Totally positive functions in sampling theory and time-frequency analysis

Duration: 49 mins 45 secs

Share this media item:

Embed this media item:

<iframe width="_width_" height="_height_" src="https://sms.cam.ac.uk/media/3009680/embed" frameborder="0" scrolling="no" allowfullscreen></iframe>

Choose size:

About this item

Available Formats

About this item

Description:	Groechenig, K Friday 21st June 2019 - 09:50 to 10:40


Created:	2019-06-21 13:15
Collection:	Approximation, sampling, and compression in high dimensional problems
Publisher:	Isaac Newton Institute
Copyright:	Groechenig, K
Language:	eng (English)


Abstract:	Totally positive functions play an important role in approximation theory and statistics. In this talk I will present recent new applications of totally positive functions (TPFs) in sampling theory and time-frequency analysis. (i) We study the sampling problem for shift-invariant spaces generated by a TPF. These spaces arise the span of the integer shifts of a TPF and are often used as a substitute for bandlimited functions. We give a complete characterization of sampling sets for a shift-invariant space with a TPF generator of Gaussian type in the style of Beurling. (ii) A related problem is the question of Gabor frames, i.e., the spanning properties of time-frequency shifts of a given function. It is conjectured that the lattice shifts of a TPF generate a frame, if and only if the density of the lattice exceeds 1. At this time this conjecture has been proved for two important subclasses of TPFs. For rational lattices it is true for arbitrary TPFs. So far, TPFs seem to be the only window functions for which the fine structure of the associated Gabor frames is tractable. (iii) Yet another question in time-frequency analysis is the existence of zeros of the Wigner distribution (or the radar ambiguity function). So far all examples of zero-free ambiguity functions are related to TPFs, e.g., the ambiguity function of the Gaussian is zero free.

Abstract:

Totally positive functions play an important role in approximation theory and statistics. In this talk I will present recent new applications of totally positive functions (TPFs) in sampling theory and time-frequency analysis. (i) We study the sampling problem for shift-invariant spaces generated by a TPF. These spaces arise the span of the integer shifts of a TPF and are often used as a substitute for bandlimited functions. We give a complete characterization of sampling sets for a shift-invariant space with a TPF generator of Gaussian type in the style of Beurling. (ii) A related problem is the question of Gabor frames, i.e., the spanning properties of time-frequency shifts of a given function. It is conjectured that the lattice shifts of a TPF generate a frame, if and only if the density of the lattice exceeds 1. At this time this conjecture has been proved for two important subclasses of TPFs. For rational lattices it is true for arbitrary TPFs. So far, TPFs seem to be the only window functions for which the fine structure of the associated Gabor frames is tractable. (iii) Yet another question in time-frequency analysis is the existence of zeros of the Wigner distribution (or the radar ambiguity function). So far all examples of zero-free ambiguity functions are related to TPFs, e.g., the ambiguity function of the Gaussian is zero free.

Available Formats

Format	Quality	Bitrate	Size
MPEG-4 Video	640x360	1.93 Mbits/sec	723.82 MB	View	Download
WebM	640x360	551.01 kbits/sec	200.85 MB	View	Download
iPod Video	480x270	522.1 kbits/sec	190.24 MB	View	Download
MP3	44100 Hz	249.76 kbits/sec	91.10 MB	Listen	Download
Auto *	(Allows browser to choose a format it supports)