Hilbert-space techniques for spectral representation in terms
of overcomplete bases

D 2002

Hilbert-space techniques for spectral representation in terms of overcomplete bases

VESELÝ, Vítězslav

Základní údaje

Originální název

Hilbert-space techniques for spectral representation in terms of overcomplete bases

Autoři

VESELÝ, Vítězslav
Ed. I. Horová.

Vydání

Brno (Czech Rep.), Proceedings of the Summer School DATASTAT'2001, Folia Fac. Sci. Nat. Univ. Masaryk. Brunensis, Mathematica 11, s. 259-273, 2002

Nakladatel

Masaryk University

Další údaje

Jazyk

angličtina

Typ výsledku

Stať ve sborníku

Obor

10101 Pure mathematics

Stát vydavatele

Česká republika

Utajení

není předmětem státního či obchodního tajemství

Kód RIV

RIV/00216224:14310/02:00007145

Organizační jednotka

Přírodovědecká fakulta

ISBN

80-210-3028-3

Klíčová slova anglicky

functional approximation; kernel operators; frame and wavelet expansions; pseudoinverse operators

Štítky

frame and wavelet expansions, functional approximation, kernel operators, pseudoinverse operators

Příznaky

Recenzováno

Změněno: 15. 1. 2007 17:18, doc. RNDr. Vítězslav Veselý, CSc.

Anotace

V originále

Topics associated with the representation of objects from a separable Hilbert space in terms of an a priori given overcomplete system (dictionary) of its generators (atoms) are handled. First the procedure of finding such a representation is formulated and solved using the Hilbert-space technique of linear bounded operators and their generalized inverse. Afterwards the problem of finding its sparse representation is discussed, i.e. such representation where most information on the given object is concentrated in a fewest possible number of its nonzero (spectral) coefficients in that representation. This may be rephrased as a procedure for finding a subbasis which is in a certain sense optimal for the given object in the scope of the prescribed overcomplete system. In general the common approach based on Moore-Penrose pseudoinverse does not yield the desired sparse solutions. That is why alternate procedures are discussed, in particular from the point of view of their numerical stability and computational feasibility.

Návaznosti

MSM 143100001, záměr

Název: Funkcionální diferenciální rovnice a matematicko-statistické modely

Investor: Ministerstvo školství, mládeže a tělovýchovy ČR, Funkcionální diferenciální rovnice a matematicko-statistické modely

Citovat

VESELÝ, Vítězslav. Hilbert-space techniques for spectral representation in terms of overcomplete bases. Ed. I. Horová. In Proceedings of the Summer School DATASTAT'2001, Folia Fac. Sci. Nat. Univ. Masaryk. Brunensis, Mathematica 11. Brno (Czech Rep.): Masaryk University, 2002, s. 259-273. ISBN 80-210-3028-3.

@inproceedings{483061,
   author = {Veselý, Vítězslav},
   address = {Brno (Czech Rep.)},
   booktitle = {Proceedings of the Summer School DATASTAT'2001, Folia Fac. Sci. Nat. Univ. Masaryk. Brunensis, Mathematica 11},
   keywords = {functional approximation; kernel operators; frame and wavelet expansions; pseudoinverse operators},
   language = {eng},
   location = {Brno (Czech Rep.)},
   isbn = {80-210-3028-3},
   pages = {259-273},
   publisher = {Masaryk University},
   title = {Hilbert-space techniques for spectral representation in terms of overcomplete bases},
   year = {2002}
}

TY  - CONF
ID  - 483061
AU  - Veselý, Vítězslav
PY  - 2002
TI  - Hilbert-space techniques for spectral representation in terms of overcomplete bases
PB  - Masaryk University
CY  - Brno (Czech Rep.)
SN  - 8021030283
KW  - functional approximation
KW  - kernel operators
KW  - frame and wavelet expansions
KW  - pseudoinverse operators
N2  - Topics associated with the representation of objects from a separable Hilbert space in terms of an a priori given overcomplete system (dictionary) of its generators (atoms) are handled. First the procedure of finding such a representation is formulated and solved using the Hilbert-space technique of linear bounded operators and their generalized inverse. Afterwards the problem of finding its sparse representation is discussed, i.e. such representation where most information on the given object is concentrated in a fewest possible number of its nonzero (spectral) coefficients in that representation. This may be rephrased as a procedure for finding a subbasis which is in a certain sense optimal for the given object in the scope of the prescribed overcomplete system. In general the common approach based on Moore-Penrose pseudoinverse does not yield the desired sparse solutions. That is why alternate procedures are discussed, in particular from the point of view of their numerical stability and computational feasibility.
ER  -

VESELÝ, Vítězslav. Hilbert-space techniques for spectral representation in terms of overcomplete bases. Ed. I. Horová. In \textit{Proceedings of the Summer School DATASTAT'2001, Folia Fac. Sci. Nat. Univ. Masaryk. Brunensis, Mathematica 11}. Brno (Czech Rep.): Masaryk University, 2002, s.~259-273. ISBN~80-210-3028-3.

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