Paley-Wiener Theorem Research Articles

Paley-Wiener Theorems for the Two-Sided Quaternion Fourier Transform

In this paper we establish new real Paley-Wiener theorems for the two-sided quaternion Fourier transform. Next, we prove the Roe’s theorem in the context of the quaternion-valued functions. Finally we study the tempered distributions with spectral gaps.

Paley–Wiener theorem for functions in Lp(ℝn)

ABSTRACTIn this paper we examine necessary and sufficient conditions on the sequences of norm of the derivatives of functions in such that their spectrum (the support of their Fourier transform) is contained in a given compact set in .

On the Paley–Wiener theorem in the Mellin transform setting

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.

Nonexistence results for relaxation spectra with compact support

In this paper we consider the problem of recovering the (transformed) relaxation spectrum h from the (transformed) loss modulus g by inverting the integral equation , where denotes convolution, using Fourier transforms. We are particularly interested in establishing properties of h, having assumed that the Fourier transform of g has entire extension to the complex plane. In the setting of square integrable functions, we demonstrate that the Paley–Wiener theorem cannot be used to show the existence of non-trivial relaxation spectra with compact support. We prove a stronger result for tempered distributions: there are no non-trivial relaxation spectra with compact support. Finally we establish necessary and sufficient conditions for the relaxation spectrum h to be strictly positive definite.

The $\mathfrak {v}$-radial Paley–Wiener theorem for the Helgason Fourier transform on Damek–Ricci spaces

The $\mathfrak {v}$-radial Paley–Wiener theorem for the Helgason Fourier transform on Damek–Ricci spaces

Paley-Wiener-type theorem for polynomial ultradifferentiable functions

The image of the space of ultradifferentiable functions with compact supports under Fourier-Laplace transformation is described. An analogue of Paley-Wiener theorem for polynomial ultradifferentiable functions is proved.

On the range of the Opdam–Cherednik transform and Roe’s theorem in the Cherednik setting

In this paper, we establish new real Paley–Wiener theorems for the Opdam–Cherednik transform on \(\mathbb {R}^{d}\). Next, we study the generalized tempered distributions with spectral gaps. Finally, the Roe’s theorem is established in the context of the Cherednik operators.

Well‐posedness and unique continuation property for the generalized Ostrovsky equation with low regularity

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Ostrovsky equation as follows: urn:x-wiley:mma:media:mma3709:mma3709-math-0385 with initial data in the modified Sobolev space . Using Fourier restriction norm method, Tao's [k,Z]−multiplier method and the contraction mapping principle, we show that the local well‐posedness is established for the initial data with (k = 2) and is established for the initial data with (k = 3). Using these results and conservation laws, we also prove that the IVP is globally well‐posed for the initial data with s = 0(k = 2,3). Finally, using complex variables technique and Paley–Wiener theorem, we prove the unique continuation property for the IVP benefited from the ideas of Zhang ZY. et al., On the unique continuation property for the modified Kawahara equation, Advances in Mathematics (China), http://advmath.pku.edu.cn/CN/10.11845/sxjz.2014078b. Copyright © 2015 John Wiley & Sons, Ltd.

The cocenter of the graded affine Hecke algebra and the density theorem

We determine a basis of the (twisted) cocenter of graded affine Hecke algebras with arbitrary parameters. In this setting, we prove that the kernel of the (twisted) trace map is the commutator subspace (the Density theorem) and that the image is the space of good forms (the trace Paley–Wiener theorem).

Paley–Wiener Theorems of Generalized Fourier Transform Associated with a Cherednik Type Operator on the Real Line

We consider a singular differential-difference operator \(\Lambda \) on the real line which generalizes the Cherednik operator associated with the reflection group \(\mathbb {Z}_2\) on \(\mathbb {R}\). We establish the Paley–Wiener theorems for the generalized Fourier transform on \(\mathbb {R}\) tied to \(\Lambda \).

Boundary conditions, semigroups, quantum jumps, and the quantum arrow of time

Experiments on quantum systems are usually divided into preparation of states and the registration of observables. Using the traditional mathematical methods (the Hilbert space and Schwartz space of distribution theory), it is not possible to distinguish mathematically between observables and states. The Hilbert space as well as Schwartz space boundary conditions for the dynamical equations lead by mathematic theorems (Stone-von Neumann) to unitary group with —∞ < t < ∞. But in the experimental set-up, one clearly distinguishes between the preparation of a state and the registration of an observable in that state. Furthermore, a state must be prepared first before an observable can be measured in this state (causality). This suggests time asymmetric boundary conditions for the dynamical equations of quantum theory. Such boundary conditions have been provided by Hardy space in the Lax-Phillips theory for electromagnetic and acoustic scattering phenomena. The Paley-Wiener theorem for Hardy space then leads to semi-group and time asymmetry in quantum physics. It introduces a finite “beginning of time” t0 for a time asymmetric quantum theory, which have been observed as an ensemble of finite times t(i)0, the onset times of dark periods in the quantum jump experiments on a single ion.

Radiation fields for semilinear wave equations

We define the radiation fields of solutions to critical semilinear wave equations in R 3 \mathbb {R}^3 and use them to define the scattering operator. We also prove a support theorem for the radiation fields with radial initial data. This extends the well-known support theorem for the Radon transform to this setting and can also be interpreted as a Paley-Wiener theorem for the distorted nonlinear Fourier transform of radial functions.

Real Paley–Wiener theorems and Roe's theorem associated with the Opdam–Cherednik transform

We reconsider Roe's theorem associated with Jacobi–Cherednik operators. We first prove two new real Paley–Wiener theorems for the Opdam–Cherednik transform, characterizing those functions whose transform has support inside or outside intervals. As a corollary, we get a characterization of those functions whose transform has support at the endpoints, which we use to prove our version of Roe's theorem.

Segal–Bargmann transform and Paley–Wiener theorems on Heisenberg motion groups

AbstractWe consider the Heisenberg motion groups ℍ𝕄 = ℍ

Roe's theorem revisited

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.

On the mathematics underlying dispersion relations

The history of mathematical methods underlying the study of dispersion relations in physics is discussed. In particular, some misconceptions connected with a theorem known in the physics literature as Titchmarsh’s Theorem are addressed. It is pointed out that the aforementioned theorem is a compilation of two well-known theorems in mathematics, the Paley-Wiener theorem and the Marcel Riesz theorem.

On approximation properties of one trigonometric system

We obtain the criteria of completeness and minimality of a system. We also obtain an analog of the Paley-Wiener theorem related to this system.

Real Paley–Wiener theorems for Fourier series

Consider an integrable function f on the torus Tm, with Fourier coefficients f˜(n), n∈Zm. For functions with support in a neighbourhood of the origin, we describe the support by means of lp-growth properties of difference operators acting on the Fourier coefficients f˜(n). We then generalize the result to compact Lie groups, using a certain Weyl group invariant difference operator. We also consider l2-Bernstein inequalities for the difference operators.

The biradial Paley–Wiener theorem for the Helgason Fourier transform on Damek–Ricci spaces

We prove the Paley–Wiener theorem for the Helgason Fourier transform of smooth compactly supported biradial functions on a Damek–Ricci space S=NA.

Real Paley-Wiener Theorems for the Multivariable Bessel Transform

Using the harmonic analysis assoicated with the Bessel operator Lα on Ωn = (0,+∞). We establish real Paley-Wiener theorems and we give a necessary and sufficient condition on a function f in order to have a multivariable Bessel transform vanishing on neigberhood of the origin, on symmetric body and a on polynomial domain.