Pointwise Fourier inversion of distributions

We show that,given a tempered distribution T whose Fourier transform is a function of polynomial growth and a point x in Rn at which T has the value τ(in the sense of Lojasiewicz),the Fourier integral of T at x is summable in Bochner-Riesz means to τ.     

Bochner Schwartz Type Theorem for Conditionally Positive Definite Fourier Hyperfunctions*

We prove the Bochner–Schwartz type theorem for conditionally positive definite Fourier hyperfunctions which generalizes the result of Gelfand-Vilenkin in their treatise Generalized functions, vol. IV for distributions.     

Approximating Fourier transformation of orbital integrals

In this paper, we are concerned with orbital integrals on a class of real reductive Lie groups with non-compact Iwasawa K-component. The class contains all connected semisimple Lie groups with infinite center. We establish that any given orbital integral over general orbits with compactly supported continuous functions for a group G in is convergent. Moreover, it is essentially the limit of corresponding orbital integrals for its quotient groups in Harish-Chandra's class. Thus the study of orbital integrals for groups in class reduces to those of Harish-Chandra's class. The abstract theory for this limiting technique is developed in the general context of locally compact groups and linear functionals arising from orbital integrals. We point out that the abstract theory can be modified easily to include weighted orbital integrals as well. As an application of this limiting technique, we deduce the explicit Plancherel formula for any group in class .     

Almost everywhere convergence of inverse Fourier transforms

We show that if , then the inverse Fourier transform of converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when and . We also consider sequential convergence for general elements of .

     

Nonlinear inversion of a band-limited Fourier transform

We consider the problem of reconstructing a compactly supported function with singularities either from values of its Fourier transform available only in a bounded interval or from a limited number of its Fourier coefficients. Our results are based on several observations and algorithms in [G. Beylkin, L. Monzón, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal. 19 (1) (2005) 17–48]. We avoid both the Gibbs phenomenon and the use of windows or filtering by constructing approximations to the available Fourier data via a short sum of decaying exponentials. Using these exponentials, we extrapolate the Fourier data to the whole real line and, on taking the inverse Fourier transform, obtain an efficient rational representation in the spatial domain. An important feature of this rational representation is that the positions of its poles indicate location of singularities of the function. We consider these representations in the absence of noise and discuss the impact of adding white noise to the Fourier data. We also compare our results with those obtained by other techniques. As an example of application, we consider our approach in the context of the kernel polynomial method for estimating density of states (eigenvalues) of Hermitian operators. We briefly consider the related problem of approximation by rational functions and provide numerical examples using our approach.     

Classical Fourier Analysis over Homogeneous Spaces of Compact Groups

This paper introduces a unified operator theory approach to the abstract Fourier analysis over homogeneous spaces of compact groups. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset space of H in G andμ be the normalized G-invariant measure on G/ H associated to the Weil's formula.Then, we present a generalized abstract framework of Fourier analysis for the Hilbert function space L~2(G/H,μ).     

关于正定厄米特矩阵的一个定理

本文推广了文献 [2 ]给出的一个不等式 ,得到以下结果 :设 Ai,Bi,… ,Ci( i=1 ,2 ,… ,k)都是 n阶正定厄米特矩阵 ,α,β,… ,γ都是正实数 ,且 α+β+… +γ=p≥1 ,则∑ki=1|Ai|α|Bi|β… |Ci|γ <∑ki=1Ai α ∑ki=1Bi β… ∑ki=1Ci γ     

Sparse generalized Fourier transforms

Block-diagonalization of sparse equivariant discretization matrices is studied. Such matrices typically arise when partial differential equations that evolve in symmetric geometries are discretized via the finite element method or via finite differences. By considering sparse equivariant matrices as equivariant graphs, we identify a condition for when block-diagonalization via a sparse variant of a generalized Fourier transform (GFT) becomes particularly simple and fast. Characterizations for finite element triangulations of a symmetric domain are given, and formulas for assembling the block-diagonalized matrix directly are presented. It is emphasized that the GFT preserves symmetric (Hermitian) properties of an equivariant matrix. By simulating the heat equation at the surface of a sphere discretized by an icosahedral grid, it is demonstrated that the block-diagonalization is beneficial. The gain is significant for a direct method, and modest for an iterative method. A comparison with a block-diagonalization approach based upon the continuous formulation is made. It is found that the sparse GFT method is an appropriate way to discretize the resulting continuous subsystems, since the spectrum and the symmetry are preserved. AMS subject classification (2000)  43A30, 65T99, 20B25     

On the order of summability of the Fourier inversion formula

In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesàro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order k, then the distribution is the (k+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order k+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.     

A generalized Fourier transform and convolution on time scales

In this paper, we develop some important Fourier analysis tools in the context of time scales. In particular, we present a generalized Fourier transform in this context as well as a critical inversion result. This leads directly to a convolution for signals on two (possibly distinct) time scales as well as several natural classes of time scales which arise in this setting: dilated, closed under addition, and additively idempotent. We explore the properties of these time scales and demonstrate the utility of these concepts in discrete convolution, Mellin convolution, and transformations of a random variable.     

A distributional convolution for a generalized finite Fourier transformation

In this paper we define a generalized finite Fourier transformation in distribution spaces. Also we investigate a distributional convolution for this finite integral transformation.

     

Cardinal Hermite interpolation using positive definite functions

In this paper, we study cardinal Hermite interpolation by using positive definite functions. Among other things, we establish a procedure that employs the multiquadrics for cardinal Hermite interpolation.     

广义H-矩阵的一组新判定条件

利用矩阵指标集的k-级划分和子矩阵的谱半径,给出了正定条件下广义H-矩阵的一组判定条件,当块矩阵退化为点矩阵时,这些条件即为非奇异H-矩阵的充分条件.这些结果改进了近期的相关结果,并用数值算例说明本文判定条件的有效性.     

A Cauchy-Schwarz type inequality for bilinear integrals on positive measures

If is Borel measurable, define for -finite positive Borel measures on the bilinear integral expression

We give conditions on such that there is a constant , independent of and , with

Our results apply to a much larger class of functions than known before.

     

A simple approach to asymptotic expansions for Fourier integrals of singular functions

In this work, we are concerned with the derivation of full asymptotic expansions for Fourier integrals as s → ∞, where s is real positive, [a, b] is a finite interval, and the functions f(x) may have different types of algebraic and logarithmic singularities at x = a and x = b. This problem has been treated in the literature by techniques involving neutralizers and Mellin transforms. Here, we derive the relevant asymptotic expansions by a method that employs simpler and less sophisticated tools.     

周期函数Fourier级数展开式唯一性的简单证明与推广

对于周期函数f(x)按不同的周期展开对应不同的Fourier级数,这些表面上不同的式子是否一致引起了人们的注意[1],[2].本文应用Parseval等式给出一个关于这种唯一性的简单证明,并把这一种性质推广到高维情况的多重Fourier级数.     

A generalization of the Fujisawa-Kuh global inversion theorem

We discuss the problem of global invertibility of nonlinear maps defined on the finite dimensional Euclidean space via differential tests. We provide a generalization of the Fujisawa-Kuh global inversion theorem and introduce a generalized ratio condition which detects when the pre-image of a certain class of linear manifolds is non-empty and connected. In particular, we provide conditions that also detect global injectivity.