Lalit Kumar Vashisht

University of Delhi | DU · Department of Mathematics (Faculty of Mathematical Sciences )

We characterize scaling functions of nonstationary matrix-valued multiresolution analysis in the matrix-valued function space \(L^2(\mathbb{R}, \mathbb{C}^{l \times l})\), l is a natural number. This is inspired by the work of Novikov, Protasov and Skopina on nonstationary multiresolution analysis of the space \(L^2(\mathbb{R})\). Using a sequence...

In this paper, we study Gabor frames in the matrix-valued signal space [Formula: see text], where [Formula: see text] is a locally compact abelian group which is metrizable and [Formula: see text]-compact, and [Formula: see text] is a positive integer. First, we give sufficient conditions on scalars in an infinite combination of vectors (from a giv...

Stable analysis and reconstruction of vectors in closed subspaces of Hilbert spaces can be studied by Gavruta’s type frame conditions which are related with the concept of atomic systems in separable Hilbert spaces. In this work, first we give Gavruta’s type frame conditions for a class of Hilbert–Schmidt operators (in short, C_2 class), where a bo...

Stable analysis and reconstruction of vectors in closed subspaces of Hilbert spaces can be studied by Gavruta's type frame conditions which are related with the concept of atomic systems in separable Hilbert spaces. In this work, first we give Gavruta's type frame conditions for a class of Hilbert-Schmidt operators (in short, C_2 class); where a bo...

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 prove the existence of Gabor frames from the Weyl--Heisenberg group and wavelet frames for the...

The relationship between the frame bounds of frames (Gabor) for the space $L^2(\mathbb{R})$ with several generators from the Weyl-Heisenberg group and the scalars linked to the sum of frames is examined in this paper. We give sufficient conditions for the finite sum of frames of the space $L^2(\mathbb{R})$ from the Weyl-Heisenberg group, with expli...

We study nonstationary frames of matrix-valued Gabor systems and wavelet systems in the matrix-valued function space [Formula: see text]. First, we show that a diagonal matrix-valued window function constitutes a frame for [Formula: see text] whenever each diagonal entry constitutes a frame for the space [Formula: see text]. This is not true for ar...

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 e...

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...

We study the construction of Gabor frames and wavelet frames for Weyl–Heisenberg group and extended affine group by using contraction between the affine group and the Weyl–Heisenberg group due to Subag, Baruch, Birman and Mann. Firstly, we give construction of Gabor frames with several generators from a unitary irreducible representation associated...

Gabor frames have interested many mathematicians and physicists due to their potential applications in time-frequency analysis, in particular, signal processing. A Gabor system is a collection of vectors which is obtained by applying modulation and shift operators to non-zero functions in signal spaces. In many applications, for example, signal pro...

Gabor frames have interested many mathematicians and physicists due to their potential applications in time-frequency analysis, in particular, signal processing. A Gabor system is a collection of vectors which is obtained by applying modulation and shift operators to non-zero functions in signal spaces. In many applications, for example, signal pro...

Dual frames are generalized Riesz bases which have potential applications in signal processing. In this paper, the construction of dual frames of matrix-valued wave packet systems in the matrix-valued function space L^2(R^d, C^{sx r}) from dual pairs of atomic wave packet frames in L^2(R^d) is studied. A class of matrix-valued dual generators from...

We study the construction of Gabor frames and wavelet frames for Weyl-Heisenberg group and extended affine group by using contraction between the affine group and the Weyl-Heisenberg group due to Subag, Baruch, Birman and Mann. Firstly, we give construction of Gabor frames with several generators from a unitary irreducible representation associated...

Motivated by the work of Frazier; and Gabardo and Nashed, we study \(P^{th}\)-stage nonuniform discrete wavelet frames (\(P^{th}\)-stage NUDW frames, in short) for \(\ell ^2(\Lambda )\), a nonuniform discrete space. In nonuniform discrete wavelet frames, the translation set is not necessary a group but a spectrum which is based on the theory of spe...

Parseval frames have attracted engineers and physicists due to their potential applications in signal processing. In this paper, we study the construction of nonuniform Parseval wavelet frames for the Lebesgue space L^2(R), where the related translation set is not necessary a group. The main purpose of this paper is to prove the unitary extension p...

In a vector-valued nonuniform multiresolution analysis (VNUMRA, in short), the corresponding translation set may not be a subgroup of R, but a spectrum associated with one-dimensional spectral pair. In this work, first, we give necessary and sufficient conditions for a family of functions related to VNUMRA to be an orthonormal system and complete s...

In this paper, we study matrix-valued wave packet frames for the matrix-valued function space L^2(R^d,C^(s×r)). An interplay between matrix-valued wave packet frames and its associated atomic wave packet frames is discussed. This is inspired by examples which show that frame properties cannot be carried from matrix-valued wave packet scaling functi...

The main purpose of this paper is to provide a characterization of scaling functions for non-uniform multiresolution analysis (NUMRA, in short). Some necessary and sufficient conditions for scaling functions of wavelet NUMRA in the frequency domain are also obtained.

