Showing papers on "Birnbaum–Orlicz space published in 1980"

Polynomial approximation of functions in Sobolev spaces

Abstract: Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.

On generalized harmonic analysis

Abstract: Motivated by Wiener's work on generalized harmonic analysis, we consider the Marcinkiewicz space 6XP(R) of functions of bounded upper averagep power and the space St(R) of functions of bounded upper p variation. By identifying functions whose difference has norm zero, we show that St(R), 1

Some approximation properties in Orlicz-Sobolev spaces

Abstract: : We prove that weak derivatives in general Orlicz spaces are globally strong derivatives with respect to the modular convergence. Other approximation theorems involving the modular convergence are presented, which improve known density results of interest in the existence theory for strongly nonlinear boundary value problems. (Author)

On geometry of Orlicz spaces

Abstract: In terms of the function Φ it is established when Orlicz space LΦ does not contain l ∞ n uniformly and when it has some type or cotype.

Paranormal operators on Banach spaces

Abstract: In this note we show that a paranormal operator T on a Banach space satisfies Weyl's theorem. This is accomplished by showing that (i) every isolated point of its spectrum is an eigenvalue and the corresponding eigenspace has invariant complement, (ii) for α ≠ 0, Ker(T-α) ⊥ Ker (T-β) (in the sense of Birkhoff) whenever β ≠ α.

Non-Separable Banach Spaces

Abstract: Publisher Summary This chapter describes the different aspects of the nonseparable Banach spaces. Two Banach spaces are isomorphic if they are linearly homeomorphic. Some very basic results about the classical Banach spaces results that naturally lead to the nonseparable are presented in the chapter. Nonseparable analogues of the theorem of Pelczynski have led to a nearly complete solution to the problem of isomorphic classification of dual spaces. Compact extremally disconnected spaces are sometimes called Stonian spaces, because they occur as the Stone spaces of complete Boolean algebras. Typical examples are the Stone-Cech compactifications of discrete spaces and the maximal ideal spaces of the algebras. An optimistic conjecture is that any Banach space a bounded subset of cardinality that contained no weak Cauchy should have a subset equivalent to the usual basis. It is found that the classes of spaces whose dual balls exhibit a lack of weak sequential compactness in a dramatic fashion are the nonreflexive Grothendieck spaces.

Automorphisms in generalized spaces

Abstract: A survey of results on the theory of automorphisms in classical spaces with connections, Finsler spaces, and spaces of support elements are presented.

On the spaces Fs p.q of iiardy-soblow type: equivalent quasi-norys , multipliers, spaces on doilains , regular elliptic botjndary vaiue prblems

Abstract: Smary. The paper is the continuation of [22] . It deals with Hardy-Sobolev spaces in the Euclidean nspace and in domains, as well as with geaeral regular elliptic boundary value problems in these spaces.

Second order stochastic processes and the dilation theory in Banach spaces

Abstract: © Gauthier-Villars, 1980, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section B » (http://www.elsevier.com/locate/anihpb) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On the dimension of injective Banach spaces

Abstract: The purpose of this note is to give an affirmative answer, assuming the generalized continuum hypothesis, to a problem of H. Rosenthal on the cardinality of the dimension on injective Banach spaces. The problem in question is contained in [4, Problem 7.a]; in this connection we prove the following result. Theorem 1. Assume the G.C.H. If X is an infinite dimensional injective Banach space with dim X = a, then a" = a. We start with some preliminaries. We denote cardinals by a, ß; w denotes the cardinality of natural numbers. We denote by a" the cardinality of the family of countable subsets of a. For a cardinal a, we denote by cf(a) the least cardinal ß such that a is the cardinal sum of ß many cardinals, each smaller than a. A cardinal a is regular if a = cf(a), and singular if cf(a) < a. The least cardinal strictly greater than ß is denoted by ß +. The cardinality of a set A is denoted by \A\. The generalised continuum hypothesis (G.C.H.) is the statement that a+ = 2" for all infinite cardinals a. A real Banach space X is injective if for every Banach space Y and every bounded linear isomorphism T: X -» Y, there is a bounded linear projection P:Y^> T(X). If T is a set, we denote by lx(Y) the Banach space of real-valued functions on Y which are absolutely summable. If X is a Banach space we denote with dim X the least cardinal a such that there is a family F = {x{: £ < a} of elements of X with the property that X is the closed linear span of F. Lemma 2. Let X be an injective Banach space with dim X = a. Then I l(a) is isomorphic to a subspace of X*. Proof. Since A' is a complemented subspace of C(S) for some compact space 5, X* is a complemented subspace of LX(X) for some measure X. So the conclusion is a direct consequence of Theorem 2.5 of [3]. Proof of Theorem 1. Let us assume that the conclusion is false. Then there is an injective Banach space X with dim X = a and a" > a. Under the G.C.H., a" > a means that cf(a) = w and since lx(N) is isomorphic to a subspace of X [5] it follows that a > cf(a). We choose a sequence (oc,: tj < w) of regular cardinals such that a, = w+, «„+, > 2^ and 2„

Banach spaces of pseudomeasures on compact groups with emphasis on homogeneous spaces

