Riesz basis of wavelets constructed from trigonometric B-splines

In this paper, we construct a class of compactly supported wavelets by taking trigonometric B-splines as the scaling function. The duals of these wavelets are also constructed. With the help of these duals, we show that the collection of dilations and translations of such a wavelet forms a Riesz basis of 𝕃2(ℝ). Moreover, when a particular differential operator is applied to the wavelet, it also generates a Riesz basis for a particular generalized Sobolev space. Most of the proofs are based on three assumptions which are mild generalizations of three important lemmas of Jia et al. [Compactly supported wavelet bases for Sobolev spaces, Appl. Comput. Harmon. Anal. 15 (2003) 224–241].

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CONSTRUCTION OF COMPACTLY SUPPORTED WAVELETS FROM TRIGONOMETRIC B-SPLINES

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s021969131100433x ◽

2011 ◽

Vol 09 (05) ◽

pp. 843-865 ◽

Cited By ~ 3

Author(s):

M. K. JENA

Keyword(s):

Convex Hull ◽

Autocorrelation Function ◽

Scaling Function ◽

Partition Of Unity ◽

B Spline ◽

B Splines ◽

Compactly Supported ◽

Odd Order ◽

Subdivision Algorithm ◽

The Scaling Function

We construct a class of semiorthogonal wavelets by taking a normalized trigonometric B-spline of any order as the scaling function. The construction is based on generalized Euler–Frobenius polynomial and generalized autocorrelation function. We also show that the odd order normalized trigonometric B-spline satisfies convex hull property as well as partition of unity property. Moreover, we also present a subdivision algorithm for the convolution of normalized trigonometric B-splines. Several examples of wavelet are also provided.

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THE CONSTRUCTION OF A CLASS OF TRIVARIATE NONSEPARABLE COMPACTLY SUPPORTED WAVELETS

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691309002908 ◽

2009 ◽

Vol 07 (03) ◽

pp. 255-267 ◽

Cited By ~ 2

Author(s):

YONGDONG HUANG ◽

SHOUZHI YANG ◽

ZHENGXING CHENG

Keyword(s):

General Theory ◽

Mild Condition ◽

Scaling Function ◽

Moment Condition ◽

Compactly Supported ◽

Vanishing Moment ◽

The Scaling Function

In this paper, under a mild condition, the construction of compactly supported [Formula: see text]-wavelets is obtained. Wavelets inherit the symmetry of the corresponding scaling function and satisfy the vanishing moment condition originating in the symbols of the scaling function. An example is also given to demonstrate the general theory.

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Maximum Vanishing Moment of Compactly Supported B-spline Wavelets

Asian Journal of Probability and Statistics ◽

10.9734/ajpas/2019/v3i330095 ◽

2019 ◽

pp. 1-8

Author(s):

Kanchan Lata Gupta ◽

B. Kunwar ◽

V. K. Singh

Keyword(s):

Scaling Function ◽

Simple Procedure ◽

Smoothness Property ◽

Vanishing Moments ◽

B Spline ◽

B Splines ◽

Compactly Supported ◽

Spline Wavelets ◽

Vanishing Moment ◽

Rule Order

Spline function is of very great interest in field of wavelets due to its compactness and smoothness property. As splines have specific formulae in both time and frequency domain, it greatly facilitates their manipulation. We have given a simple procedure to generate compactly supported orthogonal scaling function for higher order B-splines in our previous work. Here we determine the maximum vanishing moments of the formed spline wavelet as established by the new refinable function using sum rule order method.

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Compactly supported wavelet bases for Sobolev spaces

Applied and Computational Harmonic Analysis ◽

10.1016/j.acha.2003.08.003 ◽

2003 ◽

Vol 15 (3) ◽

pp. 224-241 ◽

Cited By ~ 40

Author(s):

Rong-Qing Jia ◽

Jianzhong Wang ◽

Ding-Xuan Zhou

Keyword(s):

Sobolev Spaces ◽

Wavelet Bases ◽

Compactly Supported

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M-CHANNEL MRA AND APPLICATION TO ANISOTROPIC SOBOLEV SPACES

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691305000865 ◽

2005 ◽

Vol 03 (02) ◽

pp. 153-166

Author(s):

ELENA CORDERO

Keyword(s):

Tensor Product ◽

Sobolev Spaces ◽

Unit Interval ◽

Biorthogonal Wavelet ◽

Bernstein Type ◽

Wavelet Bases ◽

Compactly Supported ◽

Anisotropic Sobolev Spaces ◽

Dilation Factor ◽

Bernstein Type Inequalities

In this paper we construct compactly supported biorthogonal wavelet bases of the interval, with dilation factor M. Next, the natural MRA on the cube arising from the tensor product of a multilevel decomposition of the unit interval is developed. New Jackson and Bernstein type inequalities are proved, providing a characterization for anisotropic Sobolev spaces.