Casazza and Christensen introduced the notion of the reconstruction property in separable Banach spaces, which is related to some deep concepts in Banach space theory. We discuss some types of operators associated with the reconstruction property and its related Banach frames in Banach spaces. It is shown that if a separable Banach space \(\mathcal...

The excess of a frame is the greatest number of elements that can be removed (from a given frame) yet leave a set which is a frame for the underlying space. We present a characterization of retro Banach frames in Banach spaces with finite excess. A sufficient condition for the existence of a retro Banach frame with infinite excess is obtained.

Bemrose et al. introduced the concept of weaving frames. Two frames $\{\phi_{i}\}_{i \in I}$ and $\{\psi_{i}\}_{i \in I}$ for a separable Hilbert space $\mathcal{H}$ are woven if there are positive constants $A \leq B$ such that for every subset $\sigma \subset I$, the family $\{\phi_{i}\}_{i \in \sigma} \cup \{\psi_{i}\}_{i \in \sigma^{c}}$ is a f...

We study frame properties of a matrix-valued wave packet system in the matrix-valued function space $L^2(\mathbb{R}^d, \mathbb{C}^{s\times r})$, where the lower frame condition is controlled by a bounded linear operator $\mathcal{K}$ on $L^2(\mathbb{R}^d, \mathbb{C}^{s\times r})$ (lower $\mathcal{K}$-frame condition, in short). There are many diffe...

Two frames $\{\phi_{i}\}_{i \in I}$ and $\{\psi_{i}\}_{i \in I}$ for a separable Hilbert space $H$ are woven if there are positive constants $A \leq B$ such that for every subset $\sigma \subset I$, the family $\{\phi_{i}\}_{i \in \sigma} \cup \{\psi_{i}\}_{i \in \sigma^{c}}$ is a frame for $H$ with frame bounds $A, B$. Bemrose et al. introduced we...

We consider matrix-valued wave packet systems in the matrix-valued function space L^2(R^d,C^(sxr))(d,s,r>=1). Some results on matrix-valued wave packet Bessel sequences have been extensively discussed in view to generate frames from Bessel sequences in L^2(R^d,C^(sxr)). Necessary conditions for matrix-valued Bessel sequences in terms of an estimate...

Generalized frames (in short g-frames) are natural generalization of standard frames in separable Hilbert spaces. Motivated by the concept of weaving frames in separable Hilbert spaces by Bemrose et al. in the context of distributed signal processing, we study weaving properties of g-frames. Firstly, we present necessary and sufficient conditions f...

The purpose of this paper is to first show relations between wave packet frame bounds and the scalars associated with finite sum of matrix-valued wave packet frames for the matrix-valued function space $L^2(\mathbb{R}^d, \mathbb{C}^{s\times r})$. A sufficient condition with explicit wave packet frame bounds for finite sum of matrix-valued wave pack...

A WH-packet is a system of vectors which is analogous to Aldroubi’s model for explicit expression of vectors (including frame vectors) in terms of a series associated with a given frame. In this paper, we study frame properties of WH-packet type system for matrix-valued wave packet frames in the function space L2(Rd,Cs×r)L2(ℝd,ℂs×r). A necessary an...

Gavruta introduced $K$-frames for Hilbert spaces to study atomic systems with respect to a bounded linear operator. There are many differences between K-frames and standard frames, so we study weaving properties of K-frames. Two frames $\{\phi_{i}\}_{i \in I}$ and $\{\psi_{i}\}_{i \in I}$ for a separable Hilbert space $\mathcal{H}$ are woven if the...

Two frames {φ i } i∈I and {ψ i } i∈I for a separable Hilbert space H are woven if there are positive constants A ≤ B such that for every subset σ ⊂ I, the family {φ i } i∈σ ∪ {ψ i } i∈σ c is a frame for H with frame bounds A, B. Bemrose et al. introduced weaving frames in separable Hilbert spaces and observed that weaving frames has potential appli...

This paper studies weaving properties of a family of operators which are analysis and synthesis systems with frame-like properties for closed subspaces of a separable Hilbert space \(\mathcal {H}\), where the lower frame condition is controlled by a bounded operator on \(\mathcal {H}\). In short, this family of operators is called a \(\Theta \)-g-f...

We present necessary and sufficient conditions for frames in real or complex locally convex commutative separable topological algebras.

We present necessary and sufficient conditions with explicit frame bounds for a discrete wavelet system of the form {DaTkϕ}a∈U(N),k∈IN to be a frame for the unitary space C^N. It is shown that the canonical dual of a discrete wavelet frame for C^N has the same structure. This is not true (well known) for canonical dual of a wavelet frame for L^2(R)...

In this paper, first we consider a wave packet system of the form fDaj TbkEcm gj;k;m2Z; and we present sufficient conditions for the extension of a pair of wave packet Bessel sequence to wave packet dual frames for L²(ℝ). In the second part, we provide necessary and sufficient conditions for the finite extension of a pair of Hilbert Bessel sequence...

