scholarly journals Frames associated with shift invariant spaces on positive half line

2021 ◽  
Vol 13 (1) ◽  
pp. 23-44
Author(s):  
Owais Ahmad ◽  
Mobin Ahmad ◽  
Neyaz Ahmad
Keyword(s):  
Sufficient Conditions ◽  
Gabor Frames ◽  
Half Line ◽  
Necessary Condition ◽  
Wavelet Frames ◽  
Invariant Space ◽  
Unified Approach ◽  
Shift Invariant Spaces ◽  
Invariant Spaces ◽  
Positive Half

Abstract In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

Frames associated with shift-invariant spaces on local fields

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3097-3110 ◽  
Author(s):  
Firdous Shah ◽  
Owais Ahmad ◽  
Asghar Rahimi
Keyword(s):  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Gabor Frames ◽  
Necessary Condition ◽  
Wavelet Frames ◽  
Unified Approach ◽  
Shift Invariant Spaces ◽  
Invariant Spaces

In this paper, we present a unified approach to the study of shift-invariant systems to be frames on local fields of positive characteristic. We establish a necessary condition and three sufficient conditions under which the shift-invariant systems on local fields constitute frames for L2(K). As an application of these results, we obtain some known conclusions about the Gabor frames and wavelet frames on local fields.

MULTI-CHANNEL SAMPLING ON SHIFT-INVARIANT SPACES WITH FRAME GENERATORS

2012 ◽  
Vol 10 (01) ◽  
pp. 1250003 ◽  
Author(s):  
A. G. GARCIA ◽  
J. M. KIM ◽  
K. H. KWON ◽  
G. J. YOON
Keyword(s):  
Continuous Function ◽  
Sufficient Conditions ◽  
Sampling Theory ◽  
Invariant Space ◽  
Frame Sequence ◽  
Shift Invariant Spaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces ◽  
Sampling Expansions ◽  
Shift Invariant Space

Let φ be a continuous function in L2(ℝ) such that the sequence {φ(t - n)}n∈ℤ is a frame sequence in L2(ℝ) and assume that the shift-invariant space V(φ) generated by φ has a multi-banded spectrum σ(V). The main aim in this paper is to derive a multi-channel sampling theory for the shift-invariant space V(φ). By using a type of Fourier duality between the spaces V(φ) and L2[0, 2π] we find necessary and sufficient conditions allowing us to obtain stable multi-channel sampling expansions in V(φ).

scholarly journals Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Fengjuan Zhu ◽  
Qiufu Li ◽  
Yongdong Huang
Keyword(s):  
Scaling Function ◽  
Minimum Energy ◽  
Sufficient Conditions ◽  
Reconstruction Algorithm ◽  
Wavelet Function ◽  
Necessary Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Numerical Examples ◽  
Dilation Matrix

In order to characterize the bivariate signals, minimum-energy bivariate wavelet frames with arbitrary dilation matrix are studied, which are based on superiority of the minimum-energy frame and the significant properties of bivariate wavelet. Firstly, the concept of minimum-energy bivariate wavelet frame is defined, and its equivalent characterizations and a necessary condition are presented. Secondly, based on polyphase form of symbol functions of scaling function and wavelet function, two sufficient conditions and an explicit constructed method are given. Finally, the decomposition algorithm, reconstruction algorithm, and numerical examples are designed.

scholarly journals Sharp Results on Sampling with Derivatives in Shift-Invariant Spaces and Multi-Window Gabor Frames

2019 ◽  
Vol 51 (1) ◽  
pp. 1-25
Author(s):  
Karlheinz Gröchenig ◽  
José Luis Romero ◽  
Joachim Stöckler
Keyword(s):  
Gabor Frames ◽  
Shift Invariant Spaces ◽  
Invariant Spaces

