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Stephen J. Dilworth

Publications and source records attributed to Stephen J. Dilworth.

18 recordsLinked to original sources
arXiv2025

Summability Methods for the Greedy Algorithm in Banach spaces

For the past 25 years, one of the most studied algorithms in the field of Nonlinear Approximation Theory has been the Thresholding Greedy Algorithm. In this paper, we propose new summability methods for this algorithm, generating two new types of greedy-like bases - namely Cesàro quasi-greedy and de la Vallée-Poussin-quasi-greedy bases. We analyze the connection between these types of bases and the well-known quasi-greedy bases, and leave some open problems for future research. In addition, as a consequence of our techniques for handling these summability methods, we answer a question posed by P. Wojtaszczyk in [16], by giving a categorial proof of equivalence between the uniform boundedness of the greedy sums and the convergence of the thresholding greedy algorithm.

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arXiv2023

Cycle spaces: invariant projections and applications to transportation cost

The paper starts with discussion of applications of cycle spaces to transportation cost. After a short survey of the known results on cycle spaces, we turn to the study of minimal projections onto cycle spaces in the corresponding $\ell_1$-spaces. This study is naturally related to the study of invariant projections on the cycle space, which, in turn, are determined by the properties of representations of the automorphism group of the corresponding graph. The main focus is on discrete tori and Hamming graphs.

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arXiv2023

New characterizations of the unit vector basis of $c_0$ or $\ell_p$

Motivated by Altshuler's famous characterization of the unit vector basis of $c_0$ or $\ell_p$ among symmetric bases, we obtain similar characterizations among democratic bases and among bidemocratic bases. We also prove a separate characterization of the unit vector basis of $\ell_1$.

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arXiv2021

On uniqueness and plentitude of subsymmetric sequences

We explore the diversity of subsymmetric basic sequences in spaces with a subsymmetric basis. We prove that the subsymmetrization $Su(T^*)$ of Tsirelson's original Banach space provides the first known example of a space with a unique subsymmetric basic sequence that is additionally non-symmetric. Contrastingly, we provide a criterion for a space with a subsymmetric basis to contain a continuum of nonequivalent subsymmetric basic sequences and apply it to $Su(T^*)^*$. Finally, we provide a criterion for a subsymmetric sequence to be equivalent to the unit vector basis of some $\ell_p$ or $c_0$.

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arXiv2021

Analysis on Laakso graphs with application to the structure of transportation cost spaces

This article is a continuation of our article in [Canad. J. Math. Vol. 72 (3), (2020), pp. 774--804]. We construct orthogonal bases of the cycle and cut spaces of the Laakso graph $\mathcal{L}_n$. They are used to analyze projections from the edge space onto the cycle space and to obtain reasonably sharp estimates of the projection constant of $\operatorname{Lip}_0(\mathcal{L}_n)$, the space of Lipschitz functions on $\mathcal{L}_n$. We deduce that the Banach-Mazur distance from TC$(\mathcal{L}_n)$, the transportation cost space of $\mathcal{L}_n$, to $\ell_1^N$ of the same dimension is at least $(3n-5)/8$, which is the analogue of a result from [op. cit.] for the diamond graph $D_n$. We calculate the exact projection constants of $\operatorname{Lip}_0(D_{n,k})$, where $D_{n,k}$ is the diamond graph of branching $k$. We also provide simple examples of finite metric spaces, transportation cost spaces on which contain $\ell_\infty^3$ and $\ell_\infty^4$ isometrically.

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arXiv2020

Higher projective tensor products of $c_0$

Let $m,n$ be positive integers with $m<n$. Under certain assumptions on the Banach space $X$, we prove that the $n$-fold projective tensor product of $X$, $\widehat{\otimes}^n_πX$, is not isomorphic to any subspace of any quotient of the $m$-fold projective tensor product, $\widehat{\otimes}_π^m X$. In particular, we prove that $\widehat{\otimes}^n_πc_0$ is not isomorphic to any subspace of any quotient of $\widehat{\otimes}_π^m c_0$.

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arXiv2020

A dichotomy for subsymmetric basic sequences with applications to Garling spaces

Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positioning and develop new tools which lead to a dichotomy theorem that holds for general spaces with subsymmetric bases. As an illustration of how to use this dichotomy theorem we obtain the classification of all subsymmetric sequences in certain types of spaces. To be more specific, we show that Garling sequence spaces have a unique symmetric basic sequence but no symmetric basis and that these spaces have a continuum of subsymmetric basic sequences.

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arXiv2018

Non-superreflexivity of Garling sequence spaces and applications to the existence of special types of conditional bases

In this paper we settle in the negative the problem of the superreflexivity of Garling sequence spaces by showing that they contain a complemented subspace isomorphic to a non superreflexive mixed-norm sequence space. As a by-product of our work, we give applications of this result to the study of conditional Schauder bases and conditional almost greedy bases in this new class of Banach spaces.

