On the Approximation Properties for the SpaceH∞

Lecture Notes in Mathematics, 1989

Previous chapters were preliminary in the sense that they covered some of the basic results leading to the theory of the weighted Hardy spaces. Another important property, namely, the fact that the "norm" of a distribution in these spaces can be computed in various equivalent ways, is the content of this chapter.

Functional Analysis and Its Applications, 1983

1. If X is a Banach space of type p and of cotype q, then every its n-dimensional subspace is Cn 1/p−1/q -complemented in X (cf.

The notion of a basis in a Banach space was introduced by J. Schauder in 1927. If a Banach space has a basis then it is also separable. The problem whether every separable Banach space has a Schauder basis appeared for the first time in 1931 in the Polish edition of Banach’s book "Theory of Linear Operations". It was clear to Banach, Mazur and Schauder that this question was related to the "approximation problem". This is mentioned in Banach’s book as a remark to the chapter on compact operators. If a Banach space X has a Schauder basis it also has the approximation property, since the natural projections of X onto its finite dimensional subspaces form a bounded sequence of finite rank operators converging pointwise on X to the identity operator. The approximation problem is equivalent to whether a Banach space has the "approximation property". A Banach space X has the approximation property if the identity operator on X is the limit for the topology of uniform convergence on compact subsets of X of a sequence of finite rank operators. The approximation property in its various forms was thoroughly analyzed by A. Grothendieck in his thesis "Produits tensoriels topologiques et espaces nucléaires" published in 1955. But it was not until 1972, that P. Enflo solved both questions in the negative. He found a subspace of co, which he showed does not have the approximation property and consequently does not have a basis. Almost immediately, T. Figiel and A. M. Davie greatly simplified his proof. The later using a probabilistic lemma constructed a separable closed subspace of l1 without the approximation property. Moreover, they also showed that both co and every lp space with 1  p < 1 , (p 6= 2) has subspaces without the approximation property. In this work we present some of the equivalent properties to the approximation property due to A. Grothendieck, and we make a detailed exposition of the proof by A. M. Davie.

Mathematical Notes, 2010

The main purpose of this note is to show that the question posed in the paper [1] of D. P. Sinha D. P. and A. K. Karn (see the very end of that paper) has a negative answer, and that the answer could have been obtained, essentially, in 1985 after the papers [2], [3] by the author appeared in 1982 and 1985, respectively.

Mathematische Nachrichten, 2004

In the paper we generalize the theory of classical approximation spaces to a much wider class of spaces which are defined with the help of best approximation errors. We also give some applications. For example, we show that generalized approximation spaces can be used to find natural (in some sense) domains of definition of unbounded operators.

The notion of a basis in a Banach space was introduced by J. Schauder in 1927. If a Banach space has a basis then it is also separable. The problem whether every separable Banach space has a Schauder basis appeared for the first time in 1931 in Banach’s book "Theory of Linear Operations". It was clear to Banach, Mazur and Schauder that this question was related to the "approximation problem". This is mentioned in Banach’s book as a remark to the chapter on compact operators. If a Banach space X has a Schauder basis it also has the approximation property, since the natural projections of X onto its finite dimensional subspaces form a bounded sequence of finite rank operators converging pointwise on X to the identity operator. The approximation problem is equivalent to whether a Banach space has the "approximation property". A Banach space X has the approximation property if the identity operator on X is the limit for the topology of uniform convergence on compact subsets of X of a sequence of finite rank operators. The approximation property in its various forms was thoroughly analyzed by A. Grothendieck in his thesis "Produits tensoriels topologiques et espaces nucléaires" published in 1955. But it was not until 1972, that P. Enflo solved both questions in the negative. He found a subspace of co, which he showed does not have the approximation property and consequently does not have a basis. Almost immediately, T. Figiel and A. M. Davie greatly simplified his proof. The later using a probabilistic lemma constructed a separable closed subspace of l1 without the approximation property. Moreover, they also showed that both co and every lp space with 1  p < 1 , (p 6= 2) has subspaces without the approximation property. In this work we present some of the equivalent properties to the approximation property due to A. Grothendieck, and we make a detailed exposition of the proof by A. M. Davie.

Forum Mathematicum, 2019

This paper studies the bounded approximation property (BAP) in quasi-Banach spaces. In the first part of the paper, we show that the kernel of any surjective operator {\ell_{p}\to X} has the BAP when X has it and {0

Journal of Functional Analysis, 1989

Let BP= {f:))fj) =sup,,, (1/2T)~~,lflP)*'P< co), 1 <p< co. Then BP is the dual of a function algebra Aq on R (Beurling). In this paper, we study the harmonic extensions offin BP and in A", and the corresponding Hardy spaces H,,, HA*. It is shown that a parallel theory for L", L' and BMO, H' can be developed for the above pairs. In particular we prove that for 1 < q < 2, (HAM)* is isomorphic to the Banach space where mTf= (1/2T) I',! We also prove Burkholder, Gundy, and Silverstein's maximal function characterization for the new Hardy space HA', 1 < 4 < 2. 0 1989 Academic Press, Inc.

2013

Two related approximation problems are formulated and solved in Hardy spaces of the disc and annulus. With practical applications in mind, truncated versions of these problems are analysed, where the solutions are chosen to lie in finite-dimensional spaces of polynomials or rational functions, and are expressed in terms of truncated Toeplitz operators. The results are illustrated by numerical examples. The work has applications in systems identification and in inverse problems for PDEs. Mathematics Subject Classification (2010). Primary 30H10, 41A50, 47B35, Secondary 65J22.