Andreas Seeger, Stephen Wainger
We prove sharp L^p-L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature assumption.
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Natural geometric Fourier transforms and the associated fractional Laplacian
本研究针对$\u211d^{n}$上的任意一般几何结构，提出一对相容的几何傅里叶变换，分析其性质与经典傅里叶变换的相容性，并证明正定结构下该变换可诱导性质与经典分数阶拉普拉斯算子相似的几何分数阶拉普拉斯算子。
This paper gives a short survey of some basic results related to estimates of fractional integrals and Fourier transforms. It is closely adjoint to our previous survey papers \cite{K1998} and \cite{K2007}. The main methods used in the paper are based on nonincreasing rearrangements. We give alternative proofs of some results. We observe also that the paper represents the mini-course given by the author at Barcelona University in October, 2014.
We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function maps $L^{p}$ into a first order Sobolev space in dimensions $n \geq 5$.
In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.
The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions $e^{inωx}$ are replaced by the Mittag-Leffler functions $E_α\left (i (nωx)^α\right) ,$ over the interval $[0, M_α/ ω]$ where $0< ω<\infty $ and $M_α$ is the period of the function $E_α\left( ix^α\right),$ i.e., $E_α\left( ix^α\right)=E_α\left( i(x+M_α)^α\right).$
Maximal operators given by Fourier multipliers with dilation of fractional dimensions
本文研究了分数维膨胀下极大 Fourier 乘子算子的 $L^p$ 界，提出了基于膨胀集维数的准则，确保对任意 $p$ 均有 $L^p$ 界，并涵盖 Mikhlin 型、有限衰减与慢衰减乘子。
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as such paves a way for application to many other areas of analysis involving distributions. This includes (but is not limited to): the fractional Fourier series (i.e. an orthonormal basis for fractional derivatives), the fractional derivative of Fourier transforms, and fundamental solutions to differential equations such as the wave equation. This paper observes …
In this paper, we introduce two new non-singular kernel fractional derivatives and present a class of other fractional derivatives derived from the new formulations. We present some important results of uniformly convergent sequences of continuous functions, in particular the Comparison's principle, and others that allow, the study of the limitation of fractional nonlinear differential equations.
We study the type set of singular measures of fractional type on the Heisenbrg group.
We present sufficient conditions for the existence of positive solutions for a class of fractional singular boundary value problems in presence of Caputo fractional derivative. Further, the nonlinearity involved has singularity with respect to independent variable as well as with respect to dependent variable.
Via chirp functions from fractional Fourier transforms, the authors introduce fractional Riesz potentials related to chirp functions, establish their relations with fractional Fourier transforms, fractional Laplace operators related to chirp functions, and fractional Riesz transforms related to chirp functions, and obtain their boundedness on rotation invariant spaces related to chirp functions. Finally, the authors give the numerical image simulation of fractional Riesz potentials related to chirp functions and their applications in image processing. The main novelty of this article is to pro…
In this work, the power series solutions are given around a regular-singular point, in the case of variable coefficients for homogeneous sequential linear conformable fractional differential equations of order 2α.
In this first part of our project, we prove a classical Hardy-Littlewood-Sobolev result for a new family of fractional integral operators whose kernel has singularity appeared on the light cone in R^n+1.
In this work, we study the regularity of positive solutions for nonlinear fractional differential equation with a singular weight. We define the new Banach space and use this space to show the regularity. We also give an example with a singular weight which may not be in $L^1.$
Generalizations of the fractional Fourier transform and their analytic properties
该研究聚焦分数阶傅里叶变换的单参数二次相位积分推广，刻画了这类变换的单参数群，给出了参数在L2、逐点及几乎处处意义下连续依赖的充要条件。
In this paper, we will study the boundedness properties of multilinear Calderon--Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights.
In this paper, we study a family of fractional integral operators whose kernel carrying a critical index has singularity on the light-cone in R^n+1.
In this paper we study the local regularity of fractional integrals of Fourier series using several definitions of the Hölder exponent. We especially consider series coming from fractional integrals of modular forms. Our results show that in general cusp forms give rise to pure fractals (as opposed to multifractals). We include explicit examples and computer plots.
We study a family of convolution operators whose kernels have a singularity on the unit sphere. As a result, we prove the regarding L^p-L^q Sobolev inequalities.
