Related papers: A Fourier Frame for the Middle-Third Cantor Measur…

Let a sequence $\Lambda\subset\mathbb{C}$ be such that the corresponding system of exponential functions $\mathcal{E}(\Lambda):=\{e^{i\lambda t}\}_{\lambda\in\Lambda}$ is complete and minimal in $L^2(-\pi,\pi)$ and thus each function $f\in…

We continue developing the theory of conical and vertical square functions on $R^{n}$, where $\mu$ is a power bounded measure, possibly non-doubling. We provide new boundedness criteria and construct various counterexamples. First, we prove…

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…

In this paper we prove that given a weakly almost periodic measure $\mu$ supported inside some model set $\Lambda(W)$ with closed window $W$, then the strongly almost periodic component $\mu_S$ and the null weakly almost periodic component…

Suppose that $\mathcal{C}$ is the space of all middle Cantor sets. We characterize all triples $(\alpha,~\beta,~\lambda)\in \mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$ that satisfy $C_\alpha- \lambda C_\beta=[-\lambda,~1]. $ Also all…

Let $g \in L^2(\mathbb{R})$ be a rational function of degree $M$, i.e. there exist polynomials $P, Q$ such that $g = {{P} \over {Q}}$ and $deg(P) < deg(Q) \leq M$. We prove that for any $\varepsilon>0$ and any $M \in \mathbb{N}$ there…

Certain Bernoulli convolution measures (\mu) are known to be spectral. Recently, much work has concentrated on determining conditions under which orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure known to be…

One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in $\mathbb{R}^n$. When the support is a smooth enough manifold, an almost…

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into…

In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show…

We study measures $\mu$ on the plane with two independent Alberti representations. It is known, due to Alberti, Cs\"ornyei, and Preiss, that such measures are absolutely continuous with respect to Lebesgue measure. The purpose of this paper…

The multifractal spectrum of a Borel measure $\mu$ in $\mathbb{R}^n$ is defined as \[ f_\mu(\alpha) = \dim_H {x:\lim_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}=\alpha}. \] For self-similar measures under the open set condition the behavior of…

Let $\mu$ be a measure on $SL_{2}(\mathbb{R})$ generating a non-compact and totally irreducible subgroup, let $\chi>0$ denote its Lyapunov exponent, and let $\nu$ be the associated stationary (Furstenberg) measure for the action on the…

Let $T : \Lambda \to \Lambda$ be an expanding map on a Cantor set. For each suitably normalized H\"older continuous potential, we construct a spectral triple from which one may recover the associated Gibbs measure as a noncommutative…

Let $H$ be an infinite-dimensional separable Hilbert space and let $(X,d,\mu)$ be a metric measure space satisfying the doubling and upper Alhfors regularity conditions at small scale. We prove that every bounded continuous tight frame…

Let $C$ be the middle third Cantor set and $\mu$ be the $\frac{\log 2}{\log 3}$-dimensional Hausdorff measure restricted to $C$. In this paper we study approximations of elements of $C$ by dyadic rationals. Our main result implies that for…

Let H be a regular element of an irreducible Lie Algebra g, and let mu be the orbital measure supported on the Adjoint orbit of H. We show that the k-th power of the Fourier transform of mu is in L^2(g) if and only if k > dim g/(dim g-rank…

Let $ \Omega \subset R^d $ have finite positive Lebesgue measure, and let $ \mathcal{L}^{2}(\Omega) $ be the corresponding Hilbert space of $ \mathcal{L}^{2} $-functions on $ \Omega $. We shall consider the exponential functions $…

Let $\mu_{M,D}$ be the planar self-affine measure generated by an expansive integer matrix $M\in M_2(\mathbb{Z})$ and a non-collinear integer digit set $D=\left\{\begin{pmatrix} 0\\0\end{pmatrix},\begin{pmatrix} \alpha_{1}\\ \alpha_{2}…

Let $\mu$ be a self-similar measure satisfying the finite type condition. It is known that the set of attainable local dimensions for such a measure is a union of disjoint intervals, where some intervals may be degenerate points. Despite…