arXiv subjects

James Michael Wilson

Publications and source records attributed to James Michael Wilson.

2 recordsLinked to original sources

arXiv2023

Stability of Haar decompositions

We prove a general result implying the $L^2$ stability of Haar decompositions of $L^2({\bf R}^d)$ functions when the Haar functions are distorted by arbitrary, independent, affine changes of variable that are close to the identity. We apply our method to get fully $d$-dimensional generalizations of results of Aimar, Bernardis, Gorosito, Govil, and Zalik, on constructing frames of smooth functions which are, in many natural senses, arbitrarily close to the Haar functions. We also obtain a best-possible estimate on the $L^2$ sensitivity of dyadic averages of functions to small distortions caused by local affine changes of variable.

math.CA↗

arXiv2023

Growth and decay of Hölder moduli

If $f:{\bf R}^d\to{\bf C}$ is bounded and $f$'s Hölder $α$-modulus of continuity grows no faster than $(1+\vert x\vert)^M$ ($M\geq0$) then, for every $ε>0$, there is a $β>0$ such that $f$'s Hölder $β$-modulus grows no faster than $(1+\vert x\vert)^ε$. We use this easy fact to show that, if $\vert f\vert$ decays as fast as $(1+\vert x\vert)^{-R}$ (for $R>0$) and $f$'s $α$-Hölder modulus grows no faster than $(1+\vert x\vert)^M$, then, for every $0\leq R'< R$, there is a $β>0$ such that $f$'s $β$-Hölder modulus decays as fast as $(1+\vert x\vert)^{-R'}$. We apply this to strengthen a result of Coifman and Meyer on almost-orthogonality of vaguelet families and to derive other useful facts about vaguelets and vaguelet-like functions.

math.CA↗