On complemented copies of c 0 and ℓ ∞

We furnish examples of pairs of Banach spaces X, Y so that none of c ₀ and l _∞ live inside X ^? and Y, but they embed complementably into the space DP(X,Y) of the Dunford–Pettis operators from X into Y.     

On spaces admitting no ℓ p or c 0 spreading model

It is shown that for each separable Banach space X not admitting ? ₁ as a spreading model there is a space Y having X as a quotient and not admitting any ? _p for 1 ≤ p < ∞ or c ₀ as a spreading model. We also include the solution to a question of Johnson and Rosenthal (Studia Math 43:77–92, 1972) on the existence of a separable space not admitting as a quotient any space with separable dual.     

Solid Extensions of the Cesàro Operator on ℓ p and c 0

We introduce and study the largest Banach lattice (for the coordinate-wise order) which is a solid subspace of \({\mathbb{C}^\mathbb{N}}\) and to which the classical Cesàro operator \({\mathcal{C}\colon\ell^p \to \ell^p}\) (a positive operator) can be continuously extended while still maintaining its values in ? ^p . Properties of this optimal Banach lattice \({[\mathcal{C}, \ell^p]_s}\) are presented. In addition, all continuous convolution operators of \({[\mathcal{C}, \ell^p]_s}\) into itself are identified and the spectrum of \({\mathcal{C}\colon[\mathcal{C}, \ell^p]_s \to[\mathcal{C}, \ell^p]_s}\) is determined. A similar investigation is undertaken for the Cesàro operator \({\mathcal{C}\colon c_0\to c_0}\) .     

Norming subspaces isomorphic to ℓ 1

Norming subspaces are studied widely in the duality theory of Banach spaces. These subspaces are applied to the Borel and Baire classifications of the inverse operators. The main result of this article asserts that the dual of a Banach space X contains a norming subspace isomorphic to l₁ provided that the following two conditions are satisfied: (1) X^* contains a subspace isomorphic to l₁; and (2) X^* contains a separable norming subspace.     

Norm-one projections onto subspaces of finite codimension in ℓ1 andc 0

We study 1-complemented subspaces of the sequence spaces ₁ andc ₀. In ₁, 1-complemented subspaces of codimensionn are those which can be obtained as intersection ofn 1-complemented hyperplanes. Inc ₀, we prove a characterization of 1-complemented subspaces of finite codimension in terms of intersection of hyperplanes.Work prepared under the auspices of GNAFA-CNR (National Council of Research) and Minister of Public Instruction of Italy.     

Spaces with Dual Order-Isomorphic to ℓ1

We establish that an ordered Banach space is order-isomorphic to c₀ if and only if it is a -Dedekind complete vector lattice and its norm dual is order-isomorphic to ₁.     

Subspaces of Lp Isometric to Subspaces of ℓ p

We present three results on isometric embeddings of a (closed, linear) subspace X of L_p=L_p[0,1] into _p . First we show that if p 2N, then X is isometrically isomorphic to a subspace of _p if and only if some, equivalently every, subspace of L_p which contains the constant functions and which is isometrically isomorphic to X, consists of functions having discrete distribution. In contrast, if p 2N; and X is finite-dimensional, then X is isometrically isomorphic to a subspace of _p , where the positive integer N depends on the dimension of X, on p , and on the chosen scalar field. The third result, stated in local terms, shows in particular that if p is not an even integer, then no finite-dimensional Banach space can be isometrically universal for the 2-dimensional subspaces of L_p .     

Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2–ℓ p minimization

The iteratively reweighted ? ₁ minimization algorithm (IRL1) has been widely used for variable selection, signal reconstruction and image processing. In this paper, we show that any sequence generated by the IRL1 is bounded and any accumulation point is a stationary point of the ? ₂–? _p minimization problem with 0<p<1. Moreover, the stationary point is a global minimizer and the convergence rate is approximately linear under certain conditions. We derive posteriori error bounds which can be used to construct practical stopping rules for the algorithm.     

