非交换傅立叶变换

这是最近子因子和局部紧量子群上的非交换傅立叶变换的一系列工作的综述.简要地介绍子因子和局部紧量子群的定义及其性质,给出了Hausdorff-Young不等式,Young不等式,不确定原理,合集估计以及它们的等号成立条件.     

傅立叶变换与小波变换的比较教学

本文针对小波变换教学中小流变换概念理解困难的问题，提出了一种比较教学方法，通过分析小波变换与傅立叶变换之间的联系，并从四个方面进行对比，清楚地描述了小波变换的本质，从而对加深对小波变换的理解。     

傅里叶变换与小波分析

正1小波分析的发展历史1807年,法国的热学工程师J.B.J.Fourier提出任一函数都能展开成三角函数的无穷级数,从而开启了主要研究函数的傅里叶变换及其性质的傅里叶分析理论。1909年,Haar提出了第一个最简单的小波(Haar小波)。在1974年,法国从事石油信号处理的工程师J.Morlet首先提出小波变换的概念,且根据物理和信号处理的实际经验的需要建立了反演公式,但当时这一公式未能得到数学家的认可。直到1986年,著名数学家Y.Meyer偶然构造出一个真正的小波基,并与S.Mallat合作建立了构造小     

Fourier Transform Over Local Fields

The Fourier transform in the space of distributions on finite-dimensional vector spaces over local fields is defined, and a strong similarity between the case of archimedian and non-archimedian fields is discussed. Furthermore, the definition of the Fourier transform for the case of “functions” on finite-dimensional vector spaces over 2-dimensional local fields is introduced. Leonardo da Vinci lecture held on June 29, 2005 Received: October 2005     

傅里叶变换与处处连续无处可微函数

本文介绍如何在数学分析课程中引入傅里叶变换以及利用傅里叶变换说明魏尔斯特拉斯函数的无处可微性.     

Superfast Fourier Transform Using QTT Approximation

We propose Fourier transform algorithms using QTT format for data-sparse approximate representation of one- and multi-dimensional vectors (m-tensors). Although the Fourier matrix itself does not have a low-rank QTT representation, it can be efficiently applied to a vector in the QTT format exploiting the multilevel structure of the Cooley-Tukey algorithm. The m-dimensional Fourier transform of an n×?×n vector with n=2 ^d has $\mathcal{O}(m d^{2} R^{3})$ complexity, where R is the maximum QTT-rank of input, output and all intermediate vectors in the procedure. For the vectors with moderate R and large n and m the proposed algorithm outperforms the $\mathcal{O}(n^{m} \log n)$ fast Fourier transform (FFT) algorithm and has asymptotically the same log-squared complexity as the superfast quantum Fourier transform (QFT) algorithm. By numerical experiments we demonstrate the examples of problems for which the use of QTT format relaxes the grid size constrains and allows the high-resolution computations of Fourier images and convolutions in higher dimensions without the ‘curse of dimensionality’. We compare the proposed method with Sparse Fourier transform algorithms and show that our approach is competitive for signals with small number of randomly distributed frequencies and signals with limited bandwidth.     

Fast Fourier Transform for Fitness Landscapes

We cast some classes of fitness landscapes as problems of spectral analysis on various Cayley graphs. In particular, landscapes derived from RNA folding are realized on Hamming graphs and analyzed in terms of Walsh transforms; assignment problems are interpreted as functions on the symmetric group and analyzed in terms of the representation theory of S_n. We show that explicit computation of the Walsh/Fourier transforms is feasible for landscapes with up to 10⁸ configurations using fast Fourier transform techniques. We find that the cost function of a linear sum assignment problem involves only the defining representation of the symmetric group, while quadratic assignment problems are superpositions of the representations indexed by the partitions (n), (n−1,1), (n−2,2), and (n−2,1,1). These correspond to the four smallest eigenvalues of the Laplacian of the Cayley graph obtained by using transpositions as the generating set on S_n.     

Schubert varieties as variations of Hodge structure

We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS ‘span’ the space of all infinitesimal VHS; and (3) show that the cohomology classes dual to the Schubert VHS form a basis of the invariant characteristic cohomology associated with the infinitesimal period relation (a.k.a. Griffiths’ transversality).     

On the Fourier Transform of Causal Distributions

We extend the distributional Bochner formula [1, p. 72, Theorem 26] to certain kinds of distributions. Theorem I.1 gives a formula [Eq. (I.1.14)] which makes it possible to obtain easily the Fourier transform of distributions of the form As applications of the formula (I.1.14) we evaluate the Fourier transforms of the distributions G_α(P±i0, m, n) [Eq. (I.4.1)] and H_α(P±i0,n) [Eq. (II.1.1)]. It follows from Theorem II.3 that H_zk(P±i0,n) is, for 2K≠n+2r, r=0,1..., an elementary solution of the n-dimensional ultrahyperbolic operator iterated k times.     

