Research

BanachFrameMul: Bessel and Frame Multipliers in Banach Spaces

Objective

Another project has investigated the basic properties of frame and Bessel multipliers. This project aims to generalize this concept so that it will work with Banach spaces also.

Method

As the Gram matrix plays an important role in the investigation of multipliers, it is quite natural to look at the connection to localized frames and multipliers. The dependency of the operator class on the symbol class can be researched.

The following statements will be investigated:

  • Theorem: If G is a localized frame and a is a bounded sequence, then the frame multiplier Ta is bounded on all associated Banach spaces (the associated co-orbit spaces).
  • Theorem: If G is a localized frame and a is a bounded sequence, such that the frame multiplier Ta is invertible on the Hilbert space H, then Ta is simultaneously invertible on the associated Banach spaces.

The applications of these results to Gabor frames and Gabor multipliers will be further investigated.

Application

Although Banach spaces are more general a concept than Hilbert spaces, Banach theory has found applications. For example, if any norm other than L2 (least square error) is used for approximation, Banach theory tools have to be applied.

Partners

  • K. Gröchenig, NuHAG, Faculty of Mathematics, University of Vienna

Project-completion

This project ended on 28.02.2008 and is incorporated into the 'High Potential'-Project of the WWTF, MULAC.