Restricting Fourier transforms of measures to curves in R^2

A generalization of Laplace and Fourier transforms

In this note we propose a generalization of the Laplace and Fourier transforms which we call symmetric Laplace transform. It combines both the advantages of the Fourier and Laplace transforms. We give the definition of this generalization, some examples and basic properties. We also give the form of its inverse by using the theory of the Fourier transform. Finally, we apply the symmetric Laplace transform to a parabolic problem and to an ordinary differential equation.

Uncertainty principles for the Clifford-Fourier transform

In this paper, we estabish an analogue of Hardy's theorem and Miyachi's theorem for the Clifford-Fourier transform.

Growth and integrability of Fourier transforms on Euclidean space

A fundamental theme in classical Fourier analysis relates smoothness properties of functions to the growth and/or integrability of their Fourier transform. By using a suitable class of $L^{p}-$multipliers, a rather general inequality controlling the size of Fourier transforms for large and small argument is proved. As consequences, quantitative Riemann-Lebesgue estimates are obtained and an integrability result for the Fourier transform is developed extending ideas used by Titchmarsh in the one dimensional setting.

On the $q$-Bessel Fourier transform

In this work, we are interested by the $q$-Bessel Fourier transform with a new approach. Many important results of this $q$-integral transform are proved with a new constructive demonstrations and we establish in particular the associated $q$-Fourier-Neumen expansion which involves the $q$-little Jacobi polynomials.

The two-sided Gabor quaternion Fourier transform and some uncertainty principles

In this paper, we define a new transform called the Gabor quaternionic Fourier transform (GQFT), which generalizes the classical windowed Fourier transform to quaternion valued-signals, we give several important properties such as the Plancherel formula and inversion formula. Finally, we establish the Heisenberg uncertainty principles for the GQFT.

A Characterisation of the Euclidean Fourier transform on the Schwartz space

We obtain a characterisation of the Fourier transform on the space of Schwartz class functions on $\mathbb{R}^n.$ The result states that any appropriately additive bijection of the Schwartz space onto itself, which interchanges convolution and pointwise products is essentially the Fourier transform.

Beurling's theorem for the Clifford-Fourier transform

We give a generalization of Beurling's theorem for the Clifford-Fourier transform. Then, analogues of Hardy, Cowling-Price and Gelfand-Shilov theorems are obtained in Clifford analysis.

On restriction of the Fourier transform to hypersufaces

It is considered Fourier transform of convex analytic hypersufaces on $R^{4} $. We prove that the Fourier restriction operator associated to convex analytic hypersufaces is \textit{$(L_{p}, L_{2})$} bounded whenever $1\le p\le \frac{2h+2}{h+2}$. The result is sharp.

Henstock--Kurzweil Fourier transforms

The Fourier transform is considered as a Henstock--Kurzweil integral. Sufficient conditions are given for the existence of the Fourier transform and necessary and sufficient conditions are given for it to be continuous. The Riemann--Lebesgue lemma fails: Henstock--Kurzweil Fourier transforms can have arbitrarily large point-wise growth. Convolution and inversion theorems are established. An appendix gives sufficient conditions for interchanging repeated Henstock--Kurzweil integrals and gives an estimate on the integral of a product.

The Fourier Transform of the First Derivative of the Generalized Logistic Growth Curve

In this work, we obtain the Fourier transform of the first derivative of the generalized logistic growth curve in terms of Gamma functions and we discuss special cases.

On an apparently bilinear inequality for the Fourier transform

A bilinear inequality of Geba, Greenleaf, Iosevich, Palsson, and Sawyer for the Fourier transform is shown to be equivalent to a simpler linear inequality, and the range of exponents is extended. Related mixed-norm inequalities are discussed.

Some trigonometric integrals and the Fourier transform of a spherically symmetric exponential function

We calculate the Fourier transform of a spherically symmetric exponential function. Our evaluation is much simpler than the known one. We use the polar coordinates and reduce the Fourier transform to the integral of a rational function of trigonometric functions. Its evaluation turns out to be much easier than expected because of homogeneity and a hidden symmetry. Relationship with a Fourier integral representation formula for harmonic functions is explained.

The class of Clifford-Fourier transforms

Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called Clifford-Fourier transform (see [F. Brackx et al., The Clifford-Fourier transform. J. Fourier Anal. Appl. 11 (2005), 669--681]). The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the …

Sharp rate of average decay of the Fourier transform of a bounded set

We prove that the spherical mean of the Fourier transform of the characteristic function of a bounded convex set (without any additional assumptions) or a bounded set with a C^{3/2} boundary decays at infinity at the same rate as the Fourier transform of the characteristic function of the ball.

