Localized Bounded Below Approximate Schauder Frames are Finite Unions of Approximate Riesz Sequences

关于区间和一点的并集的矩阵Fejér-Riesz型定理

Matrix Fejér-Riesz type theorem for a union of an interval and a point

本文证实了关于无界区间和一点的并集的矩阵Fejér-Riesz型定理，通过解决截断矩阵值矩量问题。

Perturbations of frames

In this paper, we give some sufficient conditions under which perturbations preserve Hilbert frames and near-Riesz bases. Similar results are also extended to frame sequences, Riesz sequences and Schauder frames. It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.

Dilation theorem for p-approximate Schauder frames for separable Banach spaces

Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space $\mathcal{H}$ is image of a Riesz basis under an orthogonal projection from a separable Hilbert space $\mathcal{H}_1$ which contains $\mathcal{H}$ isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case.

Feichtinger Conjectures, $R_\varepsilon$-Conjectures and Weaver's Conjectures for Banach spaces

Motivated from two decades old famous Feichtinger conjectures for frames, $R_\varepsilon$-conjecture and Weaver's conjecture for Hilbert spaces (and their solution by Marcus, Spielman, and Srivastava), we formulate Feichtinger conjectures for p-approximate Schauder frames, $R_\varepsilon$-conjecture, Weaver's conjectures and Akemann-Weaver conjectures for Banach spaces. We also formulate conjectures on p-approximate Schauder frames based on the results of Casazza for frames for Hilbert spaces. We state conjectures and problems for p-approximate Schauder frames based on fundamental inequality f…

The Feichtinger Conjecture and Reproducing Kernel Hilbert Spaces

We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences, then a general bounded Bessel sequence in an arbitrary Hilbert space can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and prove that for these spaces bounded Bessel sequences of normalized kernel functions are finite unions of Riesz bas…

Riesz bases of exponentials on multiband spectra

Let $S$ be the union of finitely many disjoint intervals on the real line. Suppose that there are two real numbers $α, β$ such that the length of each interval belongs to $Z α+ Z β$. We use quasicrystals to construct a discrete set of real frequencies such that the corresponding system of exponentials is a Riesz basis in the space $L^2(S)$.

Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends

Let $M$ be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, $\RR^n \setminus B(0,R)$ for some $R > 0$, each of which carries the standard metric. Our main result is that the Riesz transform on $M$ is bounded from $L^p(M) \to L^p(M; T^*M)$ for $1 < p < n$ and unbounded for $p \geq n$ if there is more than one end. It follows from known results that in such a case the Riesz transform on $M$ is bounded for $1 < p \leq 2$ and unbounded for $p > n$; the result is new for $2 < p \leq n$. We also give some heat kernel estimates on such manifolds…

Minimal sequences and the Kadison-Singer problem

The Kadison-Singer problem asks: does every pure state on the diagonal sublgebra of the C*-algebra of bounded operators on a separable infinite dimensional Hilbert space admit a unique extension? A yes answer is equivalent to several open conjectures including Feichtinger's: every bounded frame is a finite union of Riesz sequences. We consider the special case: Feichtinger's conjecture for exponentials and prove that the set of projections onto a measurable subset of the circle group of the set of exponential functions equals a union of a finite number of Reisz sequences if and only if there e…

Frame spectral pairs and exponential bases

Given a domain $Ω\subset\Bbb R^d$ with positive and finite Lebesgue measure and a discrete set $Λ\subset \Bbb R^d$, we say that $(Ω, Λ)$ is a {\it frame spectral pair} if the set of exponential functions $\mathcal E(Λ):=\{e^{2πi λ\cdot x}: λ\in Λ\}$ is a frame for $L^2(Ω)$. Special cases of frames include Riesz bases and orthogonal bases. In the finite setting $\Bbb Z_N^d$, $d, N\geq 1$, a frame spectral pair can be similarly defined. %(Here, $\Bbb Z_N$ is the cyclic abelian group of order.) We show how to construct and obtain new classes of frame spectral pairs in $\Bbb R^d$ by "adding" frame…

关于有界四元数算子的Browder S-谱的一个注记

A note on the Browder S-spectrum of a bounded quaternionic operator

本文建立了有界四元数算子Browder S-谱的紧扰动交集刻画，并证明了有限秩限制及逐点稳定性结果。