S. K. Kaushik

Banach Frames for Conjugate Banach Spaces

Retro Banach frames for conjugate Banach spaces have been introduced and studied. It has been proved that a Banach space E is separable if and only if E∗ has a retro Banach frame. Finally, a necessary and sufficient condition for a sequence in a separable Banach space to be a retro Banach frame has been given.

ON STABILITY OF BANACH FRAMES

Some stability theorems (Paley-Wiener type) for Banach frames in Banach spaces have been derived.

ON PERTURBATION OF BANACH FRAMES

A necessary and sufficient condition for the perturbation of a Banach frame by a non-zero functional to be a Banach frame has been obtained. Also a sufficient condition for the perturbation of a Banach frame by a sequence in E* to be a Banach frame has been given. Finally, a necessary condition for the perturbation of a Banach frame by a finite linear combination of linearly independent functionals in E* to be a Banach frame has been given.

FRAMES OF SUBSPACES FOR BANACH SPACES

Frames of subspaces for Banach spaces have been introduced and studied. Examples and counter-examples to distinguish various types of frames of subspaces have been given. It has been proved that if a Banach space has a Banach frame, then it also has a frame of subspaces. Also, a necessary and sufficient condition for a sequence of projections, associated with a frame of subspaces, to be unique has been given. Finally, we consider complete frame of subspaces and prove that every weakly compactly generated Banach space has a complete frame of subspaces.

A generalization of frames in Banach spaces

Fusion Banach frames satisfying property S have been studied. A sufficient condition for the existence of a fusion Banach frame satisfying property S in weakly compactly generated Banach spaces has been given. Also, a necessary and sufficient condition for a fusion Banach frame to satisfy property S has been given. Finally, fusion Banach frames satisfying property S have been characterized in terms of closedness of certain subspaces of the dual spaces in the weak*-topology.

ON FRAME SYSTEMS IN BANACH SPACES

Banach frame systems in Banach spaces have been defined and studied. A sufficient condition under which a Banach space, having a Banach frame, has a Banach frame system has been given. Also, it has been proved that a Banach space E is separable if and only if E* has a Banach frame ({φn},T) with each φn weak*-continuous. Finally, a necessary and sufficient condition for a Banach Bessel sequence to be a Banach frame has been given.

ON NEAR EXACT BANACH FRAMES IN BANACH SPACES

Near exact Banach frames are introduced and studied, and examples demonstrating the existence of near exact Banach frames are given. Also, a sufficient condition for a Banach frame to be near exact is obtained. Further, we consider block perturbation of retro Banach frames, and prove that a block perturbation of a retro Banach frame is also a retro Banach frame. Finally, it is proved that if E and F are both Banach spaces having Banach frames, then the product space E × F has an exact Banach frame.

Wavelet packet approximation

ABSTRACT Wavelet packet expansions have been the centre of many research problems in the last few years. We begin by furnishing few results related to approximation of functions of using wavelet packets. ‘Orthogonal Coifman wavelet packet (in short OCWP) systems’ followed by ‘biorthogonal Coifman wavelet packet (in short BCWP) systems’ with the vanishing moments distributed equally between the scaling function and the wavelet packet functions have been introduced and thereby wavelet packet approximation theorem is given. It was known earlier that, ‘Hilbert transform of wavelet is again a wavelet.’ This motivated us to seek whether ‘Hilbert transform of wavelet packets are again wavelet packets’ or not? The answer to this query has been addressed and certain results have been given in this direction. Finally, Hartley-like wavelet packets have been introduced and results by Hernández and Weiss have been generalized.

An Interplay between Gabor and Wilson Frames

Wilson frames as a generalization of Wilson bases have been defined and studied. We give necessary condition for a Wilson system to be a Wilson frame. Also, sufficient conditions for a Wilson system to be a Wilson Bessel sequence are obtained. Under the assumption that the window functions and for odd and even indices of are the same, we obtain sufficient conditions for a Wilson system to be a Wilson frame (Wilson Bessel sequence). Finally, under the same conditions, a characterization of Wilson frame in terms of Zak transform is given.

Vanishing moments of wavelet packets and wavelets associated with Riesz projectors

In the present paper, we show that, under some conditions wavelet packet basis of L²(ℝ) can be used as a tool for the uniform approximation of a function f ∈ (C^M ∩ L²)(ℝ),M > 0. The necessary and sufficient condition on the wavelet packet transform for estimating the Hölder continuity of a function f has been given. Wavelets associated with Riesz projectors have been proposed and various results related to vanishing moments have been presented. Further, we use such wavelets and prove that Hölder continuity of a function aids in the decay of wavelet coefficients and thus helps in approximating it. Finally, we give some properties of wavelets associated with Riesz projectors.