Local uncertainty principles for the two-sided Gabor quaternion Fourier transform

By the important applications of Gabor transform in time-frequency analysis and signal analysis, in this paper, we consider the Gabor quaternion Fourier transform (GQFT), and we prove of it a version Benedicks-type uncertainty principle for GQFT and some local concentration uncertainty principles.

Bump Functions With Monotone Fourier Transforms Satisfying Decay Bounds

The existence of a smooth, nonnegative, compactly supported function with monotone (on the half-line) Fourier transform satisfying two-sided decay bounds is demonstrated.

The kernel of the generalized Clifford-Fourier transform and its generating function

In this paper, we study the generalized Clifford-Fourier transform introduced in [6] using the Laplace transform technique. We give explicit expressions in the even dimensional case, we obtain polynomial bounds for the kernel functions and establish a generating function.

The Fourier transform of multiradial functions

We obtain an exact formula for the Fourier transform of multiradial functions, i.e., functions of the form $Φ(x)=ϕ(|x_1|, \dots, |x_m|)$, $x_i\in \mathbf R^{n_i}$, in terms of the Fourier transform of the function $ϕ$ on $\mathbf R^{r_1}\times \cdots \times \mathbf R^{r_m}$, where $r_i$ is either 1 or 2.

A Calderón-Zygmund estimate with applications to generalized Radon transforms and Fourier integral operators

We prove a Calderón-Zygmund type estimate which can be applied to sharpen known regularity results on spherical means, Fourier integral operators and generalized Radon transforms.

The uncertainty principle for the two-sided quaternion Fourier transform

In this paper, we provide the Heisenberg's inequality and the Hardy's theorem for the two-sided quaternion Fourier transform.

On the half-Hartley transform, its iteration and composition with Fourier transforms

Employing the generalized Parseval equality for the Mellin transform and elementary trigonometric formulas, the iterated Hartley transform on the nonnegative half-axis (the iterated half-Hartley transform) is investigated in L_2. Mapping and inversion properties are discussed, its relationship with the iterated Stieltjes transform is established. Various compositions with the Fourier cosine and sine transforms are obtained. The results are applied to the uniqueness and universality of the closed form solutions for certain new singular integral and integro-functional equations.

Fourier transforms on an amalgam type space

We introduce an amalgam type space, a subspace of $L^1(\mathbb R_+).$ Integrability results for the Fourier transform of a function with the derivative from such an amalgam space are proved. As an application we obtain estimates for the integrability of trigonometric series.

On Fourier re-expansions

We study an extension to Fourier transforms of the old problem on absolute convergence of the re-expansion in the sine (cosine) Fourier series of an absolutely convergent cosine (sine) Fourier series. The results are obtained by revealing certain relations between the Fourier transforms and their Hilbert transforms.

Fourier型积分变换及其衍生变换的数学综述：窗口Fourier变换、小波变换与Stockwell变换

A mathematical survey on Fourier type integral transform and their offshoots: windowed Fourier transform, wavelet transform and Stockwell transform

本文综述了Fourier型积分变换（如FrFT、LCT、QPFT）及其衍生变换的卷积性质、不确定性原理和多分辨率分析，并详细给出了四元数情形下的对应不等式。

Uncertainty Principles for the Fourier and the Short-Time Fourier Transforms

The aim of this paper is to establish a few uncertainty principles for the Fourier and the short-time Fourier transforms. Also, we discuss an analogue of Donoho--Stark uncertainty principle and provide some estimates for the size of the essential support of the short-time Fourier transform.

On the Finite Dimensionality of Spaces of Absolutely Convergent Fourier Transforms

We extend the result of K. Karlander [Math. Scand. 80 (1997)] regarding finite dimensionality of spaces of absolutely convergent Fourier transforms.

An analog of the Fourier transformation for a q-harmonic oscillator

A q-version of the Fourier transformation and some of its properties are discussed.

The Fourier transform and the wave equation

We solve the Cauchy problem for the $n$-dimensional wave equation using elementary properties of the Fourier transform.

Generalized Fourier transform on Chébli-Triméche hypergroups

In this paper, we prove the Hardy-Littlewood inequality for the generalized Fourier transform on Chébli-Triméche hypergroups and we study in the particular case of the Jacobi hypergroup the integrability of this transform on Besov-type spaces.

Restriction theorem for the Fourier-Dunkl transform I: Cone surface

In this article, we define the Fourier-Dunkl transform, which generalizes the Fourier transform. We prove Strichartz's restriction theorem for the Fourier-Dunkl transform for a cone-hyper-surface and its generalisation to the family of orthonormal functions. As an application of this restriction theorem, we derive the Strichartz inequality associated with the square root of Dunkl Laplacian for the family of orthonormal functions.

Cosine products, Fourier transforms, and random sums

We investigate several infinite product of cosines and find the closed form using the Fourier transform. The answers provide limiting distributions for some elementary probability experiments.