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arXiv2018

Lipschitz free spaces on finite metric spaces

Main results of the paper: (1) For any finite metric space $M$ the Lipschitz free space on $M$ contains a large well-complemented subspace which is close to $\ell_1^n$. (2) Lipschitz free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs. Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of graphs which are not necessarily graph automorphisms; (b) In the case of such recursive families of graphs as Laakso graphs we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

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arXiv2018

Weight-almost greedy bases

We introduce the notion of a \textit{weight-almost greedy} basis and show that a basis for a real Banach space is $w$-almost greedy if and only if it is both quasi-greedy and $w$-democratic. We also introduce the notion of \textit{weight-semi-greedy} basis and show that a $w$-almost greedy basis is $w$-semi-greedy and that the converse holds if the Banach space has finite cotype.

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arXiv2016

$ξ$-asymptotically uniformly smooth, $ξ$-asymptotically uniformly convex, and $(β)$ operators

For each ordinal $ξ$, we define the notions of $ξ$-asymptotically uniformly smooth and $w^*$-$ξ$-asymptotically uniformly convex operators. When $ξ=0$, these extend the notions of asymptotically uniformly smooth and $w^*$-asymptotically uniformly convex Banach spaces. We give a complete description of renorming results for these properties in terms of the Szlenk index of the operator, as well as a complete description of the duality between these two properties. We also define the notion of an operator with property $(β)$ of Rolewicz which extends the notion of property $(β)$ for a Banach space. We characterize those operators the domain and range of which can be renormed so that the operator has property $(β)$ in terms of the Szlenk index of the operator and its adjoint.

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arXiv2015

Almost isometric constants for partial unconditionality

We discuss optimal constants of certain projections on subsequences of weakly null sequences. Positive results yield constants arbitrarily close to $1$ for Schreier type projections, and arbitrarily close to $1$ for Elton type projections under the assumption that the weakly null sequence admits no subsequence generating a $c_0$ spreading model. As an application, we prove that a weakly null sequence admitting a spreading model not equivalent to the $c_0$ basis has a quasi-greedy subsequence with quasi-greedy constant arbitrarily close to $1$.

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arXiv2015

Equivalent norms with the property $(β)$ of Rolewicz

We extend to the non separable setting many characterizations of the Banach spaces admitting an equivalent norm with the property $(β)$ of Rolewicz. These characterizations involve in particular the Szlenk index and asymptotically uniformly smooth or convex norms. This allows to extend easily to the non separable case some recent results from the non linear geometry of Banach spaces.

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arXiv1999

The distribution of vector-valued Rademacher series

Let $X=\sum ε_n x_n$ be a Rademacher series with vector-valued coefficients. We obtain an approximate formula for the distribution of the random variable $||X||$ in terms of its mean and a certain quantity derived from the K-functional of interpolation theory. Several applications of the formula are given.

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arXiv1996

On Various Modes of Scalar Convergence in L_0(X)

A sequence $\{f_n\}$ of strongly-measurable functions taking values in a Banach space $\X$ is scalarly null aė\. (resp. scalarly null in measure) if $x^*f_n \rightarrow0$ aė\. (resp. $x^*f_n \rightarrow 0$ in measure) for every $x^*\in \X^*$. Let $1\le p\le \infty$. The main questions addressed in this paper are whether an $L_p(\X)$-bounded sequence that is scalarly null aė\. will converge weakly aė\. (or have a subsequence which converges weakly aė\.), and whether an $L_p(\X)$-bounded sequence that is scalarly null in measure will have a subsequence that is scalarly null aė. The answers to these and other similar questions depend upon $p$ and upon the geometry of $\X$.

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arXiv1994

Nowhere Weak Differentiability of the Pettis Integral

For an arbitrary infinite-dimensional Banach space $\X$, we construct examples of strongly-measurable $\X$-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly differentiable; thus, for these functions the Lebesgue Differentiation Theorem fails rather spectacularly. We also relate the degree of nondifferentiability of the indefinite Pettis integral to the cotype of $\X$, from which it follows that our examples are reasonably sharp. This is an expanded version of a previously posted paper with the same name.

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arXiv1993

The Fourier transform of order statistics with applications to Lorentz spaces

We present a formula for the Fourier transforms of order statistics in $\Bbb R^n$ showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in $\Bbb R^n.$ For $a_1\geq ... \geq a_n\ge0$ and $q>0,$ denote by $\ell_{w,q}^n$ the $n$-dimensional Lorentz space with the norm $\|(x_1,...,x_n)\| = (a_1 (x_1^{*})^q +...+ a_n (x_n^{*})^q)^{1/q}$, where $(x_1^{*},...,x_n^{*})$ is the non-increasing permutation of the numbers $|x_1|,...,|x_n|.$ We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spaces into $L_q$ \cite{10} to prove that, for $n\geq 3$ and $q\leq 1,$ the space $\ell_{w,q}^n$ is isometric to a subspace of $L_q$ if and only if the numbers $a_1,...,a_n$ form an arithmetic progression. For $q>1,$ all the numbers $a_i$ must be equal so that $\ell_{w,q}^n = \ell_q^n.$ Consequently, the Lorentz function space $L_{w,q}(0,1)$ is isometric to a subspace of $L_q$ if and only if {\it either} $0<q<\infty$ and the weight $w$ is a constant function (so that $L_{w,q}= L_q$), {\it or} $q\le 1$ and $w(t)$ is a decreasing linear function. Finally, we relate our results to the theory of positive definite functions.

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