These notes have a dual goal. On the one hand we shall give an overview of the recently introduced class of scalable frames. These are (finite) frames with the property that each frame vector can be rescaled in such a way that the resulting frames are tight. This process can be viewed as a preconditioning method for finite frames. In particular, we shall: (1) describe the class of scalable fram...

Regular Gabor frames for L2(Rd) are obtained by applying time-frequency shifts from a lattice in Λ Rd× R̂d to some decent so-called Gabor atom g, which typically is something like a summability kernel in classical analysis, or a Schwartz function, or more generally some g∈ S0(R). There is always a canonical dual frame, generated by the dual Gabor atom g̃. The paper promotes a numerical approach f...

We analyse the relation between innnite-dimensional frame theory and nite-dimensional models for frames as they are used for numerical algorithms. Special emphasis in this paper is on perfect reconstruction oversampled lter banks, also known as shift-invariant frames. For certain nite-dimensional models it is shown that the corresponding nite dual frame provides indeed an approximation of the d...

Coding of multiple proteins by overlapping reading frames is not a feature one would associate with eukaryotic genes. Indeed, codependency between codons of overlapping protein-coding regions imposes a unique set of evolutionary constraints, making it a costly arrangement. Yet in cases of tightly coexpressed interacting proteins, dual coding may be advantageous. Here we show that although dual ...

To overcome the limitation induced by the fixed timefrequency resolution over the whole time-frequency plane of Gabor frames, we propose a simple extension of the Gabor theory leading to the construction of frames with timefrequency resolution evolving over time or frequency. We describe the construction of such frames and give the explicit formulation of the canonical dual frame for some condi...

The category-theoretic nature of general frames for modal logic is explored. A new notion of “modal map” between frames is defined, generalizing the usual notion of bounded morphism/p-morphism. The category Fm of all frames and modal maps has reflective subcategories CHFm of compact Hausdorff frames, DFm of descriptive frames, and UEFm of ultrafilter enlargements of frames. All three subcategor...

The category-theoretic nature of general frames for modal logic is explored. A new notion of “modal map” between frames is defined, generalizing the usual notion of bounded morphism/p-morphism. The category Fm of all frames and modal maps has reflective subcategories CHFm of compact Hausdorff frames, DFm of descriptive frames, and UEFm of ultrafilter enlargements of frames. All three subcategor...