arXiv subjects

Daniel Freeman

Publications and source records attributed to Daniel Freeman.

30 records · Page 2Linked to original sources

arXiv2020

A Schauder basis for $L_2$ consisting of non-negative functions

We prove that $L_2(\mathbb{R})$ contains a Schauder basis of non-negative functions. Similarly, $L_p(\mathbb{R})$ contains a Schauder basic sequence of non-negative functions such that $L_p(\mathbb{R})$ embeds into the closed span of the sequence. We prove as well that if $X$ is a separable Banach space with the bounded approximation property, then any set in $X$ with dense span contains a quasi-basis (Schauder frame) for $X$. Furthermore, if $X$ is a separable Banach lattice with a bibasis then any set in $X$ with dense span contains a u-frame.

math.FA↗

arXiv2018

Continuous Schauder frames for Banach spaces

We introduce the notion of a continuous Schauder frame for a Banach space. This is both a generalization of continuous frames and coherent states for Hilbert spaces and a generalization of unconditional Schauder frames for Banach spaces. As a natural example, we prove that any wavelet for $L_p(\R)$ with $1<p<\infty$ generates a continuous wavelet Schauder frame. Furthermore, we generalize the properties shrinking and boundedly complete to the continuous Schauder frame setting, and prove that many of the fundamental James theorems still hold in this general context.

math.FA↗

arXiv2017

Intelligent Perioperative System: Towards Real-time Big Data Analytics in Surgery Risk Assessment

Surgery risk assessment is an effective tool for physicians to manage the treatment of patients, but most current research projects fall short in providing a comprehensive platform to evaluate the patients' surgery risk in terms of different complications. The recent evolution of big data analysis techniques makes it possible to develop a real-time platform to dynamically analyze the surgery risk from large-scale patients information. In this paper, we propose the Intelligent Perioperative System (IPS), a real-time system that assesses the risk of postoperative complications (PC) and dynamically interacts with physicians to improve the predictive results. In order to process large volume patients data in real-time, we design the system by integrating several big data computing and storage frameworks with the high through-output streaming data processing components. We also implement a system prototype along with the visualization results to show the feasibility of system design.

cs.SE↗

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↗

arXiv2015

Weaving Schauder frames

We extend the concept of weaving Hilbert space frames to the Banach space setting. Similar to frames in a Hilbert space, we show that for any two approximate Schauder frames for a Banach space, every weaving is an approximate Schauder frame if and only if there is a uniform constant $C\geq 1$ such that every weaving is a $C$-approximate Schauder frame. We also study weaving Schauder bases, where it is necessary to introduce two notions of weaving. On one hand, we can ask if two Schauder bases are woven when considered as Schauder frames with their biorthogonal functionals, and alternatively, we can ask if each weaving of two Schauder bases remains a Schauder basis. We will prove that these two notions coincide when all weavings are unconditional, but otherwise they can be different. Lastly, we prove two perturbation theorems for approximate Schauder frames.

math.FA↗

arXiv2015

Classes of operators determined by ordinal indices

We introduce and study the Bourgain index of an operator between two Banach spaces. In particular, we study the Bourgain $\ell_p$ and $c_0$ indices of an operator. Several estimates for finite and infinite direct sums are established. We define classes determined by these indices and show that some of these classes form operator ideals. We characterize the ordinals which occur as the index of an operator and establish exactly when the defined classes are closed. We study associated indices for non-preservation of $\ell_p^ξ$ and $c_0^ξ$ spreading models and indices characterizing weak compactness of operators between separable Banach spaces. We also show that some of these classes are operator ideals and discuss closedness and distinctness of these classes.

math.FA↗

arXiv2014

The stabilized set of $p$'s in Krivine's theorem can be disconnected

For any closed subset $F$ of $[1,\infty]$ which is either finite or consists of the elements of an increasing sequence and its limit, a reflexive Banach space $X$ with a 1-unconditional basis is constructed so that in each block subspace $Y$ of $X$, $\ell_p$ is finitely block represented in $Y$ if and only if $p \in F$. In particular, this solves the question as to whether the stabilized Krivine set for a Banach space had to be connected. We also prove that for every infinite dimensional subspace $Y$ of $X$ there is a dense subset $G$ of $F$ such that the spreading models admitted by $Y$ are exactly the $\ell_p$ for $p\in G$.

math.FA↗

arXiv2013

Uniformly factoring weakly compact operators

Let $X$ and $Y$ be separable Banach spaces. Suppose $Y$ either has a shrinking basis or $Y$ is isomorphic to $C(2^\mathbb{N})$ and $A$ is a subset of weakly compact operators from $X$ to $Y$ which is analytic in the strong operator topology. We prove that there is a reflexive space with a basis $Z$ such that every $T \in A$ factors through $Z$. Likewise, we prove that if $A \subset L(X, C(2^\mathbb{N}))$ is a set of operators whose adjoints have separable range and is analytic in the strong operator topology then there is a Banach space $Z$ with separable dual such that every $T \in A$ factors through $Z$. Finally we prove a uniformly version of this result in which we allow the domain and range spaces to vary.

math.FA↗

arXiv2012

Moving finite unit tight frames for $S^n$

Frames for $\R^n$ can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word "frame" has a different meaning in the context of differential geometry and topology. A moving frame for the tangent bundle of a smooth manifold is a basis for the tangent space at each point which varies smoothly over the manifold. It is well known that the only spheres with a moving basis for their tangent bundle are $S^1$, $S^3$, and $S^7$. On the other hand, after combining the two separate meanings of the word "frame", we show that the $n$-dimensional sphere, $S^n$, has a moving finite unit tight frame for its tangent bundle if and only if $n$ is odd. We give a procedure for creating vector fields on $S^{2n-1}$ for all $n\in\N$, and we characterize exactly when sets of such vector fields form a moving finite unit tight frame.

math.FA↗

arXiv2012

Upper and lower estimates for schauder frames and atomic decompositions

We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite dimensional decompositions of Banach spaces can be extended and modified to Schauder frames. We show as well that if a separable infinite dimensional Banach space has a Schauder frame, then it also has a Schauder frame which is not shrinking.

math.FA↗

arXiv2010

The universality of $\ell_1$ as a dual space

Let $X$ be a Banach space with a separable dual. We prove that $X$ embeds isomorphically into a $\cL_\infty$ space $Z$ whose dual is isomorphic to $\ell_1$. If, moreover, $U$ is a space so that $U$ and $X$ are totally incomparable, then we construct such a $Z$, so that $Z$ and $U$ are totally incomparable. If $X$ is separable and reflexive, we show that $Z$ can be made to be somewhat reflexive.

math.FA↗

arXiv2007

Weakly null sequences with upper estimates

We prove that if $(v_i)$ is a normalized basic sequence and X is a Banach space such that every normalized weakly null sequence in X has a subsequence that is dominated by $(v_i)$, then there exists a uniform constant $C\geq1$ such that every normalized weakly null sequence in X has a subsequence that is C-dominated by $(v_i)$. This extends a result of Knaust and Odell, who proved this for the cases in which $(v_i)$ is the standard basis for $\ell_p$ or $c_0$.

math.FA↗