It is proved that every banach space with a separable dual embeds into a space with a shrinking basis. Casazza, han and larson 9also carried out a study of atomic decompositio ns and banach frames. This goal motivates the need of understanding the correspondence of frames, riesz bases, and rkhs inbanach spaces. The novelty of the work is that, it provides a systematic approach of approximate controllability of nonautonomous nonlinear evolution systems in re. Frames, riesz bases, and sampling expansions in banach spaces. Expository discussions of these connections can be found in dau92 and hw89. Oleg reinov frames for linear operators in banach spaces. Request pdf on p frames and reconstruction series in separable banach spaces it is well known that a frame gi for a hilbert space allows every element f to be represented as f. An analysis of pframes in general banach spaces appeared in 6. We show that a very strong axiom, 1 1 comprehension, is needed to prove such. Further, we investigate equivalence between several perturbation conditions. A family of local atoms in a banach space has been introduced and it has been generalized to an atomic system for operators in banach spaces, which has been further led to introduce new frames for operators by dastourian and janfada, by making use of semiinner products. Throughout, f will denote either the real line r or the complex plane c. Since every banach space with separable dual admits an equivalent lur and c1 smooth.
We determine stability conditions with sharp bounds and discuss the necessity of some of them. The author begins with a discussion of weak topologies, weak compactness and isomorphisms of banach spaces before proceeding to the more detailed study of particular spaces. In this paper we consider perturbation of several frameconcepts in separable banach spaces. For recent development in frames for banach spaces one may refer to 1017. Reconstruction property and frames in banach spaces. In section 3, p,yoperator frames and p,yoperator riesz bases for a banach space will be defined and discussed. Stoeva october 28, 20 abstract in this paper we consider perturbation of xdbessel sequences, xd frames, banach frames, atomic decompositions and xdriesz bases in separable banach spaces. X to be represented as an unconditionally convergent series g. Frames, riesz bases, and sampling expansions in banach. Banach spaces from the metric we also get a topology notion of open set. X has the reconstruction property for xwith respect to a sequence ff kg.
On pframes and reconstruction series in separable banach. X to be represented by an unconditionally convergent series f i. Banach spaces with a schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients say is dense. P frames on banach spaces are a straight extension of frames on hilbert spaces. Oct 17, 2004 p frames in separable banach spaces p frames in separable banach spaces christensen, ole. Also, in 10, the authors gave a c algebra structure to the set of all bessel sequences for h. As we will see, this problem does not arise in hilbert spaces, which always have an orthonormal basis. Gr ochenig has given the nontrivial extension of frames to banach spaces gr o91. In particular, this leads to a somewhat unwanted example of a banach frame for a hilbert space, which is not a hilbert frame.
For a long time all known examples of banach spaces were found to have such a basis. A product of at most continuum many separable spaces is separable willard 1970, p. A countable product of secondcountable spaces is second countable, but an uncountable. A sequence fi in a separable hilbert space h is called a frame if there exist. Frames for linear operators in banach spaces oleg reinov oleg reinov frames for linear operators in banach spaces. Finally, we define the concept of frame operator for these kinds of frames in banach spaces and then we establish a perturbation result in this framework. A short course on non linear geometry of banach spaces 3 we nish this very short section by mentioning an important recent result by g. Equivalence between some perturbation conditions is investigated. I would be appreciated if someone could point out that how really a separable and nonseparable space differ. Approximation by smooth functions with no critical points on. In this paper, by considering conditions on p frames, properties of the p frame mapping like weakly continuous.
Such frames play an important role in gabor and wavelet analysis. The harder question is whether a separable banach space necessarily has a schauder basis. Perturbation of frames in banach spaces asianeuropean. We characterize banach frames and xdframes in separable banach spaces, and relate them to series expansions in banach spaces. Weusebx,y to denote the banach space of all bounded linear operators from x into y. A schauder basis in a banach space x is a sequence e n n. Godefroykalton 2003 let xand ybe separable banach spaces and suppose that f. We investigate the strength of set existence axioms needed for separable banach space theory. Frame expansions in separable banach spaces university of. An infinitedimensional separable banach space is called separably injective if, for every separable space x containing e, there is a projection p of x onto e. Frame expansions in separable banach spaces sciencedirect.
Unlike the other kinds of frames, the mapping of p frames, because of the nonlinearity of the duality mapping, has lost its linearity and acts as a nonlinear mapping from banach space x to its dual. Separable banach space an overview sciencedirect topics. The book is intended to be used with graduate courses in banach space theory, so the prerequisites are a background in functional, complex and real analysis. The aim of this paper is to generalize this result of adr to a huge class of locally convex spaces, including arbitrary products of separable fr echet spaces. Most of the standard properties of alternate dual frames follow immediately from this classi.
