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Darrin Speegle

Publications and source records attributed to Darrin Speegle.

13 recordsLinked to original sources
arXiv2026

From Fractional to Set Tilings for Pairs of Lattices

We generalize a theorem of Isbell asserting that every countably infinite doubly stochastic matrix has a positive generalized diagonal. As an application, we prove a support-reduction theorem for simultaneous lattice tilings. Namely, if a nonnegative measurable function bounded above by one tiles Euclidean space by translations along two full-rank lattices with integer multiplicities, then its pointwise support contains a possibly nonmeasurable set whose indicator function satisfies the same two tiling identities. The proof reduces the problem on each orbit of the group generated by the two lattices to an infinite matrix rounding theorem with integer row and column margins. This matrix theorem gives a \(0\)-\(1\) matrix with prescribed integer margins and support contained in the support of the original matrix. The result is motivated by simultaneous tiling questions arising in harmonic analysis and wavelet-set constructions.

math.CA↗
arXiv2025

Are MSF wavelets minimally supported?

Larson's problem asks ``Must the support of the Fourier transform of a wavelet contain a wavelet set?". We give an affirmative answer to a non-measurable variant of this question by proving that the Fourier transform of a wavelet must contain a possibly non-measurable wavelet set. We also provide background results on Larson's problem and propose two new related problems.

math.CA↗
arXiv2025

Strictly Expansive Matrices

If $A$ is an integer valued, strictly expansive matrix, then there exists an orthonormal $A$-wavelet whose Fourier transform is compactly supported and smooth. We show that strongly connected diagonally dominant integer matrices are strictly expansive, and that integer matrices with determinant two are not strictly expansive with respect to particularly nice sets.

math.CA↗
arXiv2021

Simultaneous dilation and translation tilings of $\mathbb R^n$

We solve the wavelet set existence problem. That is, we characterize the full-rank lattices $Γ\subset \mathbb R^n$ and invertible $n \times n$ matrices $A$ for which there exists a measurable set $W$ such that $\{W + γ: γ\in Γ\}$ and $\{A^j(W): j\in \mathbb Z\}$ are tilings of $\mathbb R^n$. The characterization is a non-obvious generalization of the one found by Ionascu and Wang, which solved the problem in the case $n = 2$. As an application of our condition and a theorem of Margulis, we also strengthen a result of Dai, Larson, and the second author on the existence of wavelet sets by showing that wavelet sets exist for matrix dilations, all of whose eigenvalues $λ$ satisfy $|λ| \ge 1$. As another application, we show that the Ionascu-Wang characterization characterizes those dilations whose product of two smallest eigenvalues in absolute value is $\ge 1$.

math.CA↗
arXiv2017

The discretization problem for continuous frames

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 Hilbert space may be sampled to obtain a frame.

math.FA↗
arXiv2016

Improved bounds in Weaver and Feichtinger Conjectures

We sharpen the constant in the $KS_2$ conjecture of Weaver \cite{We}, which was validated by Marcus, Spielman, and Srivastava \cite{MSS} in their solution of the Kadison--Singer problem. We then apply this result to prove optimal asymptotic bounds on the size of partitions in the Feichtinger conjecture.

math.FA↗
arXiv2010

Orthonormal dilations of non-tight frames

We establish dilation theorems for non-tight frames with additional structure, i.e., frames generated by unitary groups of operators and projective unitary representations. This generalizes previous dilation results for Parseval frames due to Han and Larson and Gabardo and Han. We also extend the dilation theorem for Parseval wavelets, due to Dutkay, Han, Picioroaga, and Sun , by identifying the optimal class of frame wavelets for which dilation into an orthonormal wavelet is possible.

math.FA↗
arXiv2010

Spanning and independence properties of frame partitions

We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that in finite dimensional Hilbert spaces, Parseval frames with norms bounded away from 1 can be decomposed into a number of sets whose complements are spanning, where the number of these sets only depends on the norm bound. We also prove, assuming the Kadison-Singer conjecture is true, that this holds for infinite dimensional Hilbert spaces. Further, we prove a stronger result for Parseval frames whose norms are uniformly small, which shows that in addition to the spanning property, the sets can be chosen to be independent, and the complement of each set to contain a number of disjoint, spanning sets.

math.FA↗
arXiv2008

Uniform partitions of frames of exponentials into Riesz sequences

The Feichtinger Conjecture, if true, would have as a corollary that for each set $E\subset \T$ and $Λ\subset \Z$, there is a partition $Λ_1,...,Λ_N$ of $\Z$ such that for each $1\le i \le N$, $\{\exp(2πi xλ): λ\in Λ_i\}$ is a Riesz sequence. In this paper, sufficient conditions on sets $E\subset \T$ and $Λ\subset \R$ are given so that $\{\exp(2πi xλ) 1_E: λ\in Λ\}$ can be uniformly partitioned into Riesz sequences.

math.FA↗
arXiv2007

A Decomposition Theorem for frames and the Feichtinger Conjecture

In this paper we study the Feichtinger Conjecture in frame theory, which was recently shown to be equivalent to the 1959 Kadison-Singer Problem in $C^{*}$-Algebras. We will show that every bounded Bessel sequence can be decomposed into two subsets each of which is an arbitrarily small perturbation of a sequence with a finite orthogonal decomposition. This construction is then used to answer two open problems concerning the Feichtinger Conjecture: 1. The Feichtinger Conjecture is equivalent to the conjecture that every unit norm Bessel sequence is a finite union of frame sequences. 2. Every unit norm Bessel sequence is a finite union of sets each of which is $ω$-independent for $\ell_2$-sequences.

math.FA↗
arXiv2006

Explicit cross-sections of singly generated group actions

We consider two classes of actions on $\mathbb{R}^n$ - one continuous and one discrete. For matrices of the form $A = e^B$ with $B \in M_n(\R)$, we consider the action given by $γ\to γA^t$. We characterize the matrices $A$ for which there is a cross-section for this action. The discrete action we consider is given by $γ\to γA^k$, where $A\in GL_n(\R)$. We characterize the matrices $A$ for which there exists a cross-section for this action as well. We also characterize those $A$ for which there exist special types of cross-sections; namely, bounded cross-sections and finite measure cross-sections. Explicit examples of cross-sections are provided for each of the cases in which cross-sections exist. Finally, these explicit cross-sections are used to characterize those matrices for which there exist MSF wavelets with infinitely many wavelet functions. Along the way, we generalize a well-known aspect of the theory of shift-invariant spaces to shift-invariant spaces with infinitely many generators.

math.FA↗
arXiv2003

Groups, Wavelets, and Wavelet Sets

Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. This article gives an overview over some well known results about the continuous and discrete wavelet transforms and groups acting on $\mathbb{R}^n$. We also show how this action can give rise to wavelets, and in particular, MSF wavelets)in $L^2(\mathbb{R}^n)$.

math.FA↗