A Weierstrass-like theorem for real separable Hilbert spaces

Journal of Computational and Applied Mathematics, 1997

By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper ~ertain Banach spaces. A relation between these spaces and the space (Co, S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.

2001

It is shown that if a separable real Banach space X admits a separating analytic Ž Ž . function with an additional condition property K , concerning uniform behaviour . of radii of convergence then every uniformly continuous mapping on X into any real Banach space Y can be approximated by analytic operators. In particular, the result applies to c . ᮊ 2001 Academic Press 0

Journal of Approximation Theory, 1996

The intention of this paper is to study a family of positive linear approximation operators relating to most of the well known Bernstein-type operators. These operators depend on a parameter. We give some characterization theorems to show that the operators corresponding to different parameters can be quite different. The direct and converse results make use of the Ditzian Totik modulus of smoothness.

2011

Recall that a Banach space X has the Dunford-Pettis property if every weakly compact operator defined on X takes weakly compact sets into norm compact sets. Some valuable characterisations of Banach spaces with the Dunford-Pettis property are: X has the DP P if and only if for all Banach spaces Y , every weakly compact operator from X to Y sends weakly convergent sequences onto norm convergent sequences (i.e. it requires that weakly compact operators on X are completely continuous) and this is equivalent to "if (x n) and (x * n) are sequences in X and X * respectively and lim n x n = 0 weakly and lim n x * n = 0 weakly then lim n x * n x n = 0". A striking application of the Dunford-Pettis property (as was observed by Grothendieck) is to prove that if X is a linear subspace of L ∞ (µ) for some finite measure µ and X is closed in some L p (µ) for 1 ≤ p < ∞, then X is finite dimensional. The fact that the well known spaces L 1 (µ) and C(Ω) have this property (as was proved by Dunford and Pettis) was a remarkable achievement in the early history of Banach spaces and was motivated by the study of integral equations and the hope to develop an understanding of linear operators on L p (µ) for p ≥ 1. In fact, it played an important role in proving that for each weakly compact operator T : L 1 (µ) → L 1 (µ) or T : C(Ω) → C(Ω), the operator T 2 is compact, a fact which is important from the point of view that there is a nice spectral theory for compact operators and operators whose squares are compact. There is an extensive literature involving the Dunford-Pettis property. Almost all the articles and books in our list of references contain some information about this property, but there are plenty more that could have been listed. The reader is for instance referred to [4], [5], [7], [8],[10], [17] and [24] for information on the role of the DP P in different areas of Banach space theory. In this dissertation, however, we are motivated by the two papers [7] and i a larger family of Banach spaces, but also to find an answer to the question: "When will every symmetric bilinear separately compact map X × X → c 0 be p-convergent?"

Journal of Soviet Mathematics, 1984

The initiators of the spectral theory of linear operators have so strongly associated, in the mathematical consciousness, the term "self-adjoint" with the term "operator" that up to now, when mathematicians think of a linear transformation of a sufficiently general form, many of them prefer to say "NON-self-adjoint," which, obviously, reminds us of the anecdotal "... but it is only you I love." The problems which constitute this chapter are not of this kind. Almost all of them belong to the spectral theory which tends to be mixed with complex analysis and in any case it borrows from the latter much more than the Liouville and Stone-Weierstrass theorems, to which the classical approach has been basically restricted. The mentioned intermixing constitutes probably the most characteristic feature of the present state of the theory, and the origin of this interpenetration has to be sought in the works of the late thirties (H. Wold, A. I. Pleater).

1985

Journal of Functional Analysis, 1982

Mediterranean Journal of Mathematics, 2014

In this paper we characterize compact linear operators from Banach function spaces to Banach spaces by means of approximations with bounded homogeneous maps. To do so, we undertake a detailed study of such maps, proving a factorization theorem and paying special attention to the equivalent strong domination property involved. Some applications to compact maximal extensions of operators are also given. Banach function space and p-th power and compact operator and homogeneous operator. 46E30 and 47B38 and 46B42 and 46B28

Resultate der Mathematik, 1996

In this paper we give positive answers to some problems posed by H.Gonska, respectively A.Lupa §. We show here that there exist positive linear operators H n : C[O, 1] -+ II n satisfying I(HnJ)(x) -l(x)l:s C W2 (1;yx~-X)) .

Journal of Functional Analysis, 1967

Journal of Functional Analysis, 1976

1. Let E be a complex Banach space. It is well known that C(E\C), the space of continuous scalar-valued functions on E endowed with the compact-open topology, always has the approximation property, since there are continuous partitions of unity. However, for the space H{E\C) of holomorphic scalar-valued functions on E 9 the situation is more complicated. In §2 of this note, we describe this situation. Briefly, there is an exact analogy between the question of approximation by finite rank linear mappings on compact sets and the question of approximation by finite rank holomorphic mappings on compact sets.

Journal of Mathematical Analysis and Applications, 1998

In this paper we introduce and study the notions of isotropic mapping and essential kernel. In addition some theorems on the Borel graph and Baire mapping for polynomial operators are proved. It is shown that a polynomial functional from an infinite dimensional complex linear space into the field of complex numbers vanishes on some infinite dimensional affine subspace.

Arkiv för matematik, 1989

Mathematische Nachrichten, 2000

We prove that the Banach space of bounded analytic functions and the disc algebra have the uniform bounded approximation property of order p for 1 < p < ∞.

Demonstratio Mathematica

In this paper we prove basic results in the approximation of vector-valued functions by polynomials with coefficients in normed spaces, called generalized polynomials. Thus we obtain : estimates in terms of Ditzian-Totik L p-moduli of smoothness for approximation by Bernstein-Kantorovich generalized polynomials and by other kinds of operators like the Szasz-Mirakian operators, Baskakov operators, Post-Widder operators and their Kantorovich analogues and inverse theorems for these operators. Applications to approximation of random functions and of fuzzy-number-valued functions are given.