship between the growth conditions of monogenic extensions of square-integrable functionsfin terms of pointwise bounds or bounds on integral averages on the one hand, and the support of the Fourier transformor its annihilation by certain higher-dimensional analogues of the signum function on the other. We review known results involving a function’s monogenic extension and their classical Fourier transform. These results are extended to the Clifford-Fourier transform of Brackx, De Schepper and Sommen. The equivalence of the pointwise bounds and the bounds on the integral averages is observed as a consequence. [ABSTRACT FROM AUTHOR]

In this paper our aim is to establish the Paley–Wiener Theorem for the Weinstein Transform. Furthermore, some applications are presents. In particular some properties for the generalized translation operator associated with the Weinstein operator are proved, an integral representation and a series representation for a function in the Paley–Wiener classes are investigated. [ABSTRACT FROM AUTHOR]

In this article, we present a survey of some new results obtained in [2,8]. First, we give a geometric Paley–Wiener theorem for the Dunkl transform in the crystallographic case. Next we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension. We also investigate the Lp → Lq boundedness properties of the Riesz potentials Iακ and the related fractional maximal function Mκ,α associated to the Dunkl transform. Finally we prove the Lp-boundedness, (1

In this paper we establish a version of the Paley–Wiener theorem of Fourier analysis in the frame of Mellin transforms. We provide two different proofs, one involving complex analysis arguments, namely the Riemann surface of the logarithm and Cauchy theorems, and the other one employing a Bernstein inequality here derived for Mellin derivatives. [ABSTRACT FROM AUTHOR]

We find necessary and sufficient conditions for a non-negative Hankel quadratic form to be closable. We also describe the domain of the corresponding closed form. This allows us to define unbounded non-negative Hankel operators under minimal assumptions on their matrix elements. The results obtained supplement the classical Widom condition for a Hankel operator to be bounded. [ABSTRACT FROM AUTHOR]

We study the propagation speed property for a two-component Degasperis–Procesi system proposed by M. Popowicz. First, we rederive the system from the Euler equation with constant vorticity in shallow water regime. Then, we investigate the propagation behavior of compactly supported solutions, namely whether solutions which are initially compactly supported will retain this property through their lifespan. Finally, we give an exponential decay structure result on the first component function to the system. [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 give a Liouville theorem for entire solutions and Laurent series expansions for solutions with isolated singularities of the heat equation. [ABSTRACT FROM AUTHOR]