scholarly journals ON THE CHARACTERIZATION OF SCALING FUNCTIONS ON NON-ARCHEMEDEAN FIELDS

2021 ◽  
Vol 7 (1) ◽  
pp. 3
Author(s):  
Ishtaq Ahmed ◽  
Owias Ahmad ◽  
Neyaz Ahmad Sheikh
Keyword(s):  
Multiresolution Analysis ◽  
Scaling Function ◽  
Real Life ◽  
Discrete Subgroup ◽  
Local Fields ◽  
Mathematical Tool ◽  
Spectral Pair ◽  
Spectral Pairs ◽  
The Given

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool.  This gap was filled by Gabardo and Nashed [11]   by establishing a constructive algorithm based on the theory of spectral pairs for constructing non-uniform wavelet basis in \(L^2(\mathbb R)\). In this setting, the associated translation set \(\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z\) is no longer a discrete subgroup of \(\mathbb R\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we characterize the scaling function for non-uniform multiresolution analysis on local fields of positive characteristic (LFPC). Some properties of wavelet scaling function associated with non-uniform multiresolution analysis (NUMRA) on LFPC are also established.

scholarly journals Dual wavelets associated with nonuniform MRA

2018 ◽  
Vol 50 (2) ◽  
pp. 119-132
Author(s):  
Mohd Younus Bhat
Keyword(s):  
Positive Integer ◽  
Multiresolution Analysis ◽  
Discrete Subgroup ◽  
Scaling Functions ◽  
One Dimensional ◽  
Spectral Pair ◽  
Spectral Pairs ◽  
The Given

A generalization of Mallats classical multiresolution analysis, based on thetheory of spectral pairs, was considered in two articles by Gabardo and Nashed. In thissetting, the associated translation set is no longer a discrete subgroup of R but a spectrumassociated with a certain one-dimensional spectral pair and the associated dilation is aneven positive integer related to the given spectral pair. In this paper, we construct dualwavelets which are associated with Nonuniform Multiresolution Analysis. We show thatif the translates of the scaling functions of two multiresolution analyses are biorthogonal,then the associated wavelet families are also biorthogonal. Under mild assumptions onthe scaling functions and the wavelets, we also show that the wavelets generate Rieszbases

scholarly journals Construction of nonuniform periodic wavelet frames on non-Archimedean fields

2020 ◽  
Vol 74 (2) ◽  
pp. 1
Author(s):  
Owais Ahmad
Keyword(s):  
Real Life ◽  
Discrete Subgroup ◽  
Extension Principle ◽  
Mathematical Tool ◽  
Wavelet Frames ◽  
Spectral Pair ◽  
Periodic Wavelet ◽  
Spectral Pairs ◽  
Unitary Extension ◽  
Periodic Wavelet Frames

In real life applications not all signals are obtained by uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. Gabardo and Nashed, and Gabardo and Yu filled this gap by the concept of nonuniform multiresolution analysis and nonuniform wavelets based on the theory of spectral pairs for which the associated translation set \(\Lambda= \{0,r/N\}+2\mathbb{Z}\) is no longer a discrete subgroup of \(\mathbb{R}\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we introduce a notion of nonuniform periodic wavelet frame on non-Archimedean field. Using the Fourier transform technique and the unitary extension principle, we propose an approach for the construction of nonuniform periodic wavelet frames on non-Archimedean fields.

scholarly journals Frame Multiresolution Analysis on Local Fields of Positive Characteristic

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Firdous A. Shah
Keyword(s):  
Multiresolution Analysis ◽  
Positive Characteristic ◽  
Scaling Function ◽  
Local Fields ◽  
Wavelet Frames ◽  
Shift Invariant Spaces ◽  
Frame Multiresolution Analysis ◽  
Invariant Spaces ◽  
The Scaling Function

We present a notion of frame multiresolution analysis on local fields of positive characteristic based on the theory of shift-invariant spaces. In contrast to the standard setting, the associated subspace V0 of L2(K) has a frame, a collection of translates of the scaling function φ of the form φ(·-u(k)):k∈N0, where N0 is the set of nonnegative integers. We investigate certain properties of multiresolution subspaces which provides the quantitative criteria for the construction of frame multiresolution analysis (FMRA) on local fields of positive characteristic. Finally, we provide a characterization of wavelet frames associated with FMRA on local field K of positive characteristic using the shift-invariant space theory.

scholarly journals Scaling functions on the spectrum

2018 ◽  
Vol 10 (2) ◽  
pp. 340-346
Author(s):  
◽  
Firdous A. Shah
Keyword(s):  
Multiresolution Analysis ◽  
Short Communication ◽  
Discrete Subgroup ◽  
Scaling Functions ◽  
Fourier Domain ◽  
One Dimensional ◽  
Spectral Pair ◽  
Classic Theory ◽  
Spectral Pairs ◽  
Fundamental Equations

Abstract A generalization of Mallat’s classic theory of multiresolution analysis based on the theory of spectral pairs was considered by Gabardo and Nashed [4] for which the translation set Λ = {0, r/N}+2 ℤ is no longer a discrete subgroup of ℝ but a spectrum associated with a certain one-dimensional spectral pair. In this short communication, we characterize the scaling functions associated with such a nonuniform multiresolution analysis by means of some fundamental equations in the Fourier domain.

