scholarly journals Vector-valued nonuniform multiresolution analysis related to Walsh function

2020 ◽  
Vol 8 (1) ◽  
pp. 206-219
Author(s):  
Abdullah
Keyword(s):  
Multiresolution Analysis ◽  
Walsh Function ◽  
Half Line ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Vector Valued ◽  
Positive Half

In this paper, we introduce vector-valued nonuniform multiresolution analysis on positive half-line related to Walsh function. We obtain the necessary and sufficient condition for the existence of associated wavelets.

Vector-valued nonuniform multiresolution analysis on local fields

2015 ◽  
Vol 13 (04) ◽  
pp. 1550029 ◽  
Author(s):  
Firdous Ahmad Shah ◽  
M. Younus Bhat
Keyword(s):  
Multiresolution Analysis ◽  
Orthonormal Basis ◽  
Positive Characteristic ◽  
Local Fields ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Vilenkin Groups ◽  
Refinement Mask ◽  
Necessary And Sufficient ◽  
Vector Valued

A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis (VNUMRA) where the associated subspace V0 of L2(K, ℂM) has an orthonormal basis of the form {Φ (x - λ)}λ∈Λ where Λ = {0, r/N} + 𝒵, N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime and 𝒵 = {u(n) : n ∈ ℕ0}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(ξ) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups.

A New Procedure for Designing a Kind of Orthogonal Vector-Valued Multiscaled Wavelets and Wavelet Wraps

2011 ◽  
Vol 204-210 ◽  
pp. 1733-1736
Author(s):  
Hong Wei Gao
Keyword(s):  
Multiresolution Analysis ◽  
Filter Bank ◽  
Matrix Theory ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Orthogonal Vector ◽  
Novel Method ◽  
Necessary And Sufficient ◽  
Vector Valued

In this paper, notion of vector-valued multiresolution analysis is introduced. So does the notion of orthogonal vector-valued wavelets A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is presented by using paraunitary vector filter bank theory and matrix theory. A novel method for constructing a kind of orthogonal shortly supported vector -valued wavelets is presented.

WAVELETS ASSOCIATED WITH NONUNIFORM MULTIRESOLUTION ANALYSIS ON POSITIVE HALF-LINE

2012 ◽  
Vol 10 (02) ◽  
pp. 1250018 ◽  
Author(s):  
MEENAKSHI ◽  
P. MANCHANDA ◽  
A. H. SIDDIQI
Keyword(s):  
Multiresolution Analysis ◽  
Scaling Function ◽  
Half Line ◽  
Necessary And Sufficient Condition ◽  
Locally Compact Abelian Groups ◽  
Spectral Pairs ◽  
Compact Abelian Groups ◽  
Necessary And Sufficient ◽  
The Scaling Function ◽  
Positive Half

Gabardo and Nashed have studied nonuniform multiresolution analysis based on the theory of spectral pairs in a series of papers, see Refs. 4 and 5. Farkov,3 has extended the notion of multiresolution analysis on locally compact Abelian groups and constructed the compactly supported orthogonal p-wavelets on L2(ℝ+). We have considered the nonuniform multiresolution analysis on positive half-line. The associated subspace V0 of L2(ℝ+) has an orthonormal basis, a collection of translates of the scaling function φ of the form {φ(x ⊖ λ)}λ∈Λ+ where Λ+ = {0, r/N} + ℤ+, N > 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N - 1 such that r and N are relatively prime and ℤ+ is the set of non-negative integers. We find the necessary and sufficient condition for the existence of associated wavelets and derive the analogue of Cohen's condition for the nonuniform multiresolution analysis on the positive half-line.

scholarly journals Semi-smooth Points in Some Classical Function Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Spaces Of Continuous Functions ◽  
Classical Function ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

An Optimization Algorithm for Orthogonal Trivariate Wavelets with Short Support

2012 ◽  
Vol 461 ◽  
pp. 835-839
Author(s):  
Ke Zhong Han
Keyword(s):  
Filter Bank ◽  
Scaling Function ◽  
Wavelet Packets ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Orthogonal Vector ◽  
Multi Scale ◽  
Dilation Factor ◽  
Necessary And Sufficient ◽  
Vector Valued

Wavelet analysis is nowadays a widely used tool in applied mathe-matics. The advantages of wavelet packets and their promising features in various application have attracted a lot of interest and effort in recent years.. The notion of vector-valued binary wavelets with two-scale dilation factor associated with an orthogonal vector-valued scaling function is introduced. The existence of orthogonal vector-valued wavelets with multi-scale is discussed. A necessary and sufficient condition is presented by means of vector-valued multiresolution analysis and paraunitary vector filter bank theory. An algorithm for constructing a sort of orthogonal vector-valued wave-lets with compact support is proposed, and their properties are investigated.

