zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Generalized frames and their redundancy. (English) Zbl 0976.42022

In this paper the authors prove that continuously indexed frames in separable Hilbert spaces with no redundancy (or excess) are equivalent to discretely indexed sets. The result applies to general not necessarily separated Hilbert spaces, by restricting the analysis to closed countably generated subspaces. More specific, let H be a Hilbert space and (M,S,μ) a measure space. Then a generalized frame in H indexed by M is a family h={h m H; mM} such that: (a) fH, mTf(m):=h m ,f is measurable; (b) there are 0<A,B< such that fH, Af H 2 Tf L 2 (M;μ) 2 Bf H 2 . Recall also a measurable subset E of H is called an atom if 0<μ(E)< and E contains no measurable subset F such that 0<μ(F)<μ(E). The main result reads (the reviewer takes the liberty to fix a typo):

Theorem 2.2: Let h be a generalized frame in H indexed by (M,S,μ) and assume ImT=L 2 (M,dμ). Then for every countable subset L of H, there exists a countable collection {E i ;iΛ} of disjoint measurable sets such that f ˜=c fi χ i for all f in the closed linear span of L, where {c fi ;iΛ} is a set of complex numbers depending on f, and χ i denotes the characteristic function of E i . In particular, if H is an infinite dimensional separable space, then L 2 (M;μ) is isometrically isomorphic to the weighted space l w 2 consisting of all sequences {c i } with {c i } 2 = i |c i | 2 w i <, where w i =μ(E i ) for a fixed collection of disjoint μ-atoms {E 1 ,E 2 ,...}.

MSC:
42C40Wavelets and other special systems