Naslov
Iterations of the generalized Gram–Schmidt procedure for generating Parseval frames

Autori
Berić, Tomislav

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, kratko priopćenje, znanstveni

Skup
6. hrvatski matematički kongres

Mjesto i datum
Zagreb, Hrvatska, 14.06.2016. - 17.06.2016

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Nije recenziran

Ključne riječi
frames, Parseval frames, Gram-Schmidt procedure

Sažetak
A sequence $(f_i)_{; ; i \in I}; ; $ in a Hilbert space $H$ is called a frame for $H$ if there exist constants $0 < A \le B < \infty$ such that $$ A \| f \|^2 \le \sum_{; ; i \in I}; ; | \left< f, f_i \right> |^2 \le B \| f \|^2, \quad \text{; ; for all }; ; f \in H. $$ Among all frames, those for which $A = B = 1$, called Parseval frames, have proved to be most useful in applications since they provide a simple reconstruction formula $$ f = \sum_{; ; i \in I}; ; \left< f, f_i \right> f_i, \quad \text{; ; for all }; ; f \in H. $$ In this paper we investigate some properties of the generalized Gram--Schmidt procedure (GGSP) for generating Parseval frames which was first introduced by Casazza and Kutyniok (2007). Motivated by iterative algorithms such as the frame algorithm for vector reconstruction and the gradient descent of the frame potential used for construction of approximate unit-norm tight frames, we investigate iterations of the GGSP and its limit. We show that regardless of the starting frame, in the limit case we always get an orthonormal basis with added zeros. Moreover, the position of zero vectors is known in advance.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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