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Introduction to Riesz bases, frames and waveform dictionaries with Dr. Pierluigi Vellucci

This is a past event.

Tuesday, October 18, 2022 at 3:45pm to 4:45pm

DM - Deuxieme Maison, 409A
11200 SW 8th ST 33199, Deuxieme Maison, Miami, Florida 33199

Abstract The main feature of a basis {fk} ∞ k=1 in a Hilbert space H is that every f ∈ H can be represented by an infinite linear combination of the elements fk in the basis: f = X∞ k=1 ck(f)fk. (1) The coefficients ck(f) are unique. The key question in understanding the frames theory can be simply stated as follows: is the uniqueness of coefficients ck(f) really needed? In this talk, after recalling some basic facts about bases, we introduce the concept of frames. A frame is also a sequence of elements {fk} ∞ k=1 in H, such that each element f ∈ H can be represented as in (1). However, the frame elements are not necessarily linearly independent and the corresponding coefficients are not necessarily unique. Thus a frame might not be a basis. Related to the frames concept there is the concept of waveform dictionary. The waveform dictionary is a large set that contains more information than required. The waveform dictionary contains the wavelet dictionary (system) and the Gabor system providing a flexible collection of functions with desirable properties from both. The aim of this talk is just to review the arguments for generalizing the basis concept to frames and waveform dictionaries.

Event Type

Academics, Lectures & conferences


Students, Faculty & Staff


Graduate Students

Department of Mathematics and Statistics


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