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ELLIPTIC SCALING FUNCTIONS AS COMPACTLY SUPPORTED MULTIVARIATE ANALOGS OF THE B-SPLINES

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691314500180 ◽

2014 ◽

Vol 12 (02) ◽

pp. 1450018 ◽

Cited By ~ 4

Author(s):

VICTOR G. ZAKHAROV

Keyword(s):

Differential Operators ◽

Scaling Function ◽

Null Space ◽

Partition Of Unity ◽

Total Positivity ◽

Scaling Functions ◽

Algebraic Polynomials ◽

B Splines ◽

Compactly Supported ◽

Elliptic Differential Operators

In the paper, we present a family of multivariate compactly supported scaling functions, which we call as elliptic scaling functions. The elliptic scaling functions are the convolution of elliptic splines, which correspond to homogeneous elliptic differential operators, with distributions. The elliptic scaling functions satisfy refinement relations with real isotropic dilation matrices. The elliptic scaling functions satisfy most of the properties of the univariate cardinal B-splines: compact support, refinement relation, partition of unity, total positivity, order of approximation, convolution relation, Riesz basis formation (under a restriction on the mask), etc. The algebraic polynomials contained in the span of integer shifts of any elliptic scaling function belong to the null-space of a homogeneous elliptic differential operator. Similarly to the properties of the B-splines under differentiation, it is possible to define elliptic (not necessarily differential) operators such that the elliptic scaling functions satisfy relations with these operators. In particular, the elliptic scaling functions can be considered as a composition of segments, where the function inside a segment, like a polynomial in the case of the B-splines, vanishes under the action of the introduced operator.

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Design of High Dimensional Nonseparable Compactly Supported Wavelets with Special Dilation Matrix

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.542-543.547 ◽

2012 ◽

Vol 542-543 ◽

pp. 547-550

Author(s):

Lan Li

Keyword(s):

Mild Condition ◽

Scaling Function ◽

New Method ◽

High Dimensional ◽

Moment Condition ◽

Compactly Supported ◽

Vanishing Moment ◽

Dilation Matrix ◽

The Scaling Function

In this paper, a new method to construct the compactly supported M- wavelet under a mild condition are given. The constructed wavelet satisfies the vanishing moment condition which is originated from the symbols of the scaling function.

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Construction of a Class of Nonseparable Compactly Supported Wavelets with Special Dilation Matrix in L2(R4)

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.393-395.659 ◽

2011 ◽

Vol 393-395 ◽

pp. 659-662

Author(s):

Na Li

Keyword(s):

Mild Condition ◽

Scaling Function ◽

Moment Condition ◽

Compactly Supported ◽

Vanishing Moment ◽

Dilation Matrix ◽

Novel Method ◽

The Scaling Function

In this paper, a novel method to construct the compactly supported wavelet under a mild condition. The constructed wavelet satisfies the vanishing moment condition which is originated from the symbols of the scaling function.

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Analysis of Compactly Supported Nonstationary Biorthogonal Wavelet Systems Based on Exponential B-Splines

Abstract and Applied Analysis ◽

10.1155/2011/593436 ◽

2011 ◽

Vol 2011 ◽

pp. 1-17 ◽

Cited By ~ 1

Author(s):

Yeon Ju Lee ◽

Jungho Yoon

Keyword(s):

Riesz Bases ◽

Refinable Functions ◽

Biorthogonal Wavelet ◽

Wavelet Bases ◽

Refinable Function ◽

B Splines ◽

Compactly Supported ◽

Mathematical Properties ◽

The Stability

This paper is concerned with analyzing the mathematical properties, such as the regularity and stability of nonstationary biorthogonal wavelet systems based on exponential B-splines. We first discuss the biorthogonality condition of the nonstationary refinable functions, and then we show that the refinable functions based on exponential B-splines have the same regularities as the ones based on the polynomial B-splines of the corresponding orders. In the context of nonstationary wavelets, the stability of wavelet bases is not implied by the stability of a refinable function. For this reason, we prove that the suggested nonstationary wavelets form Riesz bases for the space that they generate.

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