Riesz basis of exponentials for a union of cubes in R^{d}

多重铺砌测度的指数 Riesz 基

Riesz bases of exponentials for multi-tiling measures

本文研究由具有指数 Riesz 基的测度平移拼接而成的多重铺砌测度，给出其容许指数 Riesz 基的充要条件，并据此证明正方形边界不存在一类自然候选形式的指数 Riesz 基。

Riesz Bases of Root Vectors of Indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions, I

We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigenparameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space.

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 $L^2(S)$.

Three problems on exponential bases

We consider three special and significant cases of the following problem. Let D be a (possibly unbounded) set of finite Lebesgue measure in R^d. Find conditions on D for which the standard exponential basis on the unit cube of R^d is a frame, a Riesz sequence, or a Riesz basis on L^2(D).

Riesz basis for strongly continuous groups

Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a Riesz basis (or unconditional basis) of the Hilbert space. Furthermore, we show that none of the conditions can be weakened. However, if the eigenvalues (counted with multiplicity) can be grouped into subsets of at most $K$ elements, and the distance between the groups is (uniformly) bounded away from zero, then the spectral projections associated to the gr…

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…

区间并集上指数基的示例

Examples of exponential bases on Union of intervals

本研究构造了满足间隙条件、总长为1的线段并集及$\text{R}^d$中部分立方体并集的显式指数基，还证明了推广Kadec $\frac{1}{4}$定理的线段并集稳定性结果。

Spectral properties of some unions of linear spaces

We consider \textit{additive spaces}, consisting of two intervals of unit length or two general probability measures on ${\mathbb R}^1$, positioned on the axes in ${\mathbb R}^2$, with a natural additive measure $ρ$. We study the relationship between the exponential frames, Riesz bases, and orthonormal bases of $L^2(ρ)$ and those of its component spaces. We find that the existence of exponential bases depends strongly on how we position our measures on ${\mathbb R}^1$. We show that non-overlapping additive spaces possess Riesz bases, and we give a necessary condition for overlapping spaces. We…

Exponential bases on two dimensional trapezoids

We discuss existence and stability of Riesz bases of exponential type of L^2(T) for special domains T called trapezoids. We construct exponential bases on L^2(T) when T is a finite union of rectangles with the same height. We also generalize our main theorems in dimension d\ge 3.

Syndetic Sets, Paving, and the Feichtinger Conjecture

We prove that if a Bessel sequence in a Hilbert space, that is indexed by a countably infinite group in an invariant manner, can be partitioned into finitely many Riesz basic sequences, then each of the sets in the partition can be chosen to be syndetic. We then apply this result to prove that if a Fourier frame for a measurable subset of a higher dimensional cube can be partitioned into Riesz basic sequences, then each subset can be chosen to be a syndetic subset of the corresponding higher dimensional integer lattice. Both of these results follow from a result about syndetic pavings of eleme…

Fractional Multiresolution Analysis and Associated Scaling Functions in $L^2(\mathbb R)$

In this paper, we show how to construct an orthonormal basis from Riesz basis by assuming that the fractional translates of a single function in the core subspace of the fractional multiresolution analysis form a Riesz basis instead of an orthonormal basis. In the definition of fractional multiresolution analysis, we show that the intersection triviality condition follows from the other conditions. Furthermore, we show that the union density condition also follows under the assumption that the fractional Fourier transform of the scaling function is continuous at $0$. At the culmination, we pro…

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…