Integral representations and transforms of N-functions. II

We study the problem of reconstruction of a convolution type integral transform from its behavior for power functions. We solve some of the problems posed in Part I of the article.     

Integral equations requiring small numbers of Krylov-subspace iterations for two-dimensional smooth penetrable scattering problems

This paper presents a class of boundary integral equations for the solution of problems of electromagnetic and acoustic scattering by two-dimensional homogeneous penetrable scatterers with smooth boundaries. The new integral equations, which, as is established in this paper, are uniquely solvable Fredholm equations of the second kind, result from representations of fields as combinations of single and double layer potentials acting on appropriately chosen regularizing operators. As demonstrated in this text by means of a variety of numerical examples (that resulted from a high-order Nyström computational implementation of the new equations), these “regularized combined equations” can give rise to important reductions in computational costs, for a given accuracy, over those resulting from previous iterative boundary integral equation solvers for transmission problems.     

Integral representations and convergence of the renewal density

We derive integral representations for the renewal density u associated with a square integrable probability density p on [0,∞) having finite expected value μ. These representations express u in terms of the real and the imaginary part of the Fourier transform of p, considered as a function on the lower complex half plane. We use them to give simple global integrability conditions on p under which limt→∞(u(t)−p(t))=1/μ.     

一个积分等式的新证

通过傅立叶积分公式和筛选性质给出了关于积分等式∫ -∞ ^∞e^t^2/2dt=√2π的一种新证明方法．     

Approximation of the Stieltjes Integral and Applications in Numerical Integration

Some inequalities for the Stieltjes integral and applications in numerical integration are given. The Stieltjes integral is approximated by the product of the divided difference of the integrator and the Lebesgue integral of the integrand. Bounds on the approximation error are provided. Applications to the Fourier Sine and Cosine transforms on finite intervals are mentioned as well.     

Boundedness of pseudodifferential operators on modulation spaces

We study classes of pseudodifferential operators which are bounded on large collections of modulation spaces. The conditions on the operators are stated in terms of the L^p,q estimates for the continuous Gabor transforms of their symbols. In particular, we show how these classes are related to the class of operators of Gröchenig and Heil, which is bounded on all modulation spaces.     

傅里叶变换中一个含参变量广义积分的计算

本文利用含参变量广义积分计算中常用的求导和变量替换等技巧等给出了傅里叶变换中一个重要的含参变量广义积分的计算方法.     

Representation of Fourier Integral Operators Using Shearlets

Traditional methods of time-frequency and multiscale analysis, such as wavelets and Gabor frames, have been successfully employed for representing most classes of pseudodifferential operators. However, these methods are not equally effective in dealing with Fourier Integral Operators in general. In this article, we show that the shearlets, recently introduced by the authors and their collaborators, provide very efficient representations for a large class of Fourier Integral Operators. The shearlets are an affine-like system of well-localized waveforms at various scales, locations and orientations, which are particularly efficient in representing anisotropic functions. Using this approach, we prove that the matrix representation of a Fourier Integral Operator with respect to a Parseval frame of shearlets is sparse and well-organized. This fact recovers a similar result recently obtained by Candès and Demanet using curvelets, which illustrates the benefits of directional multiscale representations (such as curvelets and shearlets) in the study of those functions and operators where traditional multiscale methods are unable to provide the appropriate geometric analysis in the phase space. The second author was supported in part by a National Science Foundation grant DMS 0604561.     

On the Sobolev boundedness results of the product of pseudo-differential operators involving a couple of fractional Hankel transforms

This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed.     

Global L 2-Boundedness Theorems for a Class of Fourier Integral Operators

We prove sharp interpolatory estimates between directional Haar projections and Riesz Transforms. We apply those to prove a conjecture of L. Tartar that arose within the theory of compensated compactness. To provide a frame of reference for the analytical estimates addressed in this paper we briefly review the core ideas of compensated compactness as developed by Murat and Tartar.     

Certain fractional calculus and integral transform results of incomplete ℵ-functions with applications

In this paper, we study certain interesting and useful properties of incomplete $\aleph$-functions. The incomplete $\aleph$-function is an extension of the $\aleph$-function. We find several useful classical integral transforms of these functions. Further, we examine the fractional calculus with the incomplete $\aleph$-functions and point out several special cases. Finally, we give the applications of incomplete $\aleph$-functions in detecting glucose supply in human blood.     

关于粗糙奇异积分算子的一点注记

研究R^n上一类沿多项式曲线的奇异积分算子，在一些相当弱的尺寸条件下建立了这些算子的L^p有界性．     

Composition of Fourier Integral Operators with Fold and Blowdown Singularities

ABSTRACT

The purpose of this work is to present results about the composition of Fourier integral operators with certain singularities, for which the composition is not again a Fourier integral operator. The singularities considered here are folds and blowdowns. We prove that for such operators, the Schwartz kernel of F*F belongs to a class of distributions associated to two cleanly intersection Lagrangians. Such Fourier integral operators appear in integral geometry, inverse acoustic scattering theory and Synthetic Aperture Radar imaging, where the composition calculus can be used as a tool for finding approximate inversion formulas and for recovering images.     

Local smoothing for operators failing the cinematic curvature condition

In this paper, we examine a class of averaging operators which exhibit local smoothing. That is, viewed as a function of space and time variables, the operators yield more smoothing than the fixed-time estimates. Sogge showed in a more general setting that if these operators satisfy a cinematic curvature condition, they will exhibit some local smoothing [C.D. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991) 231-251]. Here we translate this condition into the setting of averaging operators in the plane. We prove that cinematic curvature is not necessary for local smoothing to occur, exhibiting a class of operators which fail the cinematic curvature condition but still satisfy a local smoothing estimate. Furthermore, the amount of local smoothing exhibited by these operators is strictly less than that conjectured for operators satisfying the cinematic curvature condition.     

星像函数的某些积分算子（英）

ＣｅｒｔａｉｎＩｎｔｅｇｒａｌＯｐｅｒａｔｏｒｓｆｏｒＳｔａｒｌｉｋｅＦｕｎｃｔｉｏｎｓＢ．Ａ．ＵｒａｌｅｇａｄｄｉＣ．Ｓｏｍａｎａｔｈａ（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＫａｒｎａｔａｋＵｎｉｖｅｒｓｉｔｙ，Ｄｈａｒｗａｄ５８０００３，Ｉｎｄｉａ）Ｃｅｒ...     

Integral models of representations of current groups

We suggest a new construction of nonlocal representations of the current group. Instead of the Fock space, which is usually used in this situation, we consider the direct integral of countable tensor products of representations over the trajectories of some stochastic process. The construction substantially uses the invariance of the so-called infinite-dimensional Lebesgue measure.     

On the Algebra of Pair Integral Operators with Homogeneous Kernels

In this paper, we study the Banach algebra generated by multidimensional pair integral operators with homogeneous kernels. We describe necessary and sufficient conditions for operators from the algebra to be Fredholm and present a formula for calculating the index.     

锥中调和函数的积分表示

本文证明了锥内一类调和函数h, 若其正部h⁺ = max{h,0} 满足一种增长条件, 则h 能被其边界值的积分表示. 同时证明了其负部h^- = max{-h,0} 也能被类似的一种增长条件所控制. 所得结论推广了解析函数和调和函数在上半空间中关于积分表示的相关结果.