Compensator conditions for stochastic ordering of point processes

1991 ◽  
Vol 28 (04) ◽  
pp. 751-761 ◽  
Author(s):  
A. Kwieciński ◽  
R. Szekli
Keyword(s):  
Probability Space ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Stochastic Ordering ◽  
Half Line ◽  
Simple Point ◽  
Martingale Approach ◽  
Random Elements ◽  
Markov Point Processes ◽  
Positive Half

Sufficient conditions are given under which two simple point processes on the positive half-line can be stochastically compared as random elements of D(0,∞) or R∞ + Using a martingale approach to point processes, the conditions are proposed via a compensator function family. Appropriate versions of the processes being compared are constructed on the same probability space. The results are illustrated by replacement policies and semi-Markov point processes.

scholarly journals Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions

2017 ◽  
Vol 211 (3) ◽  
pp. 1119-1148 ◽  
Author(s):  
Karlheinz Gröchenig ◽  
José Luis Romero ◽  
Joachim Stöckler
Keyword(s):  
Gabor Frames ◽  
Sampling Theorems ◽  
Totally Positive ◽  
Shift Invariant Spaces ◽  
Positive Functions ◽  
Invariant Spaces

REFINABLE SHIFT INVARIANT SPACES IN ℝd

2005 ◽  
Vol 03 (03) ◽  
pp. 321-345
Author(s):  
CARLOS A. CABRELLI ◽  
SIGRID B. HEINEKEN ◽  
URSULA M. MOLTER
Keyword(s):  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Invariant Space ◽  
Refinement Equation ◽  
Compactly Supported Function ◽  
Dilation Matrix ◽  
Jordan Basis ◽  
Shift Invariant Spaces ◽  
Invariant Spaces ◽  
Shift Invariant Space

Let φ : ℝd → ℂ be a compactly supported function which satisfies a refinement equation of the form [Formula: see text] where Γ ⊂ ℝd is a lattice, Λ is a finite subset of Γ, and A is a dilation matrix. We prove, under the hypothesis of linear independence of the Γ-translates of φ, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of L = [cAi-j]i,j∈Γ and a finite-dimensional subspace [Formula: see text] in the shift-invariant space generated by φ. We provide a basis of [Formula: see text] and show that its elements satisfy a property of homogeneity associated to the eigenvalues of L. If the function φ has accuracy κ, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than κ. These latter functions are associated to eigenvalues that are powers of the eigenvalues of A-1. Furthermore we show that the dimension of [Formula: see text] coincides with the local dimension of φ, and hence, every function in the shift-invariant space generated by φ can be written locally as a linear combination of translates of the homogeneous functions.

SHIFT INVARIANT SPACES FOR LOCAL FIELDS

2011 ◽  
Vol 09 (03) ◽  
pp. 417-426 ◽  
Author(s):  
A. AHMADI ◽  
A. ASKARI HEMMAT ◽  
R. RAISI TOUSI
Keyword(s):  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Open Subgroup ◽  
Local Fields ◽  
Locally Compact Abelian Group ◽  
Compact Open Subgroup ◽  
Parseval Frame ◽  
Shift Invariant Spaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces

This paper is an investigation of shift invariant subspaces of L2(G), where G is a locally compact abelian group, or in general a local field, with a compact open subgroup. In this paper we state necessary and sufficient conditions for shifts of an element of L2(G) to be an orthonormal system or a Parseval frame. Also we show that each shift invariant subspace of L2(G) is a direct sum of principle shift invariant subspaces of L2(G) generated by Parseval frame generators.

scholarly journals The necessary and sufficient conditions for wavelet frames in Sobolev space over local fields

2021 ◽  
Vol 39 (3) ◽  
pp. 81-92
Author(s):  
Ashish Pathak ◽  
Dileep Kumar ◽  
Guru P. Singh
Keyword(s):  
Sobolev Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Local Fields ◽  
Necessary Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Necessary And Sufficient

In this paper we construct wavelet frame on Sobolev space. A necessary condition and sufficient conditions for wavelet frames in Sobolev space are given.

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