Related papers: Frame measures for infinitely many measures

A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This approach concludes finally the problem of the…

A notion of the limit spectral measure of a metric triple (i.e., a metric measure space) is defined. If the metric is square integrable, then the limit spectral measure is deterministic and coinsides with the spectrum of the integral…

Let $Q$ be a fundamental domain of some full-rank lattice in ${\Bbb R}^d$ and let $\mu$ and $\nu$ be two positive Borel measures on ${\Bbb R}^d$ such that the convolution $\mu\ast\nu$ is a multiple of $\chi_Q$. We consider the problem as to…

In this paper we show that if $\mu$ is any locally and uniformly $\alpha$-dimensional measure supported on a $\alpha$-quasi-regular set $E$, then $L^2(\mu)$ admits a frame of exponentials. In particular, for the uniform middle third Cantor…

We study spectral properties of the self-affine measure $\mu_{M,\mathcal {D}}$ generated by an expanding integer matrix $M\in M_n(\mathbb{Z})$ and a consecutive collinear digit set $\mathcal {D}=\{0,1,\dots,q-1\}v$ where $v\in…

It is well-known that the Beurling dimension of the spectra of certain singularly continuous spectral measures possesses an intermediate property. In this paper, we establish that for a class of self-affine spectral measures $\mu$, both the…

We study sparsity and spectral properties of dual frames of a given finite frame. We show that any finite frame has a dual with no more than $n^2$ non-vanishing entries, where $n$ denotes the ambient dimension, and that for most frames no…

One of the goals of this article is to define a an unified setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure. We first remark that some…

We construct measure which determines a two-variable mean in a very natural way. Using that measure we can extend the mean to infinite sets as well. E.g. we can calculate the geometric mean of any set with positive Lebesgue measure. We also…

We analyze certain parametrized families of one-dimensional maps with infinitely many critical points from the measure-theoretical point of view. We prove that such families have absolutely continuous invariant probability measures for a…

Motivated to generalize the Fourier frame concept to Banach spaces we introduce (p, q)-Bessel/frame measures for a given finite measure on LCA groups. We also present a general way of constructing (p, q)-Bessel/frame measures for a given…

If $\mu$ is a finite complex measure in the complex plane $\C$ we denote by $C^\mu$ its Cauchy integral defined in the sense of principal value. The measure $\mu$ is called reflectionless if it is continuous (has no atoms) and $C^\mu=0$ at…

We study finitely additive extensions of the asymptotic density to all the subsets of natural numbers. Such measures are called density measures. We consider a class of density measures constructed from free ultrafilters on $\mathbb{N}$ and…

Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of…

We characterize when a coherent state or continuous frame for a Hilbert space may be sampled to obtain a frame, which solves the discretization problem for continuous frames. In particular, we prove that every bounded continuous frame for a…

Scalable frames are frames with the property that the frame vectors can be rescaled resulting in tight frames. However, if a frame is not scalable, one has to aim for an approximate procedure. For this, in this paper we introduce three…

In this paper, we study the spectrality of infinite convolutions generated by infinitely many admissible pairs which may not be compactly supported, where the spectrality means the corresponding square integrable function space admits a…

The paper studies finite extensions of Bessel sequences in infinite-dimensional Hilbert spaces. We provide a characterization of Bessel sequences that can be extended to frames by adding finitely many vectors. We also characterize frames…

It is known [Dai and Sun, J. Funct. Anal. 268 (2015), 2464--2477] that there exist spectral measures with arbitrary Hausdorff dimensions, and it is natural to pose the question of whether similar phenomena occur for other dimensions of…

For continuous maps on a compact manifold M, particularly for those that do not preserve the Lebesgue measure m, we define the observable invariant probability measures as a generalization of the physical measures. We prove that any…