单位圆周上广义Fourier积分的收敛性

本文研究了复平面单位圆上的广义Fourier积分.利用经典的Fourier分析的结果和Carleson定理,以及复平面上解析函数在高阶导数下直角坐标和极坐标之间的关系,我们得到了前面定义的广义Fourier积分的一个收敛定理,从而推广了直线上经典Fourier积分的收敛结果.     

A generalisation of a theorem of Pati on the Nörlund summability of Fourier series

Pati generalised a theorem by Siddiqi on the harmonic summability of Fourier series, by replacing the special sequence by a more gererral sequence. Dikshit proved an analogue of it in the case of conjugate series of Fourier series and further he improved his theorem by introducing a functional factor. The problem which was unsolved as to the same improvement and generalisation can be done in case of Fourier series is tackled and proved by me in the paper attached herewith.     

Paley-Wiener and Boas Theorems for the Quaternion Fourier Transform

This paper establishes a real Paley-Wiener theorem to characterize the quaternion-valued functions whose quaternion Fourier transform has compact support by the partial derivative and also a Boas theorem to describe the quaternion Fourier transform of these functions that vanish on a neighborhood of the origin by an integral operator.     

On an αth-order fractional Radon transform and a wave type of equation

The Fourier slice theorem holds for the classical Radon transform. In this paper, we consider a fractional Radon transform for which a sort of Fourier slice theorem also holds, and then present an inversion formula. The fractional Radon transform is shown to be characterized by the multi-dimensional case of a wave type of equation in analogy to the classical Radon transform.     

有理Fourier级数在变差条件下的收敛性研究

本文把Fourier级数的一些经典结论推广到有理Fourier级数的情况下. 首先给出了有理Fourier级数和共轭有理Fourier级数在有界变差条件下的收敛速度估计. 利用此结论, 得到了类似于Fourier级数的Dirichlet-Jordan定理和W. H. Young定理. 最后, 证明了这两个定理在调和有界变差条件下也成立.     

Some results related to the Logvinenko-Sereda theorem

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.

     

Notes on Embedding Theorems for Local Hardy spaces

In this note we establish an embedding theorem (Theorem 2.4) for local Hardy spaces in the sense of GOLDBERG [G]. This result is a non-homogeneous version of the theorem of BAERNSTEIN and SAWYER (Theorem BS). Also applying this theorem we establish embedding theorem and Fourier embedding theorem (Theorem 4.2, Theorem 4.3 and Corollary 4.4) for local Hardy spaces.     

Ramanujan’s Master Theorem for the Hypergeometric Fourier Transform Associated with Root Systems

Ramanujan’s Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several definite integrals and power series, which explains why it is referred to as the “Master Theorem”. In this paper we prove an analogue of Ramanujan’s Master theorem for the hypergeometric Fourier transform associated with root systems. This theorem generalizes to arbitrary positive multiplicity functions the results previously proven by the same authors for the spherical Fourier transform on semisimple Riemannian symmetric spaces.     

Uncertainty Principles for the Generalized Fourier Transform Associated with Spherical Mean Operator

The aim of this paper is to establish an extension of qualitative and quantitative uncertainty principles for the Fourier transform connected with the spherical mean operator.     

Uncertainty principles for the weinstein transform

The Weinstein transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price theorem, Miyachi’s theorem, Beurling’s theorem, and Donoho-Stark’s uncertainty principle are obtained for the Weinstein transform.     

Fourier analysis of 2-point hermite interpolatory subdivision schemes

Two subdivision schemes with Hermite data on ℤ are studied. These schemes use 2 or 7 parameters respectively depending on whether Hermite data involve only first derivatives or include second derivatives. For a large region in the parameter space, the schemes are convergent in the space of Schwartz distributions. The Fourier transform of any interpolating function can be computed through products of matrices of order 2 or 3. The Fourier transform is related to a specific system of functional equations whose analytic solution is unique except for a multiplicative constant. The main arguments for these results come from Paley-Wiener-Schwartz theorem on the characterization of the Fourier transforms of distributions with compact support and a theorem of Artzrouni about convergent products of matrices.     

Solvability of a system of equations for the Fourier–Chebyshev coefficients when solving ordinary differential equations by the Chebyshev series method

A solvability theorem for a system of equations with respect to approximate values of Fourier–Chebyshev coefficients is formulated. This theorem is a theoretical justification for numerical solution of ordinary differential equations using Chebyshev series.     

A Converse to Kluvánek’s Theorem

I. Kluvánek extended the Whittaker-Kotel’nikov-Shannon (WKS) theorem to the abstract harmonic analysis setting. To do this, the ‘band limited’ condition on the spectrum of a continuous square-integrable function (analogue signal) required for classical WKS theorem is replaced by an ‘almost disjoint’ translates condition arising from the Fourier transform of the function vanishing almost everywhere outside a transversal of a compact quotient group. A converse of Kluvánek’s theorem is established, i.e., if the representation given by the abstract WKS theorem holds for a continuous square-integrable function with support of its Fourier transform essentially A, then A is a subset of a transversals of Γ/Λ     

Hardy's Theorem and the Short-Time Fourier Transform of Schwartz Functions

The Schwartz space of rapidly decaying test functions is characterizedby the decay of the short-time Fourier transform or cross-Wignerdistribution. Then a version of Hardy's theorem is proved forthe short-time Fourier transform and for the Wigner distribution.     

Lefschetz fixed point theorems for Fourier–Mukai functors and DG algebras

We propose some variants of Lefschetz fixed point theorem for Fourier–Mukai functors on a smooth projective algebraic variety. Independently we also suggest a similar theorem for endo-functors on the category of perfect modules over a smooth and proper DG algebra.     

Absolute matrix summability of a factored Fourier series

A theorem for a factored Fourier series is proved.     

Justification of an Approach to Application of Orthogonal Expansions for Approximate Integration of Canonical Systems of Second Order Ordinary Differential Equations

A solvability theorem for a nonlinear system of equations with respect to approximate values of Fourier—Cliebysliev coefficients is proved. This theorem is a theoretical substantiation for the numerical solution of second order ordinary differential equations using Chebyshev series and Markov quadrature formulas.     

Relations among Fourier coefficients and sum-functions

Among others a general equivalence theorem on Fourier cosine series with monotone coefficients are generalized to coefficients of rest bounded variation. Similarly some theorems of Aljan?i? are also extended, namely one of them in this generalized form is required to the proof of the equivalence theorem.     

On Hardy’s Theorem on SU(1, 1)

The classical Hardy theorem asserts that ■ and its Fourier transform ■ can not both be very rapidly decreasing.This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform.However,on SU(1,1)there are infinitely many"good"functions in the sense that ■ and its spherical Fourier transform ■ both have good decay. In this paper,we shall characterize such functions on SU(1,1).     

p-adic数域Q_p上的窗口Fourier变换

提出了 p-adic数域 Qp上的窗口 Fourier变换和逆变换 ,详细地讨论了 p-adic模函数和阶梯函数的窗口 Fourier变换和逆变换 ,最后给出了 p-adic模函数和阶梯函数的窗口 Fourier变换定理 .