We establish various relationships between bessel respectively, frames and riesz sequences respectively, riesz bases, and then some of their applications are presented. Request pdf p frames in separable banach spaces let x be a separable banach space with dual x. In the present paper, by using suns extension and some techniques in a frame for banach spaces, we shall introduce pg frames and g frames for banach spaces that allows p every element f. Frames for operators in banach spaces via semiinner. Request pdf pframes in separable banach spaces let x be a separable banach space with dual x. Zd of a separable hilbert space h is called a frame. Indeed, the standard norms of the classical separable banach spaces p n and lprn are lur and cp smooth 11. New properties of the mapping obtained by pframes on banach. Our starting point is the concept of p frames, which was introduced by aldroubi et al. I would be appreciated if someone could point out that how really a separable and non separable space differ. Reconstruction property and frames in banach spaces iii sis a bounded linear operator such that sff k fg f. Despite the fact that these do not belong to the standard classes of nice locally convex spaces, they do have the property that they satisfy a closed graph theorem for operators with values in a separable banach space see the first chapter of the monograph saks spaces and applications to functional analysis, prop.
Moreover this classical theory of feichtinger and gr ochenig has shown in particular that gabor and wavelet l2frames can in fact extend to banach frames for modulation 19, 37, 38, 32 and homogeneous besov spaces 34, 54, 55 respectively. The main body of the paper deals with frames for banach spaces. Banach frames for modulation spaces ii 3 modulation spaces by brushlet unconditional basis have been given by nielsen and borup 49 and the mapping properties of pseudodi erential operators in h ormander classes on. Without further assumptions, we prove that a p frame allows every g. Frames for operators in banach spaces via semiinner products. Function spaces a function space is a vector space whose \vectors are functions. An analysis ofp frames in general banach spaces appeared in 6. This paper is devoted to an investigation of frames and riesz bases for general banach sequence spaces. In particular, our results show that we can not expect banach frames to share all the nice properties of frames in hilbert spaces. A normed space is a pair x,kk, where xis a linear space. In 1989 grochenig 21 generalized frames to banach spaces and called them atomic decompositions. The concept of frames in banach spaces have been introduced by christension and stoera9, casazza, han and larson10, grochenig11, cao12.
One of the main purposes of this paper is to extend it to banach spaces of functions. A countable family of elements g i x is a p frame 1 p if the norm x is equivalent to the. Unlike the traditional way of considering sequences in the dual space, sequences in the original space are considered to. On p, qframes in banach spaces research india publications. Every separable banach space has a banach frame with frame. Recall that a banach space is a complete normed space, and a hilbert space is a special kind of banach space, where the norm is given by an inner product. In other words, why do we need to define separability. Frames for the solution of operator equations in hilbert spaces with. Now we are going to state frames for separable banach spaces by casazza et al. We also show that every separable banach space x has a g banach frame with bounds equal to 1. Frames for operators in banach spaces springerlink. If e is a separable banach space such that e has a complemented subspace isomorphic to jr with t uncountable then e contains a complemented, o\e, e closed subspace isomorphic to ma, the. The banach spaces that arise in applications typically have schauder bases, but en o showed in 1973 that there exist separable banach spaces that do not have any schauder bases. In a similar way a seminorm p on e gives rise to a pseudo metric.
You should have met lp and p for 1 p 1and ck the space of continuous functions on k. Given a total sequence in a hilbert space, we speak of an upper resp. In particular, a result for tight frames for hilbert spaces is obtained. In the present paper, by using suns extension and some techniques in a frame for banach spaces, we shall introduce gm. New properties of the mapping obtained by pframes on. In 9 and 3, the authors investigated the set of all bessel sequences for a separable hilbert space has a banach space and obtained important results. We are going to work with complex banach and hilbert spaces whose elements are functions. We characterize banach frames and x dframes in separable banach spaces, and relate them to series expansions in banach spaces. It follows that every separable reflexive space can be embedded in a reflexive space with a basis. B x we obtain interesting results for pframes in banach spaces.
Recently, various generalizationsof frames in banach spaces have been introduced and studied. Banach and hilbert spaces in what follows k will denote r of c. Antoine, jeanpierre ucl balazs, peter acoustic research institute, austrian acad. Frame expansions in separable banach spaces pete casazza ole christensen diana t. A countable family of elements g i x is a pframe 1. Y is an into isometry, then xis linearly isometric to a subspace of y. The definition of a separable space that i know is that a space is separable if you can find a countable dense subset of it. Fa 20 feb 2009 perturbation of frames in banach spaces diana t. Let e be a real or complex separable hilbert space with inner product, definition 1. Any continuous image of a separable space is separable willard 1970, th. The characterizations of atomic systems allow us to state some results for sampling theory in s. All vector spaces are assumed to be over the eld f.
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