Translation and modulation invariant Hilbert spaces

Let $$\mathcal H$$ H be a Hilbert space of distributions on $$\mathbf{R}^{d}$$ R d which contains at least one non-zero element of the Feichtinger algebra $$S_0$$ S 0 and is continuously embedded in $$\mathscr {D}'$$ D ′ . If $$\mathcal H$$ H is translation and modulation invariant, also in the sense of its norm, then we prove that $$\mathcal H= L^2$$ H = L 2 , with the same norm apart from a multiplicative constant.

Spectral Invariance of $$*$$-Representations of Twisted Convolution Algebras with Applications in Gabor Analysis

We show spectral invariance for faithful $$*$$ ∗ -representations for a class of twisted convolution algebras. More precisely, if G is a locally compact group with a continuous 2-cocycle c for which the corresponding Mackey group $$G_c$$ G c is $$C^*$$ C ∗ -unique and symmetric, then the twisted convolution algebra $$L^1 (G,c)$$ L 1 ( G , c ) is spectrally invariant in $${\mathbb {B}}({\mathcal {H}})$$ B ( H ) for any faithful $$*$$ ∗ -representation of $$L^1 (G,c)$$ L 1 ( G , c ) as bounded operators on a Hilbert space $${\mathcal {H}}$$ H . As an application of this result we give a proof of the statement that if $$\Delta $$ Δ is a closed cocompact subgroup of the phase space of a locally compact abelian group $$G'$$ G ′ , and if g is some function in the Feichtinger algebra $$S_0 (G')$$ S 0 ( G ′ ) that generates a Gabor frame for $$L^2 (G')$$ L 2 ( G ′ ) over $$\Delta $$ Δ , then both the canonical dual atom and the canonical tight atom associated to g are also in $$S_0 (G')$$ S 0 ( G ′ ) . We do this without the use of periodization techniques from Gabor analysis.

CHARACTERISATIONS OF PARTITION OF UNITIES GENERATED BY ENTIRE FUNCTIONS IN

Collections of functions forming a partition of unity play an important role in analysis. In this paper we characterise for any $N\in \mathbb{N}$ the entire functions $P$ for which the partition of unity condition $\sum _{\mathbf{n}\in \mathbb{Z}^{d}}P(\mathbf{x}+\mathbf{n})\unicode[STIX]{x1D712}_{[0,N]^{d}}(\mathbf{x}+\mathbf{n})=1$ holds for all $\mathbf{x}\in \mathbb{R}^{d}.$ The general characterisation leads to various easy ways of constructing such entire functions as well. We demonstrate the flexibility of the approach by showing that additional properties like continuity or differentiability of the functions $(P\unicode[STIX]{x1D712}_{[0,N]^{d}})(\cdot +\mathbf{n})$ can be controlled. In particular, this leads to easy ways of constructing entire functions $P$ such that the functions in the partition of unity belong to the Feichtinger algebra.

The random Wigner distribution of Gaussian stochastic processes with covariance inS0(ℝ2d)

The paper treats time-frequency analysis of scalar-valued zero mean Gaussian stochastic processes onℝd. We prove that if the covariance function belongs to the Feichtinger algebraS0(ℝ2d)then: (i) the Wigner distribution and the ambiguity function of the process exist as finite variance stochastic Riemann integrals, each of which defines a stochastic process onℝ2d, (ii) these stochastic processes onℝ2dare Fourier transform pairs in a certain sense, and (iii) Cohen's class, ie convolution of the Wigner process by a deterministic functionΦ∈C(ℝ2d), gives a finite variance process, and ifΦ∈S0(ℝ2d)thenW∗Φcan be expressed multiplicatively in the Fourier domain.