ℓ1-summability of higher-dimensional Fourier series

It is proved that the maximal operator of the ${?}_1$-Fejér means of a $d$-dimensional Fourier series is bounded from the periodic Hardy space $H_p\left({\mathbb{T}}^d\right)$ to $L_p\left({\mathbb{T}}^d\right)$ for all $d/(d+1)<p\le \infty$ and, consequently, is of weak type (1, 1). As a consequence we obtain that the ${?}_1$-Fejér means of a function $f\in L_1\left({\mathbb{T}}^d\right)$ converge a.e. to $f$. Moreover, we prove that the ${?}_1$-Fejér means are uniformly bounded on the spaces $H_p\left({\mathbb{T}}^d\right)$ and so they converge in norm $(d/(d+1)<p<\infty )$. Similar results are shown for conjugate functions and for a general summability method, called $\theta$-summability. Some special cases of the ${?}_1–\theta$-summation are considered, such as the Weierstrass, Picard, Bessel, Fejér, de la Vallée Poussin, Rogosinski and Riesz summations.     

Shift invariant preduals of ℓ 1(ℤ)

The Banach space ? ₁(?) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on ? ₁(?) weak*-continuous. This is equivalent to making the natural convolution multiplication on ? ₁(?) separately weak*-continuous and so turning ? ₁(?) into a dual Banach algebra. We call such preduals shift-invariant. It is known that the only shift-invariant predual arising from the standard duality between C ₀(K) (for countable locally compact K) and ? ₁(?) is c ₀(?). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak*-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c ₀. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of ?. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c ₀.     

Randomized first order algorithms with applications to ℓ 1-minimization

In this paper we propose randomized first-order algorithms for solving bilinear saddle points problems. Our developments are motivated by the need for sublinear time algorithms to solve large-scale parametric bilinear saddle point problems where cheap online assessment of the solution quality is crucial. We present the theoretical efficiency estimates of our algorithms and discuss a number of applications, primarily to the problem of ? ₁ minimization arising in sparsity-oriented signal processing. We demonstrate, both theoretically and by numerical examples, that when seeking for medium-accuracy solutions of large-scale ? ₁ minimization problems, our randomized algorithms outperform significantly (and progressively as the sizes of the problem grow) the state-of-the art deterministic methods.     

What portion of the sample makes a partial sum asymptotically stable or normal?

Summary Let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0<2 be given. We show that if at each stage n a number k _n depending on n of the lower and upper order statistics are removed from the n-th partial sum of the given random variables then under appropriate conditions on k _n the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.Work done while visiting the Bolyai Institute, Szeged University, partially supported by a University of Delaware Research Foundation Grant     

Sensitivity Analysis to the Value of p of the ℓ p Distance Weber Problem

In this paper we analyze the sensitivity of the _p distance Weber problem to the value of p. We consider each power p to be in a range and not necessarily the same for each demand point. We find the set of possible optimal locations when p is in a given range. We also find the location that minimizes the expected cost, and find the Expected Value of Perfect Information. Computational results are presented.     

Almost Euclidean subspaces of ℓ 1 N VIA expander codes

We give an explicit (in particular, deterministic polynomial time) construction of subspaces X⊆ℝ ^N of dimension (1−o(1))N such that for every x∈X,

Extensions of distance reducing mappings to piecewise congruent mappings on ℓm

In this note I will show any distance reducing mapping f: M ⁿ, where M is a finite subset of ^m (m n), can be extended to a piecewise conqruent mapping f: ^{m n}.     

On projective tensor products with complemented subspaces isomorphic to ℓ1(*)

We give new sufficient conditions in order that a projective tensor product of Banach spaces contains complemented copies of ?¹, which provides us new examples of this situation.     

Isomorphic embedding of ℓ p n , 1<p<2, into ℓ 1 (1+ε)n

In this paper we partially answer a question posed by V. Milman and G. Schechtman by proving that ℓ _p ⁿ , (C logn)^1/q(1+1/ε)-embeds into ℓ ₁ ^(1+ε)n , where 1<p<2 and 1/p+1/q=1. Supported by ISF.