The Bargmann Transform and Windowed Fourier Localization

We consider the relationship between Gabor-Daubechies windowed Fourier localization operators and Berezin-Toeplitz operators T _φ, using the Bargmann isometry β. For “window” w a finite linear combination of Hermite functions, and a very general class of functions φ, we prove an equivalence of the form by obtaining the exact formulas for the operator C and the linear differential operator D.     

Some Properties of the Spinor Fourier Transform

In this paper, the theory of the spinor Fourier transform introduced in [Batard T, Berthier M, Saint-Jean C, Clifford-Fourier Transform for Color Image Processing, Geometric Algebra Computing for Engineering and Computer Science (E. Bayro-Corrochano and G. Scheuermann Eds.), Springer, London, 2010, pp. 135–161] is further developed. While in the original paper, the transform was determined for vector-valued functions only, it now will be extended to functions taking values in the entire Clifford algebra. Next, two bases are determined under which this Fourier transform is diagonalizable. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. This problem will be tackled in the final section of this paper.     

Multiplier Theorems for the Short-Time Fourier Transform

So-called short-time Fourier transform multipliers (also called Anti-Wick operators in the literature) arise by applying a pointwise multiplication operator to the STFT before applying the inverse STFT. Boundedness results are investigated for such operators on modulation spaces and on L _p -spaces. Because the proofs apply naturally to Wiener amalgam spaces the results are formulated in this context. Furthermore, a version of the Hardy-Littlewood inequality for the STFT is derived. This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No K67642.     

A General Geometric Fourier Transform Convolution Theorem

The large variety of Fourier transforms in geometric algebras inspired the straight forward definition of “A General Geometric Fourier Transform” in Bujack et al., Proc. of ICCA9, covering most versions in the literature. We showed which constraints are additionally necessary to obtain certain features like linearity, a scaling, or a shift theorem. In this paper we extend the former results by a convolution theorem.     

对偶L_p-质心体与Fourier变换

对p>0,Lutwak,Yang和Zhang引进了R~n中一个凸体K的对偶L_p~-质心体Γ_(-p)K.本文研究Γ_(-p)KΓ_(-p)L是否必定蕴涵vol_n(K)≤vol_n(L)的问题.我们的结果是Lutwak(p=1的情形)及Grinberg和Zhang(p>1的情形)关于L_(p~-)质心体算子Γ_p的类似问题的结果的对偶形式.     

On the Windowed Fourier Transform and Wavelet Transform of Almost Periodic Functions

We present the windowed Fourier transform and wavelet transform as tools for analyzing persistent signals, such as bounded power signals and almost periodic functions. We establish the analogous Parseval-type identities. We consider discretized versions of these transforms and construct generalized frame decompositions. Finally, we bring out some relations with shift-invariant operators and linear systems.     

Syntomic cohomology as a p-adic absolute Hodge cohomology

The purpose of this paper is to interpret rigid syntomic cohomology, defined by Amnon Besser [Bes], as a p-adic absolute Hodge cohomology. This is a p-adic analogue of a work of Beilinson [Be1] which interprets Beilinson-Deligne cohomology in terms of absolute Hodge cohomology. In the process, we will define a theory of p-adic absolute Hodge cohomology with coefficients, which may be interpreted as a generalization of rigid syntomic cohomology to the case with coefficients. Received: 25 September 2000 / In final form: 23 March 2001 / Published online: 28 February 2002     

Minimal Hermite-Type Eigenbasis of the Discrete Fourier Transform

There exist many ways to build an orthonormal basis of \(\mathbb {R}^N\), consisting of the eigenvectors of the discrete Fourier transform (DFT). In this paper we show that there is only one such orthonormal eigenbasis of the DFT that is optimal in the sense of an appropriate uncertainty principle. Moreover, we show that these optimal eigenvectors of the DFT are direct analogues of the Hermite functions, that they also satisfy a three-term recurrence relation and that they converge to Hermite functions as N increases to infinity.

     

Autocorrelation Scaling and Fourier Transform of Non-Autonomous Systems

. Upper bounds for the (strong) Fourier transform,of a rather general sequence of unitary operators, are related to the uniform !-Hölder continuity of its autocorrelation measure. It is a natural generalization of the "Dynamical Bombieri-Taylor Conjecture". Immediate applications include driven quantum systems, classical and quantum harmonic oscillators, and non-autonomous twisted generalized random walks in Hilbert spaces.