In this paper we study linear and nonlinear fractional differential equations involving the Caputo fractional derivative with Mittag-Leffler non-singular kernel of order $0<α<1.$ We first obtain a new estimate of the fractional derivative of a function at its extreme points and derive a necessary condition for the existence of a solution to the linear fractional equation. The obtained sufficient condition determine the initial condition of the associated fractional initial value problem. We then derive comparison principles to the linear fractional equations. We apply these principles to obtai…
We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose "higher-order Fourier dimension" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of Additive Combinatorics and uniformity norms) extension of the Fourier dimension of Geometric Measure Theory, and can be understood as asking that the uniformity norm of a measure, restricted to a given scale, decay as the scale increases. We further obtain quantitative information about the size and $L^p$ regularity of the set of common distances of the art…
On $L^p$ extremals for Fourier extension estimate to fractional surface
本文研究分数阶曲面对应的傅里叶延拓算子，证明了α≥2时该延拓不等式在p∈[1,2]存在L^p极值函数，且p∈(1,2]时极值序列在对称下准紧。
In this paper, we investigate the uniqueness of the phase retrieval problem for the fractional Fourier transform (FrFT) of variable order. This problem occurs naturally in optics and quantum physics. More precisely, we show that if $u$ and $v$ are such that fractional Fourier transforms of order $α$ have same modulus $|F_αu|=|F_αv|$ for some set $τ$ of $α$'s, then $v$ is equal to $u$ up to a constant phase factor. The set $τ$ depends on some extra assumptions either on $u$ or on both $u$ and $v$. Cases considered here are $u$, $v$ of compact support, pulse trains, Hermite functions or linear c…
In analogy with the Poisson summation formula, we identify when the fractional Fourier transform, applied to a Dirac comb in dimension one, gives a discretely supported measure. We describe the resulting series of complex multiples of delta functions, and through either the metaplectic representation of $SL(2,\mathbb{Z})$ or the Bargmann transform, we see that the the identification of these measures is equivalent to the functional equation for the Jacobi theta functions. In tracing the values of the antiderivative in certain small-angle limits, we observe Euler spirals, and on a smaller scale…
This paper, that will appear in the Notices of the AMS, begins with a brief historical account of the beginnings of fractional calculus and the crucial roles played by Leibniz and Fourier. Fourier's definition of fractional derivative is introduced and unpacked by using the modern technique known as the method of semigroups. Recent advances on the theory of fractional derivatives are presented. Furthermore, we address some questions that have been raised by some in the scientific community. Finally, we present three different applications: population growth with memory, viscoleastic materials …
Fourier dimension of the graph of fractional Brownian motion with $H \ge 1/2$
本文证明了当Hurst指数大于1/2时，分数布朗运动图的傅里叶维度几乎必然为1，扩展了Fraser和Sahlsten(2018)关于布朗运动的结果，并验证了Fraser, Orponen和Sahlsten(2014)的部分猜想。文章引入了组合积分分部公式来计算图测度的傅里叶变换的矩，并利用该公式、Faà di Bruno公式和分数布朗运动的强局部非确定性证明了主要结果。此外，还证明了对称α-稳定过程的图在α∈[1,2]时几乎必然具有傅里叶维度1，当α=1时是Salem集。
By using quasi--derivatives, we develop a Fourier method for studying the spectral properties of one dimensional Schrödinger operators with periodic singular potentials.
We prove that, regardless of the choice of the angles $θ_1,θ_2,θ_3$, three fractional Fourier transforms $F_{θ_1}$, $F_{θ_2}$ and $F_{θ_3}$ do not solve the phase retrieval problem. That is, there do not exist three angles $θ_1$, $θ_2$, $θ_3$ such that any signal $ψ\in L^2(R)$ could be determined up to a constant phase by knowing only the three intensities $|F_{θ_1}ψ|^2$, $|F_{θ_2}ψ|^2$ and $|F_{θ_3}ψ|^2$. This provides a negative argument against a recent speculation by P. Jaming, who stated that three suitably chosen fractional Fourier transforms are good candidates for phase retrieval in in…
In this article we focus our attention on the relation between a singular linear discrete time system and a singular linear system of fractional nabla difference equations whose coefficients are square constant matrices. By using matrix pencil theory, first we give necessary and sufficient condition to obtain a unique solution for the continuous time model. After by assuming that the input vector changes only at equally space sampling instants, we shall derive the corresponding discrete time state equation which yield the values of the solutions of the continuous time model which will connect …
Restriction of Fractional Derivatives of the Fourier Transform
本文将傅里叶变换的分数阶导数限制到非零高斯曲率曲面上，在给定曲面限制正则性条件下建立其负阶 Sobolev 空间归属，推广了 GoldStol 的定理。
Sharp Fourier extension on fractional surfaces
本文研究分数曲面上的傅里叶延拓算子，利用缺失质量方法与双线性限制理论刻画了α-Strichartz不等式极值序列的预紧性，并在二维情形证明了α∈[2,α0)（α0>5）时极值的存在性。
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。
Definitions of fractional derivative of order $α$ ($0 < α\leq 1$) using non-singular kernels have been recently proposed. In this note we show that these definitions cannot be useful in modelling problems with a initial value condition (like, for example, the fractional diffusion equation) because the solutions obtained for these equations do not satisfy the initial condition (except for the integer case $α= 1$). In order to satisfy an arbitrary initial condition the definitions of fractional derivative must necessarily involve a singular kernel.
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