Fractal dimension pattern-based multiresolution analysis for rough estimator of speaker-dependent audio emotion recognition

2017 ◽  
Vol 15 (05) ◽  
pp. 1750042 ◽  
Author(s):  
Miao Cheng ◽  
Ah Chung Tsoi
Keyword(s):  
Fractal Dimension ◽  
Emotion Recognition ◽  
Multiresolution Analysis ◽  
Real Life ◽  
Emotional States ◽  
Acoustic Feature ◽  
Comparative Performance ◽  
Audio Analysis ◽  
The Arts ◽  
The Given

As a general means of expression, audio analysis and recognition have attracted much attention for its wide applications in real-life world. Audio emotion recognition (AER) attempts to understand the emotional states of human with the given utterance signals, and has been studied abroad for its further development on friendly human–machine interfaces. Though there have been several the-state-of-the-arts auditory methods devised to audio recognition, most of them focus on discriminative usage of acoustic features, while feedback efficiency of recognition demands is ignored. This makes possible application of AER, and rapid learning of emotion patterns is desired. In order to make predication of audio emotion possible, the speaker-dependent patterns of audio emotions are learned with multiresolution analysis, and fractal dimension (FD) features are calculated for acoustic feature extraction. Furthermore, it is able to efficiently learn the intrinsic characteristics of auditory emotions, while the utterance features are learned from FDs of each sub-band. Experimental results show the proposed method is able to provide comparative performance for AER.

scholarly journals Application of Average fs-Aggregate Algorithm for Multi-criteria Decision Making in Real Life Problem

2018 ◽  
pp. 1-11
Author(s):  
Rajesh Kumar Pal
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Accurate Result ◽  
Medical Science ◽  
Real Life ◽  
Mathematical Tool ◽  
Fuzzy Soft Set ◽  
Mathematical Methods ◽  
Real Life Problem ◽  
The Given

Generally in our day to day life we come across the real life situations when right decision making becomes difficult. In different fields like economics, social science, medical science, environment, where such complicacy arises and involves uncertainties. The conventional mathematical methods are unsuitable to solve such problems. To solve this problem, we are to find the various parameters related to the problem to use them to get the accurate result. So fuzzy soft set theory is the best mathematical tool for solving such problems using different techniques. In this research paper, we have taken a multi-criteria real life problem to select a suitable city which suits the best conditions as per the given parameters. For getting the solution, we used average fs-aggregation algorithm for using multi-criteria parameters.

scholarly journals SPECTRAL SELF-AFFINE MEASURES IN $\mathbb{R}^N$

2007 ◽  
Vol 50 (1) ◽  
pp. 197-215 ◽  
Author(s):  
Jian-Lin Li
Keyword(s):  
Large Class ◽  
Sierpinski Gasket ◽  
Three Dimensional ◽  
Iterated Function Systems ◽  
Sierpiński Gasket ◽  
Spectral Pair ◽  
Iterated Function ◽  
Spectral Pairs ◽  
The Given ◽  
Check Condition

AbstractThe aim of this paper is to investigate and study the possible spectral pair $(\mu_{M,D},\varLambda(M,S))$ associated with the iterated function systems $\{\phi_{d}(x)= M^{-1}(x+d)\}_{d\in D}$ and $\{\psi_{s}(x)=M^{\ast}x+s\}_{s\in S}$ in $\mathbb{R}^n$. For a large class of self-affine measures $\mu_{M,D}$, we obtain an easy check condition for $\varLambda(M,S)$ not to be a spectrum, and answer a question of whether we have such a spectral pair $(\mu_{M,D},\varLambda(M,S))$ in the Eiffel Tower or three-dimensional Sierpinski gasket. Further generalization of the given condition as well as some elementary properties of compatible pairs and spectral pairs are discussed. Finally, we give several interesting examples to illustrate the spectral pair conditions considered here.

scholarly journals Inequalities for nonuniform wavelet frames

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Firdous A. Shah
Keyword(s):  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Discrete Subgroup ◽  
Wavelet Frames ◽  
Wavelet System ◽  
One Dimensional ◽  
Spectral Pair ◽  
Spectral Pairs

Abstract Gabardo and Nashed studied nonuniform wavelets by using the theory of spectral pairs for which the translation set {\Lambda=\{0,r/N\}+2\mathbb{Z}} is no longer a discrete subgroup of {\mathbb{R}} but a spectrum associated with a certain one-dimensional spectral pair. In this paper, we establish three sufficient conditions for the nonuniform wavelet system {\{\psi_{j,\lambda}(x)=(2N)^{j/2}\psi((2N)^{j}x-\lambda),\,j\in\mathbb{Z},\,% \lambda\in\Lambda\}} to be a frame for {L^{2}(\mathbb{R})} . The proposed inequalities are stated in terms of Fourier transforms and hold without any decay assumptions on the generator of such a system.

scholarly journals Vector valued nonuniform nonstationary wavelets and associated MRA on local fields

2021 ◽  
Vol 17 (2) ◽  
pp. 19-46
Author(s):  
O. Ahmad ◽  
A. H. Wani ◽  
N. A. Sheikh ◽  
M. Ahmad
Keyword(s):  
Multiresolution Analysis ◽  
Complete Characterization ◽  
Local Fields ◽  
Dimension Function ◽  
Vector Valued

Abstract In this paper we study nonstationary wavelets associated with vector valued nonuniform multiresolution analysis on local fields. By virtue of dimension function a complete characterization of vector valued nonuniform nonstationary wavelets is obtained.

Sign in / Sign up

Export Citation Format

Share Document