arXiv subjects

Christopher Heil

Publications and source records attributed to Christopher Heil.

4 recordsLinked to original sources

arXiv2023

Overcomplete Reproducing Pairs

The Gaussian Gabor system at the critical density has the property that it is overcomplete in $L^2(\mathbf{R})$ by exactly one element, and if any single element is removed then the resulting system is complete but is not a Schauder basis. This paper characterizes systems that are overcomplete by finitely many elements. Among other results, it is shown that if such a system has a reproducing partner, then it contains a Schauder basis. While a Schauder basis provides a strong reproducing property for elements of a space, the existence of a reproducing partner only requires a weak type of representation of elements. Thus for these systems weak representations imply strong representations. The results are applied to systems of weighted exponentials and to Gabor systems at the critical density. In particular, it is shown that the Gaussian Gabor system does not possess a reproducing partner.

math.FA↗

arXiv2023

$\ell^1$-Bounded Sets

A subset $M$ of a separable Hilbert space $H$ is $\ell^1$-bounded if there exists a Riesz basis $\mathcal{F} = \{e_n\}_{n \in \mathbb{N}}$ for $H$ such that $\sup_{x \in M} \sum_{n \in \mathbb{N}} |\langle x, e_n\rangle| < \infty.$ A similar definition for $\ell^1$-frame-bounded sets is made by replacing Riesz bases with frames. This paper derives properties of $\ell^1$-bounded sets, operations on the collection of $\ell^1$-bounded sets, and the relation between $\ell^1$-boundedness and $\ell^1$-frame-boundedness. Some open problems are stated, several of which have intriguing implications.

math.FA↗

arXiv2022

Convergence of frame series

If $\{x_n\}_{n \in \mathbb{N}}$ is a frame for a Hilbert space $H,$ then there exists a canonical dual frame $\{\tilde{x_n}\}_{n \in \mathbb{N}}$ such that for every $x \in H$ we have $x = \sum \langle x, \tilde{x_n} \rangle \, x_n,$ with unconditional convergence of this series. However, if the frame is not a Riesz basis, then there exist alternative duals $\{y_n\}_{n \in \mathbb{N}}$ and synthesis-pseudo duals $\{z_n\}_{n \in \mathbb{N}}$ such that $x = \sum \langle x, y_n \rangle \, x_n,$ and $x = \sum \langle x, x_n \rangle \, z_n,$ for every $x.$ We characterize the frames for which the frame series ($x = \sum \langle x, y_n \rangle \, x_n,$) converges unconditionally for every $x$ for every alternative dual, and similarly for synthesis-pseudo duals. In particular, we prove that if $\{x_n\}_{n \in \mathbb{N}}$ does not contain infinitely many zeros then the frame series converge unconditionally for every alternative dual (or synthesis-pseudo dual) if and only if $\{x_n\}_{n \in \mathbb{N}}$ is a near-Riesz basis. We also prove that all alternative duals and synthesis-pseudo duals have the same excess as their associated frame.

math.CA↗

arXiv2008

Invariance of a Shift-Invariant Space

A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. This paper characterizes those shift-invariant subspaces S that are also invariant under additional (non-integer) translations. For the case of finitely generated spaces, these spaces are characterized in terms of the generators of the space. As a consequence, it is shown that principal shift-invariant spaces with a compactly supported generator cannot be invariant under any non-integer translations.

math.FA↗