Generalized Fourier--Feynman transforms and generalized convolution products on Wiener space II

Beyond Wiener's Lemma: Nuclear Convolution Algebras and the Inversion of Digital Filters

A convolution algebra is a topological vector space $\mathcal{X}$ that is closed under the convolution operation. It is said to be inverse-closed if each element of $\mathcal{X}$ whose spectrum is bounded away from zero has a convolution inverse that is also part of the algebra. The theory of discrete Banach convolution algebras is well established with a complete characterization of the weighted $\ell_1$ algebras that are inverse-closed and referred to as the Gelfand-Raikov-Shilov (GRS) spaces. Our starting point here is the observation that the space $\mathcal{S}(\mathbb{Z}^d)$ of rapidly de…

Wiener Amalgam Spaces with respect to Quasi-Banach Spaces

We generalize the theory of Wiener amalgam spaces on locally compact groups to quasi-Banach spaces. As a main result we provide convolution relations for such spaces. Also we weaken the technical assumption that the global component is invariant under right translations, which is even new for the classical Banach space case. To illustrate our theory we discuss in detail an example on the $ax+b$ group.

A Wiener Tauberian theorem for operators and functions

We prove variants of Wiener's Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or of two trace class operators. Our results include Wiener's Tauberian theorem as a special case. Applications of our Tauberian theorems are related to localization operators, Toeplitz operators, isomorphism theorems between Bargmann-Fock spaces and quantization schemes with consequences for Shubin's pseudodifferential operator calculus and Born-Jordan quantization. Based on the links between localization ope…

Sharpness of some properties of Wiener amalgam and modulation spaces

We prove sharp estimates for the dilation operator $f(x)\longmapsto f(λx)$, when acting on Wiener amalgam spaces $W(L^p,L^q)$. Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces $M^{p,q}$, as well as the optimality of an estimate for the Schrödinger propagator on modulation spaces.

基于卷积的短时偏移线性正则变换的理论与应用

Theory and Applications of Convolution-Based short time offset linear canonical transform

提出基于卷积的短时偏移线性正则变换，建立其连续性、正交性、反演公式等基本性质，并应用于泊松求和公式、Paley-Wiener准则和采样定理，通过数值仿真验证其信号重建优势。

On temporal regularity of stochastic convolutions in $2$-smooth Banach spaces

We show that paths of solutions to parabolic stochastic differential equations have the same regularity in time as the Wiener process (as of the current state of art). The temporal regularity is considered in the Besov-Orlicz space $B^{1/2}_{Φ_2,\infty}(0,T;X)$ where $Φ_2(x)=\exp(x^2)-1$ and $X$ is a $2$-smooth Banach space.

Dual of Hardy-amalgam spaces and norms inequalities

We characterize the dual spaces of the generalized Hardy spaces defined by replacing Lebesgue quasi-norms by Wiener amalgam ones. In these generalized Hardy spaces, we prove that some classical linear operators such as Calderón-Zygmund, convolution and Riesz potential operators are bounded.

复抽象威纳空间

Complex abstract Wiener spaces

本文提出复抽象威纳空间的框架，通过特征函数替代可测范数，证明高斯场与抽象威纳空间的等价性，并建立复与实抽象威纳空间的联系。

Multi-linear products with odd factors in pseudo-differential calculus with symbols in modulation spaces

We give sufficient conditions on the Lebesgue exponents for compositions of odd numbers of pseudo-differential operators with symbols in modulation spaces. As a byproduct, we obtain sufficient conditions for twisted convolutions of odd numbers of factors to be bounded on Wiener amalgam spaces.

与偏移线性正则变换相关的新卷积定理及应用

New convolution related theorems and applications associated with offset linear canonical transform

本文提出了与偏移线性正则变换（OLCT）相关的新型卷积与相关定理，探讨了其在乘法滤波器设计及信号恢复中的应用，并分析了L2域内OLCT的实Paley-Wiener和Boas定理。

Sharp results for the Weyl product on modulation spaces

We give sufficient and necessary conditions on the Lebesgue exponents for the Weyl product to be bounded on modulation spaces. The sufficient conditions are obtained as the restriction to $N=2$ of a result valid for the $N$-fold Weyl product. As a byproduct, we obtain sharp conditions for the twisted convolution to be bounded on Wiener amalgam spaces.

Stability of Localized Operators

Let $\ell^p, 1\le p\le \infty$, be the space of all $p$-summable sequences and $C_a$ be the convolution operator associated with a summable sequence $a$. It is known that the $\ell^p$- stability of the convolution operator $C_a$ for different $1\le p\le \infty$ are equivalent to each other, i.e., if $C_a$ has $\ell^p$-stability for some $1\le p\le \infty$ then $C_a$ has $\ell^q$-stability for all $1\le q\le \infty$. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of …

一种改进的Hartley卷积变种：代数结构、谱半径及相关问题

A refined variant of Hartley convolution: algebraic structures, spectral radius and related issues

本文提出了一种改进的Hartley卷积变种，探讨其代数结构、谱半径及相关问题，并应用于积分方程和热源问题分析。

Banach空间上的近似序列：一种丰富的方法

Approximation sequences on Banach spaces: a rich approach

研究了Banach空间中大规模卷积型算子有限截断的稳定性准则，提出了一种更简洁有力的代数方法，结合P理论、丰富序列概念和局部化技术，探讨了序列代数中的Fredholm理论及近似数的渐进行为。

Improved Epstein-Glaser renormalization in $x$-space versus differential renormalization

Renormalization of massless Feynman amplitudes in $x$-space is reexamined here, using almost exclusively real-variable methods. We compute a wealth of concrete examples by means of recursive extension of distributions. This allows us to show perturbative expansions for the four-point and two-point functions at several loop order. To deal with internal vertices, we expound and expand on convolution theory for log-homogeneous distributions. The approach has much in common with differential renormalization as given by Freedman, Johnson and Latorre; but differs in important details.