Abstract: S OF AUSTRALASIAN PHD THESES BANACH SPACES OF PSEUDOMEASURES ON COMPACT GROUPS WITH EMPHASIS ON HOMOGENEOUS SPACES JOSEPHINE A. WARD (NEE\" GOODALL) Segal algebras and homogeneous Banach algebras, which are generalizations of the group algebra L (G) of a compact, or locally compact abelian group G , have been studied in Reiter [4], [5], [6], Burnham [/], [2], [3]; and Wang [7]. In Chapter 3 of this thesis a generalized notion of homogeneous Banach space over a compact group is considered. No multiplication is assumed. If B denotes such a space then an operator * may be defined from M(G) x B to B so that it corresponds to the convolution product in familiar cases. Then l im \\\\k*b 2>||g = 0 for any b € B and any approximate identity (J: ) ,u of L (G) . In fact this property is characteristic of homogeneous Banach spaces amongst those which are translation invariant. It is the basic tool used to study homogeneous Banach spaces. Chapters h and 5 consider, in detail, the family 8. of homogeneous Banach spaces of pseudomeasures defined on a compact group. In Chapter k the elements B of B. are characterized by means of norms defined on a family of trigonometric polynomials, and also by certain subspaces of Received 15 April l°80. Thesis submitted to the Australian National University, Canberra, November 1979Degree approved: April 198O. Supervisor: Professor R.E. Edwards. I 55 156 Josephine A. Ward .E(E(G)) which are identified with B* . If B is also a subspace of M(G) then it is a homogeneous convolution algebra; these algebras are the subject of Chapter 5. A complete description of the homogeneous subalgebras of L (G) is easily given. It is proved that the closed two-sided and closed left ideal theory of a homogeneous convolution Banach algebra B is precisely the same as that of one of the homogeneous subalgebras of L (G) . Moreover, an explicit representation of the ideals is given. One is also given for the ^-representations of any symmetric B again the picture is similar to that of one of the symmetric homogeneous subalgebras of L (G) . The seemingly disjoint second chapter considers a problem concerning weighted subspaces of some pointwise algebras of integrable functions defined on a compact abelian group G . This problem leads to the study of the tensor algebras C(G) ® C(G)_ , where G is now a compact group and *1 2 F , F are subsets of E(C) . These algebras are homogeneous convolution Banach algebras in B^ . References [7] J .T. Burnham, \"Closed ideals in subalgebras of Banach algebras. I\" , Proa. Amer. Math. Soc. 32 (1972), 551-555[2] J .T. Burnham, \"Closed ideals in subalgebras of Banach algebras I I : Di tk in ' s condit ion\", Monash. Math. 78 (197M, 1-3. [3] James T. Burnham, \"Segal algebras and dense ideals in Banach algebras\", Functional analysis and its applications, 33-58 ( in te rna t iona l Conference, Madras, 1973Lecture Notes in Mathematics, 399. Springer-Verlag, Berl in, Heidelberg, New York, 197^). [4] H. Rei ter , \"Subalgebras of L(G) \", Nederl. Akad. Wetensch. Proc. Ser. A 68 (1965), 691-696. [5] H.J. Rei te r , Classical harmonic analysis and locally compact groups (Oxford Universi ty Press , Oxford, 1968). Banach spaces of pseudomeasures 157 [6] H.J. Reiter , L -algebra and Segal algebra (Lecture Notes in Mathematics, 231. Springer-Verlag, Ber l in , Heidelberg, New York, 1971). [7] H.C. Wang, Homogeneous Banach algebras (Marcel Dekker, New York, 1977).

An extension theorem for a family of functional spaces

Abstract: For Lipschitz spaces of functions (in the sense of Yu. A. Brudnyi, V. K. Shalashov, “Lipschitz spaces of functions,” Dokl. Akad. Nauk SSSR,197, No. 1, 18–20, 1971), defined in a bounded domain with a Lipschitz boundary, one gives a theorem for the existence of a linear operator of extension. As a consequence, one formulates some new results of Lipschitz spaces (interpolation, embedding, change of variables).

Euclidean harmonic analysis : proceedings of seminars held at the University of Maryland, 1979

Abstract: Some analytic problems related to statistical mechanics.- On spectral synthesis in ?n, n ? 2.- Spectral synthesis and stability in Sobolev spaces.- Fourier analysis of multilinear convolutions, Calderon's theorem, and analysis on Lipschitz curves.- The complex method for interpolation of operators acting on families of Banach spaces.- Maximal functions: A problem of A. Zygmund.- Multipliers of F(LP).

Free interpolation in some banach spaces of analytic functions

Abstract: We state a series of results regarding the interpolation in the spaces of analytic functions being the Taylor coefficient of the expansion at O, and . One asserts that a sequence with one condensation point, having a structure similar to a geometric progression, is an interpolation sequence for these spaces, i.e., the restriction operator on these sets maps these spaces onto the corresponding collection of sequences. In this case the restriction operator has a continuous right inverse which is explicitly constructed. This note is a continuation of the author's paper. Ref. Zh. Mat. 1973, 4B164.

Sobolev spaces and symbolic calculus

Abstract: We use elementary theory of distributions and geometry of Euclidean spaces to obtain a theorem on the symbolic calculus of several variables in spaces of Fourier series with weights.