Related papers: Fourier Series for Singular Measures

For a fixed singular Borel probability measure $\mu$ on $\mathbb{T}$, we give several characterizations of when an entire function is the Fourier transform of some $f \in L^2(\mu)$. The first characterization is given in terms of criteria…

For multi-variable finite measure spaces, we present in this paper a new framework for non-orthogonal $L^2$ Fourier expansions. Our results hold for probability measures $\mu$ with finite support in $\mathbb{R}^d$ that satisfy a certain…

It is known that if a finite Borel measure $\mu$ on $[0,1)$ possesses a frame of exponential functions for $L^{2}(\mu)$, then $\mu$ is of pure type. In this paper, we prove the existence of a class of finite Borel measures $\mu$ on $[0,1)$…

In this paper we study 2D Fourier expansions for a general class of planar measures $\mu$, generally singular, but assumed compactly supported in $\mathbb{R}^2$. We focus on the following question: When does $L^2(\mu)$ admit a 2D system of…

Consider a Moran-type iterated function system (IFS) \( \{\phi_{k,d}\}_{d\in D_{2p_k}, k\geq 1} \), where each contraction map is defined as \[ \phi_{k,d}(x) = (-1)^d b_k^{-1}(x + d), \] with integer sequences \( \{b_k\}_{k=1}^\infty \) and…

Let $\mu$ be a Borel probability measure on $\mathrm{SL}_2(\mathbb R)$ with a finite exponential moment, and assume that the subgroup $\Gamma_{\mu}$ generated by the support of $\mu$ is Zariski dense. Let $\nu$ be the unique…

Let $\mu$ be a probability measure on $\mathbb{T}$ that is singular with respect to the Haar measure. In this paper we study Fourier expansions in $L^2(\mathbb{T},\mu)$ using techniques from the theory of model subspaces of the Hardy space.…

A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $\mu$ is identifying all $t\in\R$ such that both \[\left\{e^{-2\pi i\lambda x}: \lambda\in\Lambda\right\}\quad\text{and}\quad \left\{e^{-2\pi…

Let $l\geq 6$ be any integer, where $l\equiv 2$ mod $4$. Suppose that $\mu(\tau)d\tau$ is a measure with bounded variation and is supported on a compact subset of the complex plane, where…

In recent papers a number of authors have considered Borel probability measures $\mu$ in $\br^d$ such that the Hilbert space $L^2(\mu)$ has a Fourier basis (orthogonal) of complex exponentials. If $\mu$ satisfies this property, the set of…

We construct a probability measure $\mu$ supported on a set of zero $2d/p$-Hausdorff measure such that $\hat{\mu}\in L_{p}(\mathbb{R}^d)$.

We prove, under certain conditions on $(\alpha,\beta)$, that each Schwartz function $f$ such that $f(\pm n^{\alpha}) = \hat{f}(\pm n^{\beta}) = 0, \forall n \ge 0$ must vanish identically, complementing a series of recent results involving…

Let $d$ be a positive integer, and let $\mu$ be a finite measure on $\br^d$. In this paper we ask when it is possible to find a subset $\Lambda$ in $\br^d$ such that the corresponding complex exponential functions $e_\lambda$ indexed by…

Based on the seminal work of Hutchinson, we investigate properties of {\em $\alpha$-weighted Cantor measures} whose support is a fractal contained in the unit interval. Here, $\alpha$ is a vector of nonnegative weights summing to $1$, and…

We completely classify Fourier summation formulas of the form $$ \int_{\mathbb{R}} \widehat{\varphi}(t) d\mu(t)=\sum_{n=0}^{\infty} a(\lambda_n)\varphi(\lambda_n), $$ that hold for any test function $\varphi$, where $\widehat\varphi$ is the…

In this paper, exact rate of approximation of functions by linear means of Fourier series and Fourier integrals and corresponding $K$-functionals are expressed via special moduli of smoothness. . Introduction is given in $\S 1$. In $\S2$…

In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call $k$-spherical measures, correspond to Fourier series exhibiting a modular-type transformation behaviour…

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in \Bbb R^{md}$, let $\pi^{{\bf a}}:\; \Sigma=\{1,\ldots, m\}^{\Bbb N}\to \Bbb R^d$ denote the…

We prove that if a multiple trigonometric series is spherically Abel summable everywhere to an everywhere finite function $f(x)$ which is bounded below by an integrable function, then the series is the Fourier series of $f(x)$ if the…

We prove the following. Let $\mu_{1},\ldots,\mu_{n}$ be Borel probability measures on $[-1,1]$ such that $\mu_{j}$ has finite $s_j$-energy for certain indices $s_{j} \in (0,1]$ with $s_{1} + \ldots + s_{n} > 1$. Then, the multiplicative…