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)
The duality condition for Weyl-Heisenberg frames. (English) Zbl 0890.42006
Feichtinger, Hans G. (ed.) et al., Gabor analysis and algorithms. Theory and applications. Boston, MA: Birkhäuser. Applied and Numerical Harmonic Analysis. 33-84, 453-488 (1998).

The first chapter of the book “Gabor analysis and algorithms” (H. G. Feichtinger and T. Strohmer, eds.) written by A. J. E. M. Janssen is concerned with the condition of duality of Gabor frames in the time domain, the frequency domain, the time-frequency domain, and, for rational time-frequency sampling factors, the Zak domain, both for the continuous-time and the discrete-time case. Many of the results are presented in the more general framework of shift-invariant systems or filter banks.

In all considered domains, the author derives formulas for the frame operator and for the frame bounds and characterizes and computes minimal dual systems.

MSC:
42C15General harmonic expansions, frames
42A65Completeness of sets of functions
42C30Completeness of sets of functions of non-trigonometric Fourier analysis