Luca Brandolini, Giacomo Gigante, Allan Greenleaf, Alexander Iosevich, Andreas Seeger, Giancarlo Travaglini
We consider Fourier transforms of densities supported on curves in R^d. We obtain sharp lower and close to sharp upper bounds for the L^q decay rates.
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Sharp Fourier decay estimates for measures supported on the well-approximable numbers
本文在良逼近数上构造了傅里叶变换近乎最优衰减的测度，从而对Kaufman的先前构造实现了对数级别的改进。
This paper builds upon two key principles behind the Bourgain-Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theorem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah-Logvinenko-Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on…
Fourier decay of measures supported on sets of numbers with consecutive partial quotients belonging to a given set
该研究探讨满足特定连分数部分商条件的无理数集上的测度，证明温和假设下这类集合恒存在Fourier变换衰减到零的支撑测度，丰富了连分数相关测度的Fourier分析结果。
Are MSF wavelets minimally supported?
本文探讨了MSF小波的支撑问题，证明了其傅里叶变换必须包含一个可能非可测的波let集，并提出了两个新问题。
We establish a broad notion of admissible tilings of frequency space which admit associated wave packet frames with elements which are smooth and compactly supported. The framework is designed to allow for tile geometries which are minimally constrained by the need to accommodate Schwartz tails on the Fourier side and goes beyond the usual scale of geometries ranging from Gabor to wavelet-type decompositions. The approach builds on techniques of Hernández, Labate and Weiss and Labate, Weiss, and Wilson as well as a classical result of Ingham characterizing the best-possible Fourier decay for f…
The existence of a smooth, nonnegative, compactly supported function with monotone (on the half-line) Fourier transform satisfying two-sided decay bounds is demonstrated.
Fourier decay for curved Frostman measures
本文研究非零曲率曲线上Frostman测度的解耦，并结合Furstenberg集合下界重新证明了Orponen关于抛物线的结果。
A measure with small support and p-summable Fourier transform
该研究构造了支撑于零2d/p-豪斯多夫测度集上的概率测度μ，证明其傅里叶变换满足Lₚ(ℝᵈ)可和性，丰富了调和分析中测度与傅里叶变换性质的关联研究。
We prove that for radial Fourier multipliers $m: \mathbb{R}^3\to\mathbb{C}$ supported compactly away from the origin, $T_m$ is restricted strong type (p,p) if $K=\hat{m}$ is in $L^p(\mathbb{R}^3)$, in the range $1<p<\frac{13}{12}$. We also prove an $L^p$ characterization for radial Fourier multipliers in four dimensions; namely, for radial Fourier multipliers $m: \mathbb{R}^4\to\mathbb{C}$ supported compactly away from the origin, $T_m$ is bounded on $L^p(\mathbb{R}^4)$ if and only if $K=\hat{m}$ is in $L^p(\mathbb{R}^4)$, in the range $1<p<\frac{36}{29}$. Our method of proof relies on a geome…
We exhibit three classes of compactly supported functions which provide reproducing kernels for the Sobolev spaces $H^δ(\R^d)$ of arbitrary order $\,δ>d/2.\,$ Our method of construction is based on a new class of oscillatory integral transforms that incorporate radial Fourier transforms and Hankel transforms.
The Fourier Ratio: Uncertainty, Restriction, and Approximation for Compactly Supported Measures
本文提出傅里叶比概念，用于研究紧支撑测度的不确定性、限制与近似问题，揭示了信号恢复与分形测度之间的差异。
The quaternion Fourier transform (QFT) satisfies some uncertainty principles similar to the Euclidean Fourier transform. In this paper, we establish Miyachi's theorem for this transform.
The two-sided quaternion Fourier transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization of Beurling's theorem, Hardy, Cowling-Price and Gelfand-Shilov theorems, is obtained for the two-sided quaternion Fourier transform.
We consider quasiradial Fourier multipliers, i.e. multipliers of the form $m(a(ξ))$ for a class of distance functions $a$. We give a necessary and sufficient condition for the multiplier transformations to be bounded on $L^p$ for a certain range of $p$. In addition, when $m$ is compactly supported in $(0,\infty)$, we give a similar result for associated maximal operators.
