Rapidly growing Fourier integrals

On Fourier multipliers with rapidly oscillating symbols

We provide asymptotically sharp bounds for the $L_p$ norms of the Fourier multipliers with the symbols $e^{iλφ(ξ/|ξ|)}$, where $λ\in \mathbb{R}$ is a large parameter.

Approximation by Fourier sums in classes of differentiable functions with high exponents of smoothness

We find asymptotic equalities for the exact upper bounds of approximations by Fourier sums of Weyl-Nagy classes $W^r_{β,p}, 1\le p\le\infty,$ for rapidly growing exponents of smoothness $r$ $(r/n\rightarrow\infty)$ in the uniform metric. We obtain similar estimates for approximations of the classes $W^r_{β,1}$ in metrics of the spaces $L_p, 1\le p\le\infty$.

On a family of differential-reflection operators: intertwining operators and Fourier transform of rapidly decreasing functions

We introduce a family of differential-reflection operators $Λ_{A, \varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is a Strum-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ For special pairs $(A,\varepsilon),$ we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). The spectral problem for the operators $Λ_{A, \varepsilon}$ is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of $Λ_{A, \varepsilon}$. As the operators $Λ_{A, \varepsilon}$ are mixture of $d/dx$ and reflection operators, w…

Assouad维数与下维数：具有快速增长频率的Weierstrass型函数图像

Assouad Dimensions and Spectra of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies

本文研究Weierstrass型函数图像的Assouad维数与下维数，给出频率快速增长时的定量条件，使图像在紧区间上维数确定，并构造出四种维数互异的图像。

Elements of harmonic analysis, 5

These notes are concerned with the Schwartz class of rapidly decreasing smooth functions on R^n, Fourier transforms, etc.

Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2)

The Howe dual pair (sl(2),O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a well-chosen element of sl(2) such that the Helmholtz relations are satisfied. In this paper we first investigate what happens when instead we consider exponentials of elements of the universal enveloping algebra of sl(2). This leads to a complete class of generalized Fourier transforms, that all satisfy properties similar to the classical FT. There is moreover a finite subset of transforms which very closely resemble the FT. We…