arXiv subjects

Pu-Ting Yu

Publications and source records attributed to Pu-Ting Yu.

10 recordsLinked to original sources

arXiv2026

Woven weighted exponentials

Let $f$ and $g$ be nonzero functions in $L^2([0,1])$. The \emph{woven weighted exponential system} (associated with $f$ and $g$) is defined by $$\Wc(f,g)=\bigset{\set{fe^{2\pi i nt}}_{n\in J} \cup \set{ge^{2\pi i nt}}_{n\in J^c}\,|\,J\subset\Z}.$$ We say that $\Wc(f,g)$ is \emph{wovenly complete}, (resp. \emph{wovenly minimal}, a \emph{woven frame}) if the weaving $\set{fe^{2\pi i nt}}_{n\in J} \cup \set{ge^{2\pi i nt}}_{n\in J^c}$ is complete, (resp. minimal, a frame) for all $J\subseteq \Z.$ In this paper, we study conditions that imply certain approximation properties of $\Wc(f,g)$, such as completeness, minimality and the frame property. We first provide a complete characterization of the woven weighted exponential systems that are wovenly complete. We also show that $\Wc(f,g)$ is a woven frame if $f/g$ is strictly positive or strictly negative over $[0,1].$ Additionally, several counterexamples are provided to show that certain seemingly correct conditions do not imply the desired approximation properties of $\Wc(f,g).$ All results presented in this paper apply equivalently to systems of regular translates and Gabor systems at critical density in $L^2(\R)$.

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arXiv2026

Uniform discretization of continuous frames

Let $H$ be an infinite-dimensional separable Hilbert space and let $(X,d,\mu)$ be a metric measure space satisfying the doubling and upper Alhfors regularity conditions at small scale. We prove that every bounded continuous tight frame $\Psi\colon X\rightarrow H$ can be sampled to obtain a frame for $H$, which is uniformly discrete and nearly tight. That is, for every $0<\epsilon<1$, there exist a sampling sequence $\{x_n\}_{n\in\mathbb{N}}$ in $X$ and $r>0$ such that $\inf_{n\neq m}d(x_n,x_m)\geq r$ and $\{\Psi(x_n)\}_{n\in\mathbb{N}}$ is a frame whose ratio of frame bounds is less than $1+\epsilon$. We apply our main result to show that for every nonzero function $g$ in $L^2(\mathbb{R}^d)$ there exists a uniformly discrete set $\Lambda$ such that the corresponding Gabor system $\{e^{2\pi ibx}g(x-a)\}_{(a,b)\in \Lambda}$ is a nearly tight frame. We also prove that if $\psi\in L^2(\mathbb{R})$ satisfies the Calder\'on admissibility condition, then there exists a uniformly discrete set $\Gamma$ such that wavelet system $\{a^{1/2}\psi(ax-b)\}_{(a,b)\in \Gamma}$ is a nearly tight frame. Analogous discretization results for exponential frames and spectral subspaces of elliptic differential operators are presented as well.

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arXiv2026

Every semi-normalized unconditional Schauder frame in Hilbert spaces contains a frame

Let $H$ be an infinite-dimensional Hilbert space. We prove that every unconditional Schauder frame for $H$ contains a subsequence that can be normalized to form a frame for $H$. As a consequence, every semi-normalized unconditional Schauder frame contains a frame for $H.$ Here we say that a sequence $\{x_n\}_{n\in \mathbb{N}}$ in a Hilbert space $H$ is an \emph{unconditional Schauder frame} for $H$ if there exists some sequence $\{y_n\}_{n\in \mathbb{R}}subseteq H$ such that $$x=\sum_{n=1}^\infty \langle x,y_n\rangle x_n\quad \text{for all }x\in H,$$ with the unconditional convergence of the series in the norm of $H.$ We say that $\{x_n\}_{n\in\mathbb{N}}$ is semi-normalized if $m\leq \|x_n\|\leq M$ for all $n\in \mathbb{N}$ for some positive constants $m,M.$ We then apply our main results to answer several open questions concerning the existence of certain unconditional Schauderf frames. For example, we prove that if a closed subspace of $L^2(\mathbb{R}^d)$ contains $\{e^{2\pi ib\cdot x}g\}_{b\in \Lambda}$ for some infinite uniformly discrete subset $\Lambda$ of $\mathbb{R}^d$ and some nonzero function $g$ in the Feichtinger algebra, it does not admit any unconditional Schauder frames of translates with finitely many generators. We will also show that no Gabor system with the critical lower Beurling density can be an unconditional Schauder frame when the window function belongs to the Feichtinger algebra. Furthermore, we present an example of a compact set of $\mathbb{R}$ which does not admit any unconditional Schauder frames of exponentials with the critical lower Beurling density. All results in this paper apply equivalently to sequences that can be rescaled to form a frame for $H.$

