Sharp Fourier type and cotype with respect to compact semisimple Lie groups

Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.

     

The amenability constant of the Fourier algebra

For a locally compact group , let denote its Fourier algebra and its dual object, i.e., the collection of equivalence classes of unitary representations of . We show that the amenability constant of is less than or equal to and that it is equal to one if and only if is abelian.

     

Demystification of the geometric Fourier transforms and resulting convolution theorems

T. Qian As it will turn out in this paper, the recent hype about most of the Clifford–Fourier transforms is not thoroughly worth the pain. Almost everyone that has a real application is separable, and these transforms can be decomposed into a sum of real valued transforms with constant multivecor factors. This fact makes their interpretation, their analysis, and their implementation almost trivial. Copyright © 2015 John Wiley & Sons, Ltd.     

Fourier type 2 operators with respect to locally compact abelian groups

A linear and bounded operator T between Banach spaces X and Y has Fourier type 2 with respect to a locally compact abelian group G if there exists a constant c > 0 such that∥T ∥₂ ≤ c∥f∥₂ holds for all X‐valued functions f ∈ L^X₂(G) where is the Fourier transform of f. We show that any Fourier type 2 operator with respect to the classical groups has Fourier type 2 with respect to any locally compact abelian group. This generalizes previous special results for the Cantor group and for closed subgroups of ?ⁿ. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vector valued Fourier analysis on unimodular groups

The notion of Fourier type and cotype of linear maps between operator spaces with respect to certain unimodular (possibly nonabelian and noncompact) group is defined here. We develop analogous theory compared to Fourier types with respect to locally compact abelian groups of operators between Banach spaces. We consider the Heisenberg group as an example of nonabelian and noncompact groups and prove that Fourier type and cotype with respect to the Heisenberg group implies Fourier type with respect to classical abelian groups. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The behavior of Fourier transforms for nilpotent Lie groups

We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-valued Fourier transforms on a nilpotent Lie group . Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on extends to be ``holomorphic' on an appropriate complexification of (a part of) . We prove the weak scalar-valued Paley-Wiener Theorem for some nilpotent Lie groups but show that it is false in general. We also prove a weak operator-valued Paley-Wiener Theorem for arbitrary nilpotent Lie groups, which in turn establishes the truth of a conjecture of Moss. Finally, we prove a conjecture about Dixmier-Douady invariants of continuous-trace subquotients of when is two-step nilpotent.

     

Characterizations of the Gelfand-Shilov spaces via Fourier transforms

We give symmetric characterizations, with respect to the Fourier transformation, of the Gelfand-Shilov spaces of (generalized) type and type . These results explain more clearly the invariance of these spaces under the Fourier transformations.

     

Operator weak amenability of the Fourier algebra

We show that for any locally compact group , the Fourier algebra is operator weakly amenable.

     

超曲面上具有非光滑密度之测度的Fourier变换

设Ｓ是Ｒ~ｎ中具有非零Ｇａｕｓｓ曲率的光滑闭曲面,ｄσ是Ｓ上的Ｌｅｂｅｓｇｕｅ诱导测度．若密度函数Ψ∈Ｃ∞（Ｓ） 则测度 ｄμψ＝ψｄσ的 Ｆｏｕｒｉｅｒ变换│d　　　　　　　 本文证明,若ψ非光滑则上述结论一般不成立;但对于ψ∈Ｌ~p（S,dσ）ｐ≥２在适当条件下, ｄμψ 仍有相应的衰减估计．应用所得结论可获得球面 Ｓ~（ｎ－１）上一种加权极大平均算子的有界性．     

Fourier transform and related integral transforms in superspace

In this paper extensions of the classical Fourier, fractional Fourier and Radon transforms to superspace are studied. Previously, a Fourier transform in superspace was already studied, but with a different kernel. In this work, the fermionic part of the Fourier kernel has a natural symplectic structure, derived using a Clifford analysis approach. Several basic properties of these three transforms are studied. Using suitable generalizations of the Hermite polynomials to superspace (see [H. De Bie, F. Sommen, Hermite and Gegenbauer polynomials in superspace using Clifford analysis, J. Phys. A 40 (2007) 10441-10456]) an eigenfunction basis for the Fourier transform is constructed.     

Clifford algebras,Fourier transforms,and quantum mechanics

In this review, we give an overview of several recent generalizations of the Fourier transform, related to either the Lie algebra or the Lie superalgebra . In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform, and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford–Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these transforms is given, with emphasis on aspects such as eigenfunctions and spectrum of the transform, characterization of the integral kernel, and connection with various special functions. Copyright © 2012 John Wiley & Sons, Ltd.     

Interpolation of vector measures

Let (Ω, Σ) be a measurable space and m ₀: Σ → X ₀ and m ₁: Σ → X ₁ be positive vector measures with values in the Banach Köthe function spaces X ₀ and X ₁. If 0 < α < 1, we define a new vector measure [m ₀, m ₁] _α with values in the Calderón lattice interpolation space X ₀ ^1?ga X ₁ ^α and we analyze the space of integrable functions with respect to measure [m ₀, m ₁] _α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of L ^p -spaces. Since each p-convex order continuous Köthe function space with weak order unit can be represented as a space of p-integrable functions with respect to a vector measure, we provide in this way a technique to obtain representations of the corresponding complex interpolation spaces. As applications, we provide a Riesz-Thorin theorem for spaces of p-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.     

Growth properties of Fourier transforms via moduli of continuity

We obtain new inequalities for the Fourier transform, both on Euclidean space, and on non-compact, rank one symmetric spaces. In both cases these are expressed as a gauge on the size of the transform in terms of a suitable integral modulus of continuity of the function. In all settings, the results present a natural corollary: a quantitative form of the Riemann-Lebesgue lemma. A prototype is given in one-dimensional Fourier analysis.     

核局部凸空间的一个新特征

Ａ．Ｐｉｅｔｓｃｈ＾［１］在讨论核局部凸空间时给出了两类矢值序列空间ｌ１［Ｘ］和ｌ１｛Ｘ｝。本文建立了矢值序列空间ｌ１［Ｘ］及ｌ１｛Ｘ｝和连续线性算子空间Ｌ（ｃ０，Ｘ）及绝对可和算子空间ＡＳ（ｃ０，Ｘ）之间的拓扑同胚关系。通过ｃ０上的矢值算子类Ｌ（ｃ０，Ｘ）和ＡＳ（ｃ０，Ｘ）及其上的拓扑等价关系，对局部凸空间Ｘ是核空间给出了一个新的特征刻划。     

块对偶Toeplitz算子的乘积

本文刻画了向量值Bergman空间上块对偶Toeplitz算子有界性和紧性,给出了块对偶Toeplitz算子的乘积是块对偶Toeplitz算子的充要条件.     

Asymptotic behavior of Fourier transforms of self-similar measures

Let be a self-similar probability measure on satisfying where 0$"> and Let be the Fourier transform of A necessary and sufficient condition for to approach zero at infinity is given. In particular, if and for then 0$"> if and only if is a PV-number and is not a factor of . This generalizes the corresponding theorem of Erdös and Salem for the case

     

Generalized fourier transform and its applications

We consider the generalized Fourier transform treated as an operator on the dual of an arbitrary locally convex space. We give a definition of this operator and establish its basic properties. Special attention is paid to cases in which the range of the generalized Fourier transform coincides with a weighted space of entire functions. The results are applied to finding the orders and types of operators in various spaces.