In this paper we study frame-like properties of a wave packet system by using hyponormal operators on $L^2(\mathbb{R})$. We present necessary and sufficient conditions in terms of relative hyponormality of operators for a system to be a wave packet frame in $L^2(\mathbb{R})$. A characterization of hyponormal operators by using tight wave packet fra...

We present some Paley-Wiener type perturbation results for frames in a real (or complex) complete locally convex separable topological vector space.

Two discrete frames $\{\phi_{i}\}_{i\in I}$ and $\{\psi_{i}\}_{i\in I}$ for a separable Hilbert space $\mathbb{H}$ are said to be woven, if there are universal positive constants $A$ and $B$ such that for every subset $\sigma \subset I$, the family $\{\phi_{i}\}_{i \in \sigma} \cup \{\psi_{i}\}_{i \in \sigma^{c}}$ is a frame for $\mathbb{H}$ with l...

In this paper, we present some classes of generalized continuous weaving frames. It is shown that if the sets of lower frame bounds of discrete frames for a Hilbert space are bounded below, then the corresponding generalized continuous frames are woven. Necessary and sufficient conditions for generalized continuous weaving frames generated by an it...

Christensen et al. proved in [Extensions of Bessel sequences to dual pairs of frames, Appl. Comput. Harmon. Anal., 34 (2013), 224--233] that in any separable Hilbert space, any pairs of Bessel sequences (even if the given Bessel system is Gabor system in $L^2(\mathbb{R})$) can be extended to a pair of dual frames. In this paper, we extend results b...

In this note, we present necessary and su�fficient conditions with explicit frame bounds for a discrete system of translates of the form {Tk gk} to be a frame for the unitary space C^N.

In this paper, we obtain lower and upper bounds on the number of parity check digits of a linear code that corrects e or less errors within a sub-block. An example of such a code is provided. We introduce blockwise-tensor product of matrices and using this, we propose classes of error locating codes (or EL-codes) that can detect e or less errors wi...

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...

Casazza and Christensen [Canad. Math. Bull., 51 (2008), 348358] introduced and studied the reconstruction property in Banach spaces. In this paper, we discuss different types of convergence of series related to the reconstruction property in Banach space. First we discuss the uniform convergence of series associated with the reconstruction property...

In this paper we give a type of a compact linear operator associated with a given boundary value problem which can generate a Banach frame for the underlying space

We present new codes by perturbation of rows of the generating matrix of a given linear code. Some properties of the perturbed linear codes are given.

In this paper we study frame-like properties of a wave packet system by using hyponormal operators on $L^2(\mathbb{R})$. We present necessary and sufficient conditions in terms of relative hyponormality of operators for a system to be a wave packet frame in $L^2(\mathbb{R})$. A characterization of hyponormal operators by using tight wave packet fra...

A family of local atoms is a collection of vectors which are analysis and synthesis systems with frame-like properties for closed subspaces of a separable Hilbert space H. In this paper, we present some perturbation results for local atoms in a subspace of a Hilbert space. Some algebraic properties of one of derivatives of local atoms are given.

The reconstruction property for Banach spaces was introduced by Casazza and Christensen. In this paper we give a type of the reconstruction property in Banach spaces which is generated by the Toeplitz matrices and we call it the Toeplitz reconstruction property. It is proved that the standard reconstructionproperty in a Banach space can generate th...

We introduce and study a redundant system of retro Banach frames consisting of eigenfunctions associated with a given boundary value problem.

Frames are redundant system which are useful in the reconstruction of certain classes of spaces. The dual of a frame (Hilbert) always exists and can be obtained in a natural way. Due to the presence of three Banach spaces in the definition of retro Banach frames (or Banach frames) duality of frames in Banach spaces is not similar to frames for Hilb...

The authors thank the referee(s) for giving constructive comments and suggestions towards the improvement of the paper. Abstract. Casazza and Christensen in [5], introduced and studied the reconstruction property in Banach spaces. In this paper sufficient conditions for the existence of the reconstruction property in Banach spaces are obtained. Cas...

Reconstruction property in Banach spaces introduced and studied by Casazza and Christensen in [1]. In this paper we introduce reconstruction property in Banach spaces which satisfy $\mathfrak{I}$-property. A characterization of reconstruction property in Banach spaces which satisfy $\mathfrak{I}$-property in terms of frames in Banach spaces is obta...

In this short note we introduce and study a particular type of Schauder frames, namely, \Phi-Schauder frames.

Retro Banach frames of type P in Banach spaces have been introduced and studied. Necessary and sufficient conditions for existence of retro Banach frames of type P are obtained. A characterization of retro Banach frames obtained from translation of retro Banach frames of type P is given. Retro Banach frames of type P in product spaces with suitable...

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 lin...

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

Examples and counter examples to distinguish various types of Banach frames have been given. It has been proved that if a Banach space has a Banach frame, then it also has a normalized tight as well as a normalized tight and exact Banach frame. In particular, it has been proved that a Banach space with a weak* separable dual has a normalized tight...

Retro Banach frames for conjugate Banach spaces are introduced and studied. It is 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 is given.