scholarly journals The product of vector-valued measures

1973 ◽  
Vol 8 (3) ◽  
pp. 359-366 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Scalar Field ◽  
Banach Spaces ◽  
Vector Measure ◽  
Sufficient Condition ◽  
Pettis Integral ◽  
Necessary And Sufficient Condition ◽  
Bilinear Map ◽  
Scalar Measure ◽  
Necessary And Sufficient ◽  
Vector Valued

Let M (N) be a σ–algebra of subsets of a set S (T) and let X, Y be Banach spaces with (,) a continuous bilinear map from X × Y into the scalar field. If μ: M → X (v: N → Y) is a vector measure and λ is the scalar measure defined on the measurable rectangles A × B, A ∈ M, B ∈ N, by λ(A×B) = 〈μ(A), v(B)〉, it is known that λ is generally not countably additive on the algebra generated by the measurable rectangles and therefore has no countably additive extension to the σ-algebra generated by the measurable rectangles. If μ (v) is an indefinite Pettis integral it is shown that a necessary and sufficient condition that λ have a countable additive extension to the σ-algebra generated by the measurable rectangles is that the function F: (s, t) → 〈f(s), g(t)〉 is integrable with respect to α × β.

scholarly journals The theory of functional least squares

1983 ◽  
Vol 34 (3) ◽  
pp. 336-355 ◽  
Author(s):  
Sándor Csörgő
Keyword(s):  
Least Squares ◽  
Sufficient Condition ◽  
Slope Parameter ◽  
Necessary And Sufficient Condition ◽  
Continuous Vector ◽  
The Family ◽  
Strong Uniform Consistency ◽  
Necessary And Sufficient ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractThe functional least squares procedure of Chambers and Heathcote for estimating the slope parameter in a linear regression model is analysed. Strong uniform consistency for the family of these estimators is proved together with a necessary and sufficient condition for weak convergence in the space of continuous vector valued functions. These results are then used to develop the asymptotic normality of an adaptive version of the functional least squares estimator with minimum limiting variance.

scholarly journals Multidimensional Tauberian theorems for vector-valued distributions

2014 ◽  
Vol 95 (109) ◽  
pp. 1-28 ◽  
Author(s):  
Stevan Pilipovic ◽  
Jasson Vindas
Keyword(s):  
Asymptotic Stability ◽  
Integral Transform ◽  
Tauberian Theorems ◽  
Sufficient Condition ◽  
Cauchy Problems ◽  
Necessary And Sufficient Condition ◽  
The Laplace Transform ◽  
Distribution Spaces ◽  
Necessary And Sufficient ◽  
Vector Valued

We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of f is given by the integral transform Mf?(x,y) = (f*?y)(x), (x,y) ? Rn ? R+, with kernel ?y(t) = y?n?(t/y). We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on {x0}?Rm. In addition, we present a new proof of Littlewood?s Tauberian theorem.

Nonuniform wavelet frames in L2(ℝ)

2015 ◽  
Vol 08 (02) ◽  
pp. 1550034 ◽  
Author(s):  
Vikram Sharma ◽  
P. Manchanda
Keyword(s):  
Multiresolution Analysis ◽  
Sufficient Condition ◽  
Wavelet Frame ◽  
Necessary And Sufficient Condition ◽  
Wavelet Frames ◽  
Spectral Pairs ◽  
Necessary And Sufficient ◽  
Funct Anal

Gabardo and Nashed, [Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal. 158 (1998) 209–241] introduced the nonuniform multiresolution analysis (NUMRA) whose translation set is not necessarily a group. The translation set is taken for elements in [Formula: see text], N ≥ 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N - 1 such that r and N are relatively prime and ℤ is the set of all integers. In this paper, we construct wavelet frame over the translation set Λ on L2(ℝ). We call it nonuniform wavelet frame. We establish necessary and sufficient condition for such wavelet frame. An example is presented at the end.

A Novel Approach for Constructing a Class of Orthogon Vector-Valued Wavelets with Finite Support

2010 ◽  
Vol 439-440 ◽  
pp. 1123-1128
Author(s):  
Shui Wang Guo ◽  
Jin Chang Shi
Keyword(s):  
Filter Bank ◽  
Matrix Theory ◽  
Sufficient Condition ◽  
Analysis Method ◽  
Necessary And Sufficient Condition ◽  
Time Frequency ◽  
Orthogonal Vector ◽  
Novel Approach ◽  
Necessary And Sufficient ◽  
Vector Valued

In this paper, the notion of orthogonal vector-valued wavelets is introduced. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is presented by using paraunitary vector filter bank theory, time-frequency analysis method and matrix theory. A new method for constructing a class of orthog- -onal finitectly supported vector-valued wavelets is presented.

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