On φ-schauder frames

In this short note we introduce and study a particular type of Schauder frames, namely, Φ-Schauder frames.Publisher's Versio

Nonstationary frames of translates and frames for the Weyl--Heisenberg group and the extended affine group

In this work, we analyze Gabor frames for the Weyl--Heisenberg group and wavelet frames for the extended affine group. Firstly, we give necessary and sufficient conditions for the existence of nonstationary frames of translates. Using these conditions, we give the existence of Gabor frames for the Weyl--Heisenberg group and wavelet frames for the extended affine group. We present a representation of functions in the closure of the linear span of a Gabor frame sequence in terms of the Fourier transform of window functions. Afterward, we give sufficient conditions with explicit frame bounds for a finite linear combination of Gabor frames to be a frame. It is illustrated that the frame bounds associated with finite linear combinations of frames can decrease the width of the frame, which increases the speed of convergence in the frame algorithm. We show that the canonical dual of frames of translates has the same structure. An approximation of inverse of the frame operator of nonstationary frames of translates is presented. It is shown that a nonstationary frame of translates is a Riesz basis if and only if it is linearly independent and satisfies approximation of the inverse frame operator. Finally, we give equivalent conditions for a nonstationary sequence of translates to be linearly independent

Matrix-Valued (Θ,Θ∗)(\Theta, \Theta^*)-Gabor Frames over LCA Groups for Hyponormal Operators

G\v avruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely $K$-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator $K$. For a locally compact abelian group G and a positive integer $n$, we study frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space $L^2(G, \mathbb{C}^{n\times n})$ , where a bounded linear operator $\Theta$ on $L^2(G, \mathbb{C}^{n\times n})$ controls not only lower but also the upper frame condition. We term such frames matrix-valued $(\Theta, \Theta^*)$-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued $(\Theta, \Theta^*)$- Gabor frames in terms of hyponormal operators. It is shown that if $\Theta$ is adjointable hyponormal operator, then $L^2(G, \mathbb{C}^{n\times n})$ admits a $\lambda$-tight $(\Theta, \Theta^*)$-Gabor frame for every positive real number $\lambda$. A characterization of matrix-valued $(\Theta, \Theta^*)$-Gabor frames is given. Finally, we show that matrix-valued $(\Theta, \Theta^*)$-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study

Unitary Extension Principle for Nonuniform Wavelet Frames in L2(R)L^2(\mathbb{R})

We study the construction of nonuniform tight wavelet frames for the Lebesgue space $L^2(\mathbb{R})$, where the related translation set is not necessary a group. The main purpose of this paper is to prove the unitary extension principle (UEP) and the oblique extension principle (OEP) for construction of multi-generated nonuniform tight wavelet frames for $L^2(\mathbb{R})$. Some examples are also given to illustrate the results

Shadow of operators on frames

Aldroubi introduced two methods for generating frames of a Hilbert space H. In one of his method, one approach is to construct frames for H which are images of a given frame for H under T ∈ B (H, H), a collection of all bounded linear operator on H. The other method uses bounded linear operator on ` 2 to generate frames of H. In this paper, we discuss construction of the retro Banach frames in Hilbert spaces which are images of given frames under bounded linear operators on Hilbert spaces. It is proved that the compact operators generated by a certain type of a retro Banach frame can construct a retro Banach frame for the underlying space. Finally, we discuss a linear block associated with a Schauder frame in Banach spaces.Publisher's Versio