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.     

Hilbert subspaces of the spaces ℓp having full measure

In the spacel _p, 1 p 2, every probability measure is supported on some Hubert subspace. For p>2 there exist measures, in particular, Gaussian measures in the spacel _p, for which each Hilbert subspace has measure zero.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad Nauk SSSR, Vol. 55, pp. 3–14, 1976.     

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.     

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.     

Euclidean Subspaces of the Complex Spaces ℓp n Constructed by Orbits of Finite Subgroups of SU(m)

A group representation method is used to construct minimal isometric embeddings ₂ ² into ₈ ¹⁰ and ₁₀ ¹² over C. The second of them yields a tight 5-design in C ². The corresponding angle set contains some irrational numbers.     

Almost Everywhere Convergence of Strong Nörlund Logarithmic Means of Walsh-Fourier Series

In this paper we study the maximal operator for a class of subsequences of strong Nörlund logarithmic means of Walsh-Fourier series. For such a class we prove the almost everywhere strong summability for every integrable function f.     

Isomorphisms and projections for quotient-spaces of ℒ1-spaces by their reflexive subspaces

Let Z₁=X₁/E₁ and Z₂=X₂/E₂, where X₁ and X₂ are ₁-spaces, E₁cX₁, E₂c X₂. In this paper we study the following questions: 1) under what conditions are Z₁ and Z₂ isomorphic; 2) under what conditions is Z₁ isomorphic to a complemented subspace of Z₂. Some results: (a) if E₁ and E₂ are reflexive and Z₁. is isomorphic to Z₂, then one of the spaces E₁ E₂ is isomorphic to the product of the other by a finite-dimensional space; (b) if is a circle), E₁=H and E₂ is reflexive and X₂=Y^* for some Y, then it is impossible to imbed Z₁ in Z₂ as a complemented subspace.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 91–101, 1977.     

A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics ℓ1 and ℓ2

Mathematical Notes - A particular class of estimates related to the Nelson–Erd?s–Hadwiger problem is studied. For two types of spaces, Euclidean and spaces with metric ?1,...     

On the optimality of a theorem of Elton on ℓ 1 n subsystems

We present examples which show that a substantial strengthening of the hypothesis in the almost isometric part of a theorem of Elton on ℓ ₁ ⁿ subsystems does not lead to a substantially stronger conclusion.     

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.     

On ℓ p -summability of the characteristic values of intergral operators onL(R N )

LetB _p (L ₂) (0<p<) denote the space of all compact linear operatorsK onL(R ^N ) with the following property: the sum ofp-th powers of all positive eigenvalues (repeated according to their multiplicity) of the nonnegative compact operator (K ^* K)^1/2 converges. We prove aB _p -criterion for an integral operatorK onL(R ^N ), for 0<p2. We require that its kernel have sufficient fractional smoothness and decay at infinity. Our proof is based on a suitable factorization of the operatorK which involves fractional powers (H+I)^– 0, 0<1, of the resolvent of the selfadjoint elliptic operatorH=(–) ^m +M _q ^2m onL(R ^N ).     

Efficient Computation of ℓ1 Regularized Estimates in Gaussian Graphical Models

In this article, I propose an effcient algorithm to compute ?₁ regularized maximum likelihood estimates in the Gaussian graphical model. These estimators, recently proposed in an earlier article by Yuan and Lin, conduct parameter estimation and model selection simultaneously and have been shown to enjoy nice properties in both large and finite samples. To compute the estimates, however, can be very challenging in practice because of the high dimensionality and positive definiteness constraint on the covariance matrix. Taking advantage of the recent advance in semidefinite programming, Yuan and Lin suggested a sophisticated interior-point algorithm to solve the optimization problem. Although a polynomial time algorithm, the optimization technique is known not to be scalable for high-dimensional problems. Alternatively, this article shows that the estimates can be computed by iteratively solving a sequence of ?₁ regularized quadratic programs. By effectively exploiting the sparsity of the graphical structure, I propose a new algorithm that can be applied to problems of larger scale. When combined with a path-following strategy, the new algorithm can be used to efficiently approximate the entire solution path of the ?₁ regularized maximum likelihood estimates, which also facilitates the choice of tuning parameter. I demonstrate the efficacy and usefulness of the proposed algorithm on a few simulations and real datasets.