Schatten类算子与相空间上函数的傅里叶限制

Fourier restriction for Schatten class operators and functions on phase space

该研究提出Schatten类算子的Fourier-Wigner傅里叶限制变体，证明其等价于相空间函数的辛傅里叶变换限制，推导了量子傅里叶延拓算子的Schatten类结果，肯定证明了Mishra与Vemuri关于测度Weyl变换的猜想。

A Characterisation of the Fourier transform on the Schwartz-Bruhat space of locally compact Abelian groups

We obtain a characterisation of the Fourier transform on the space of Schwartz-Bruhat functions on locally compact Abelian groups. The result states that any appropriately additive bijection of the Schwartz space onto itself, which interchanges convolution and pointwise products is essentially the Fourier transform. The proof of this result is very similar to that obtained by the author recently for the Euclidean Fourier transform.

The finite Fourier transform of classical polynomials

The finite Fourier transform of a family of orthogonal polynomials $A_{n}(x)$, is the usual transform of the polynomial extended by $0$ outside their natural domain. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families.

从离散到连续非线性傅里叶变换的收敛性

Convergence from the discrete to the continuous non-linear Fourier transform

该研究借助复函数论与谱问题的关联新视角，探究离散到连续非线性傅里叶变换的收敛性，指出微分算子谱问题视角下的相关结果与离散情形的收敛问题具有相似性。

赋值上的傅里叶变换即是傅里叶变换

The Fourier transform on valuations is the Fourier transform

本文通过函数上的傅里叶变换直接描述了平移不变光滑赋值空间上的同构，从而为 Alesker-Fourier 变换的主要性质提供了简单证明，并首次证明了此前仅被猜想的性质。

On the Clifford short-time Fourier transform and its properties

In this paper we investigate how the short-time Fourier transform can be extended in a Clifford setting. We prove some of the main properties of the Clifford short-time Fourier transform such as the orthogonality relation, the reconstruction property and the reproducing kernel formula. Moreover, we show the effects of modulating and translating the signal and the window function, respectively. Finally, we demonstrate the Lieb's uncertainty principle for the Clifford short-time Fourier transform.

(Anti)symmetric multivariate exponential functions and corresponding Fourier transforms

We define and study symmetrized and antisymmetrized multivariate exponential functions. They are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable. These functions are eigenfunctions of the Laplace operator on corresponding fundamental domains satisfying certain boundary conditions. To symmetric and antisymmetric multivariate exponential functions there correspond Fourier transforms. There are three types of such Fourier transforms: expansions into corresponding Fourier series, integral Fourier transforms, and multivariate finite F…

Analytic Fourier-Feynman transforms via the series approximation

In this paper, we first establish an evaluation formula to calculate Wiener integrals of functionals on Wiener space. We then apply our evaluation formula to carry out very easily calculating for the analytic Fourier-Feynman transform of the functionals. Some examples are furnished to illustrate the usefulness of the evaluation formula. Finally, using the evaluation formula, we establish the series approximation for the analytic Fourier-Feynman transform.

Fourier transform and related integral transforms in superspace

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…

Clifford algebras, Fourier transforms and quantum mechanics

In this review, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2). In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford-Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these tr…

Fourier Transformable Measures with Meyer set support and their lift to the cut and project scheme

In this paper, we prove that given a cut-and-project scheme $(G, H, \mathcal{L})$ and a compact window $W \subseteq H$, the natural projection gives a bijection between the Fourier transformable measures on $G \times H$ supported inside the strip $\cL \cap (G \times W)$ and the Fourier transformable measures supported inside $\oplam(W)$, and relate their Fourier transforms. We use this formula to relate the Fourier transforms of the measures, and explain how one can use this relation to re-derive some known results about Fourier analysis of measures with Meyer set support.

Fourier Transform of some irrational integrands

Fourier sine transforms containing irrational integrands are presented. Explicit closed form expressions are shown to be related to Lommel functions and in special cases to the Fresnel integrals. Such integrals arise in the semi-classical theory of the Chemical Potential of the atomic elements.

Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions $f$ on $\R^d$ which may be written as $P(x)\exp (Ax,x)$, with $A$ a real symmetric definite positive matrix, are characterized by integrability conditions on the product $f(x)\hat{f}(y)$. We also give the best constant in uncertainty principles of Gelf'and Shilov type. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with …

On Fourier transforms of radial functions and distributions

We find a formula that relates the Fourier transform of a radial function on $\mathbf{R}^n$ with the Fourier transform of the same function defined on $\mathbf{R}^{n+2}$. This formula enables one to explicitly calculate the Fourier transform of any radial function $f(r)$ in any dimension, provided one knows the Fourier transform of the one-dimensional function $t\to f(|t|)$ and the two-dimensional function $(x_1,x_2)\to f(|(x_1,x_2)|)$. We prove analogous results for radial tempered distributions.