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arXiv2025

Spectrum of normal operators that generate certain scalable iterative systems

Let $A\colon H\rightarrow H$ be a normal operator on an infinite-dimensional separable Hilbert space $H$ and let $S\subseteq H$ be a finite subset such that $\{A^nx\}_{n\geq 0,\,x\in S}$ can be rescaled to form a frame for $H$. That is, there exist some subsets $J_x\subseteq \mathbb{N}\cup\{0\}$ and some set of nonzero scalars $(c_{n,x})_{n\in J_x,\,x\in S}$ such that $\{c_{n,x}A^nx\}_{n\in J_x,\,x\in S}$ forms a frame for $H.$ Assume that there exist some $\eta\in\mathbb{N}$ and $\delta>0$ such that for each infinite $J_x$ there is an increasing syndetic subsequence $(n^x_{k})_{k\in \mathbb{N}}\subseteq J_x$ satisfying $|c_{n^x_{k},x}|\|A^{i^x_{k}}x\|\geq \delta$ for some non-negative integers $i^x_{k}$ with $|i^x_{k}- n^x_{k}|\leq \eta$ for all $k\in \mathbb{N}$. We prove that there exist finitely many numbers $(r_i)_{i=1}^N$ such that the continuous spectrum of $A$ is concentrated on arcs of a circle centered at origin with radius $r_i$. In particular, $A$ must be a diagonal operator if $S$ is a singleton. As an application, we establish the conjecture proposed by Aldroubi et al.\ asserting that the iterative system $\{\frac{A^nx}{\|A^nx\|}\}_{n\geq 0,\,x\in S}$ is never a frame for $H$, provided one of the following two conditions holds: (i) The continuous spectrum of $A$ contains more than $|S|-1$ points with distinct moduli; (ii) $S$ is a singleton and $A$ is not a diagonal operator

math.FA↗

arXiv2025

Operations that are incompatible with certain systems of translates in $L^2(\mathbb{R})$

We say that closed subspace $M$ of $L^2(\R)$ admits a \emph{complete set of semi-regular a-translates} if there exist some $a>0$, finitely many functions $g_1,\dots,g_N$, some subsets $J_1,\dots,J_N$ of $\Z$ and some finite subsets $\set{\al_{1j}}_{j=1}^{K_1},\dots,\set{\al_{Nj}}_{j=1}^{K_N}$ of $\R$ such that $$M=\clspan{\bigset{g_i(\cdot-ak), ~g_i(\cdot-\al_{ij})\,|\,k\in J_i,1\leq j\leq K_i\,}}_{i=1}^N.$$ Here $\cdot$ denotes a generic variable. In the first half of this paper, we study whether the properties of being closed under modulation, dilation, reflection or Fourier transform is compatible with the existence of a complete set of semi-regular $a$-translates in closed subspaces of $L^2(\R)$. Specifically, we prove that a closed subspace of $L^2(\R)$ does not admit a complete set of semi-regular $a$-translates if it is closed under modulation or if it is closed under dilation with respect to a scaling factor $b$ satisfying $|b|>1.$ We also show that no infinite-dimensional closed subspace of $L^2(\R)$ can simultaneously be closed under Fourier transform and admit a complete set of semi-regular $a$-translates with $a^2\in \Q$, whereas for any $a>0$, there do exist closed subspaces that are closed under reflection and admit a complete set of semi-regular $a$-translates. In the second half of this paper, we prove that a closed subspace of $L^2(\R)$ does not admit a frame formed by a system of translates if it contains a closed subspace that is closed under modulation and contains a nonzero function in $M^1(\R)$. In addition, we present related results concerning the incompatibility between being closed under Fourier transform and the existence of frames or Schauder bases of translates in closed subspaces of $L^2(\R)$. All results in this half can be extended to $L^2(\R^d)$ for any $d>1.$

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arXiv2025

Existence of Unconditional Frames Formed By System of Translates in Modulation Spaces

