Reproducing Kernels and Variable Bandwidth

We show that a modulation space of type () is a reproducing kernel Hilbert space (RKHS). In particular, we explore the special cases of variable bandwidth spaces Aceska and Feichtinger (2011) with a suitably chosen weight to provide strong enough decay in the frequency direction. The reproducing kernel property is valid even if () does not coincide with any of the classical Sobolev spaces because unbounded bandwidth (globally) is allowed. The reproducing kernel will be described explicitly

Frame properties of operator orbits

We consider sequences in a Hilbert space $\mathcal H$ of the form $(T^nf_0)_{n\in I},$ with a linear operator $T$, the index set being either $I = \mathbb N$ or $I = \mathbb Z$, a vector $f_0\in \mathcal H$, and answer the following two related questions: (a) {\it Which frames for $\mathcal H$ are of this form with an at least closable operator $T$?} and (b) {\it For which bounded operators $T$ and vectors $f_0$ is $(T^nf_0)_{n\in I}$ a frame for $\mathcal H$?} As a consequence of our results, it turns out that an overcomplete Gabor or wavelet frame can never be written in the form $(T^nf_0)_{n\in\mathbb N}$ with a bounded operator $T$. The corresponding problem for $I = \mathbb Z$ remains open. Despite the negative result for Gabor and wavelet frames, the results demonstrate that the class of frames that can be represented in the form $(T^nf_0)_{n\in\mathbb N}$ with a bounded operator $T$ is significantly larger than what could be expected from the examples known so far.Comment: 20 page

Design of variable bandwidth symmetric/anti-symmetric FIR single-band PCLS lowpass and highpass filters

© 2014, Springer-Verlag London. This paper proposes a reconfiguration design of variable bandwidth symmetric/anti-symmetric finite impulse response single-band peak-constrained least squares (PCLS) lowpass and highpass filters. When the bandwidth of the filter changes, a new set of filter coefficients is derived based on the old set of filter coefficients via a modified method of bisection. Since the proposed algorithm is very efficient, the new set of filter coefficients can be obtained in real time. Also, the new set of filter coefficients can be arbitrarily close to the globally optimal solution of the new PCLS optimization problem