In this paper we generalize the continuous quaternion windowed Fourier transform called the multivariate two sided continuous quaternion windowed Fourier transform. Using the two sided quaternion Fourier transform we derive several important properties.
Certain relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. The widest subspaces of the space of functions of bounded variation are indicated in which the cosine and sine Fourier transforms are integrable.
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.
Fourier decay and endpoint estimates for lacunary maximal operators
该研究针对与紧支测度相关的缺项极大算子，构造了具有多项式傅里叶衰减的测度，证明其关联算子不满足弱型(1,1)，否定了相关公开问题。
In this paper, we estabish an analogue of Hardy's theorem and Miyachi's theorem for the Clifford-Fourier transform.
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.
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.
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.
We establish estimates for restrictions to certain curves in R^2 of the Fourier transforms of some fractal measures.
The Weyl transform of a compactly supported distribution
该研究刻画了$\mathbb{R}^{2n}$上紧支撑分布的Weyl变换的性质，建立了其$p$次幂可迹性与傅里叶变换$p$次幂可积性、紧算子性与傅里叶变换无穷远消失性的等价关系。
Uncertainty Principle for distributions with Fourier transform in $L_{p,q}(\mathbb{R}^d)$
本文研究了当傅里叶变换属于 $L_{p,q}(\mathbb{R}^d)$ 时，分布的不确定性原理的精确形式，证明了在临界情况下，非零函数的存在性与参数 β 的关系。
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.
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.
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.
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.
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.
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.
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.
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.
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 …
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.
We give new formulas for finding a compactly supported function $v$ on $\mathbb{R}^d$, $d\geq 1$, from its Fourier transform $\mathcal{F} v$ given within the ball $B_r$. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions (PSWFs). In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. Related results on stability and convergence rates are also given.
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.
One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in $\mathbb{R}^n$. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let $f\in C_c^{\infty}(\mathbb{R}^n)$ and $dσ$ be the surface measure on the sphere $S^{n-1}\subset\mathbb{R}^n$. Then $$|\widehat{fdσ}(ξ)|\leq\ C\ (1+|ξ|)^{-\frac{n-1}{2}}.$$ It follows that $\widehat{fdσ}\in L^p(\mathbb{R}^n)$ for all $p>\frac{2n}{n-1}$. This result can be…
We prove several results related to the theorem of Logvinenko and Sereda on determining sets for functions with Fourier transforms supported in an interval. We obtain a polynomial instead of exponential bound in this theorem, and we extend it to the case of functions with Fourier transforms supported in the union of a bounded number of intervals. The same results hold in all dimensions.
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。
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.
We prove uniform estimates for the decay rate of the Fourier transform of measures supported on real-analytic hypersurfaces in R^3. If the surface contains the origin and is oriented such that its normal at the origin is in the direction of the z-axis and if dS denotes the surface measure for this surface, then the measures under consideration are of the form K(x,y)g(z) dS where K(x,y)g(z) is supported near the origin and both K(x,y) and g(z) are allowed to have singularities. The estimates here generalize the previously known sharp uniform estimates for when K(x,y)g(z) is smooth. The methods …
We extend the result of K. Karlander [Math. Scand. 80 (1997)] regarding finite dimensionality of spaces of absolutely convergent Fourier transforms.
We solve the Cauchy problem for the $n$-dimensional wave equation using elementary properties of the Fourier transform.
We prove a Hölder-logarithmic stability estimate for the problem of finding a sufficiently regular compactly supported function $v$ on $\mathbb{R}^d$ from its Fourier transform $\mathcal{F} v$ given on $[-r,r]^d$. This estimate relies on a Hölder stable continuation of $\mathcal{F}v$ from $[-r,r]^d$ to a larger domain. The related reconstruction procedures are based on truncated series of Chebyshev polynomials. We also give an explicit example showing optimality of our stability estimates.
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.
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.
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.
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。
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 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.
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。
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.
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。
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…
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…
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 dimension of Mandelbrot cascades on planar curves
研究平面C²曲线上的多标度Mandelbrot级数，证明其傅里叶维度达到最大值，等于测度的下点wise维度下确界。
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.
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