Let $1\leq p\leq 2$ and let $\Lambda = \{\lambda_n\}_{n\in \mathbb{N}} \subseteq \mathbb{R}$ be an arbitrary subset. We prove that for any $g\in M^p(\mathbb{R})$ with $1\leq p\leq 2$ the system of translates $\{g(x-\lambda_n)\}_{n\in \mathbb{N}}$ is never an unconditional basis for $M^q(\mathbb{R})$ for $p\leq q\leq p'$, where $p'$ is the conjugate exponent of $p.$ In particular, $M^1(\mathbb{R})$ does not admit any Schauder basis formed by a system of translates. We also prove that for any $g\in M^p(\mathbb{R})$ with $1< p\leq 2$ the system of translates $\{g(x-\lambda_n)\}_{n\in \mathbb{N}}$ is never an unconditional frame for $M^p(\mathbb{R}).$ Several results regarding the existence of unconditional frames formed by a system of translates in $M^1(\mathbb{R})$ as well as in $M^p(\mathbb{R})$ with $2<p<\infty$ will be presented as well.

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arXiv2024

Gabor frames with atoms in M^q(R) but not in M^p(R) for any 1\leq p < q \leq 2

This paper consists of two parts. In the first half, we solve the question raised by Heil as to whether the atom of a Gabor frame must be in $M^p(\mathbb{R})$ for some $1<p<2$. Specifically, for each $0<\alpha \beta \leq 1$ and $1<q\leq 2$ we explicitly construct Gabor frames $\mathcal{G}(g,\alpha,\beta)$ with atoms in $M^q(\mathbb{R})$ but not in $M^{p}(\mathbb{R})$ for any $1\leq p<q$. To construct such Gabor frames, we use box functions as the window functions and show that $$f = \sum_{k,n\in \mathbb{Z}} \langle f,M_{\beta n}T_{\alpha k} \mathcal{F}(\chi_{[0,\alpha]})\rangle M_{\beta n}T_{\alpha k} ( \mathcal{F}(\chi_{[0,\alpha]}))$$ holds for $f\in M^{p,q}(\mathbb{R})$ with unconditional convergence of the series for any $0<\alpha\beta \leq 1$, $1<p<\infty$ and $1\leq q<\infty$. In the second half of this paper, we study two questions related to unconditional convergence of Gabor expansions in modulation spaces. Under the assumption that the window functions are chosen from $M^p(\mathbb{R})$ for some $1\leq p\leq 2,$ we will prove several equivalent statements that the equation $f = \sum_{k,n\in \mathbb{Z}} \langle f, M_{\beta n}T_{\alpha k} \gamma \rangle M_{\beta n}T_{\alpha k} g$ can be extended from $L^2(\mathbb{R})$ to $M^q(\mathbb{R})$ for all $f\in M^q(\mathbb{R})$ and all $p\leq q\leq p'$ with unconditional convergence of the series. Finally, we characterize all Gabor systems $\{M_{\beta n}T_{\alpha k}g\}_{n,k\in \mathbb{Z}}$ in $M^{p,q}(\mathbb{R})$ for any $1\leq p,q<\infty$ for which $f = \sum \langle f, \gamma_{k,n} \rangle M_{\beta n}T_{\alpha k} g$ with unconditional convergence of the series for all $f$ in $M^{p,q}(\mathbb{R})$ and all alternative duals $\{\gamma_{k,n}\}_{k,n\in \mathbb{Z}}$ of $\{M_{\beta n}T_{\alpha k} g\}_{n,k\in \mathbb{Z}}$.

math.FA↗

arXiv2023

Frame-normalizable Sequences

Let $H$ be a separable Hilbert space and let $\{x_n\}$ be a sequence in $H$ that does not contain any zero elements. We say that $\{x_n\}$ is a \emph{Bessel-normalizable} or \emph{frame-normalizable} sequence if the normalized sequence $\{\frac{x_n}{\|x_n\|}\}$ is a Bessel sequence or a frame for $H$, respectively. In this paper, several necessary and sufficient conditions for sequences to be frame-normalizable and not frame-normalizable are proved. Perturbation theorems for frame-normalizable sequences are also proved. As applications, we show that the Balazs-Stoeva conjecture %\cite{BS11} holds for Bessel-normalizable sequences. Finally, we apply our results to partially answer the open question raised by Aldroubi et al.\ %\cite{ACMCP16} as to whether the iterative system $\{\frac{A^n x}{\|A^nx\|}\}_{n\geq 0,\, x\in S}$ associated with a normal operator $A\colon H\rightarrow H$ and a countable subset $S$ of $H$, is a frame for $H$. In particular, if $S$ is finite, then we are able to show that $\{\frac{A^n x}{\|A^nx\|}\}_{n\geq 0,\, x\in S}$ is not a frame for $H$ whenever $\{A^nx\}_{n\geq 0,\,x\in S}$ is a frame for $H$.

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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.

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