Minimally Supported Frequency (MSF) d-Dilation Wavelets

In this paper, we provide a geometric construction of a symmetric 2n-interval minimally supported frequency (MSF) d-dilation wavelet set with d∈(1,∞) and characterize all symmetric d-dilation wavelet sets. We also provide two special kinds of symmetric d-dilation wavelet sets, one of which has 4m-intervals whereas the other has (4m+2)-intervals, for m∈N. In addition, we construct a family of d-dilation wavelet sets that has an infinite number of components.

Construction of three interval frame scaling sets

In this paper, a construction of three interval frame scaling sets is described and certain classes of normalized tight frame wavelet sets are also constructed with the help of these three interval frame scaling sets. We also generalize this construction to obtain [Formula: see text]-interval frame scaling sets.

Scaling sets and generalized scaling sets on Cantor dyadic group

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

Superlets: time-frequency super-resolution using wavelet sets

Time-frequency analysis is ubiquitous in many fields of science. Due to the Heisenberg-Gabor uncertainty principle, a single measurement cannot estimate precisely the location of a finite oscillation in both time and frequency. Classical spectral estimators, like the short-time Fourier transform (STFT) or the continuous-wavelet transform (CWT) optimize either temporal or frequency resolution, or find a tradeoff that is suboptimal in both dimensions. Following concepts from optical super-resolution, we introduce a new spectral estimator enabling time-frequency super-resolution. Sets of wavelets with increasing bandwidth are combined geometrically in a superlet to maintain the good temporal resolution of wavelets and gain frequency resolution in the upper bands. Superlets outperform the STFT, CWT, and other super-resolution methods on synthetic data and brain signals recorded in humans and rodents, resolving time-frequency details with unprecedented precision. Importantly, superlets can reveal transient oscillation events that are hidden in the averaged time-frequency spectrum by other methods.

On wavelet induced isomorphisms for reducing subspaces

In this paper, we adapt the notion of a wavelet induced isomorphism of [Formula: see text] associated with a wavelet set, introduced in [E. J. Ionascu, A new construction of wavelet sets, Real Anal. Exchange 28(2) (2002/03) 593–610], to the case of an [Formula: see text]-wavelet set, where [Formula: see text] is a reducing subspace [X. Dai and S. Lu, Wavelets in subspaces, Michigan Math. J. 43 (1996) 81–98]. We characterize all these wavelet induced isomorphisms similar to those given in Ionascu paper and provide specific examples of this theory in the case of symmetric [Formula: see text]-wavelet sets. Examples when [Formula: see text] is the classical Hardy space are also considered.

On fixed point sets of wavelet induced isomorphisms and frame induced monomorphisms

In this paper, we study fixed point sets of wavelet induced isomorphisms for symmetric wavelet sets. Also, introducing the notion of frame induced monomorphism for a frame wavelet set, as a generalization of the wavelet induced isomorphism for a wavelet set, we provide a construction of frame wavelet sets in [Formula: see text]. We also study fixed point sets of these maps.