Smooth Parseval frames for $L^2(mathbb{R})$ and generalizations to $L^2(mathbb{R}^d)$

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نشر من قبل Emily King

تاريخ النشر 2012

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والبحث باللغة English

تأليف Emily J. King

تحليل وظيفي

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‎Shamra Academia - شمرا أكاديميا‎

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الملخص بالعربية الملخص بالإنكليزية

Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in $L^2(mathbb{R}^d)$ which only require a single generating wavelet. Although work had been done to smooth these wavelets, which are by definition discontinuous on the frequency domain, nothing had been explicitly done over $mathbb{R}^d$, $d >1$. This paper, along with another one cowritten by the author, finally addresses this issue. Smoothing does not work as expected in higher dimensions. For example, Bin Hans proof of existence of Schwartz class functions which are Parseval frame wavelets and approximate Parseval frame wavelet set wavelets does not easily generalize to higher dimensions. However, a construction of wavelet sets in $hat{mathbb{R}}^d$ which may be smoothed is presented. Finally, it is shown that a commonly used class of functions cannot be the result of convolutional smoothing of a wavelet set wavelet.

قيم البحث

240 - T. Iwabuchi , T. Matsuyama , 2016

Let $H_V=-Delta +V$ be a Schrodinger operator on an arbitrary open set $Omega$ of $mathbb R^d$, where $d geq 3$, and $Delta$ is the Dirichlet Laplacian and the potential $V$ belongs to the Kato class on $Omega$. The purpose of this paper is to show $ L^p$-boundedness of an operator $varphi(H_V)$ for any rapidly decreasing function $varphi$ on $mathbb R$. $varphi(H_V)$ is defined by the spectral theorem. As a by-product, $L^p$-$L^q$-estimates for $varphi(H_V)$ are also obtained.

تحليل وظيفي نظرية الطيف

$L^2$ estimates of trilinear oscillatory integrals of convolution type on $mathbb{R}^2$

181 - Yangkendi Deng , Zuoshunhua Shi , Dunyan Yan 2021

This paper is devoted to $L^2$ estimates for trilinear oscillatory integrals of convolution type on $mathbb{R}^2$. The phases in the oscillatory factors include smooth functions and polynomials. We shall establish sharp $L^2$ decay estimates of trili near oscillatory integrals with smooth phases, and then give $L^2$ uniform estimates for these integrals with polynomial phases.

التحليل الكلاسيكي و ODEs

On Positivity Preserving, Translation Invariant, Operators in $L^p(mathbb{R}^n)^m$

91 - Fritz Gesztesy , Michael M.H. Pang 2017

We characterize positivity preserving, translation invariant, linear operators in $L^p(mathbb{R}^n)^m$, $p in [1,infty)$, $m,n in mathbb{N}$.

تحليل وظيفي

Interpolation between $H^{p(cdot)}(mathbb R^n)$ and $L^infty(mathbb R^n)$: Real Method

151 - Ciqiang Zhuo , Dachun Yang , Wen Yuan 2017

Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into good an d bad parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(cdot)}(mathbb R^n)$ and the space $L^{infty}(mathbb R^n)$: begin{equation*} (H^{p(cdot)}(mathbb R^n),L^{infty}(mathbb R^n))_{theta,infty} =W!H^{p(cdot)/(1-theta)}(mathbb R^n),quad thetain(0,1), end{equation*} where $W!H^{p(cdot)/(1-theta)}(mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$ with $p_-:=mathopmathrm{ess,inf}_{xinrn}p(x)in(1,infty)$ is proved to coincide with the variable Lebesgue space $W!L^{p(cdot)}(mathbb R^n)$.

التحليل الكلاسيكي و ODEs تحليل وظيفي

Nonnegative $C^2(mathbb{R}^2)$ interpolation