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

2014 ◽  
Vol 12 (02) ◽  
pp. 1450018 ◽  
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.

OPERATOR-ADAPTED WAVELETS: CONNECTION WITH THE STRANG–FIX CONDITIONS

2012 ◽  
Vol 10 (01) ◽  
pp. 1250006 ◽  
Author(s):  
VICTOR G. ZAKHAROV
Keyword(s):  
Differential Operators ◽  
Null Space ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Diagonal Form ◽  
Scaling Functions ◽  
Operator Matrix ◽  
Compactly Supported ◽  
Compactly Supported Function ◽  
Constant Coefficients

In this paper, we present an explicit method to construct directly in the x-domain compactly supported scaling functions corresponding to the wavelets adapted to a sum of differential operators with constant coefficients. Here the adaptation to an operator is taken to mean that the wavelets give a diagonal form of the operator matrix. We show that the biorthogonal compactly supported wavelets adapted to a sum of differential operators with constant coefficients are closely connected with the representation of the null-space of the adjoint operator by the corresponding scaling functions. We consider the necessary and sufficient conditions (actually the Strang–Fix conditions) on integer shifts of a compactly supported function (distribution) f ∈ S'(ℝ) to represent exactly any function from the null-space of a sum of differential operators with constant coefficients.

CONSTRUCTION OF COMPACTLY SUPPORTED WAVELETS FROM TRIGONOMETRIC B-SPLINES

2011 ◽  
Vol 09 (05) ◽  
pp. 843-865 ◽  
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.

scholarly journals Maximum Vanishing Moment of Compactly Supported B-spline Wavelets

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.

CONSTRUCTION OF SMOOTH COMPACTLY SUPPORTED WINDOWS GENERATING DUAL PAIRS OF GABOR FRAMES

2013 ◽  
Vol 06 (01) ◽  
pp. 1350011 ◽  
Author(s):  
Lasse Hjuler Christiansen ◽  
Ole Christensen
Keyword(s):  
Compact Support ◽  
Partition Of Unity ◽  
Dual Pair ◽  
New Method ◽  
Gabor Frames ◽  
Scaling Functions ◽  
Dual Pairs ◽  
Compactly Supported ◽  
Compactly Supported Function ◽  
New Construction

Let g be any real-valued, bounded and compactly supported function, whose integer-translates {Tkg}k∈ℤ form a partition of unity. Based on a new construction of dual windows associated with Gabor frames generated by g, we present a method to explicitly construct dual pairs of Gabor frames. This new method of construction is based on a family of polynomials which is closely related to the Daubechies polynomials, used in the construction of compactly supported wavelets. For any k ∈ ℕ ∪ {∞} we consider the Meyer scaling functions and use these to construct compactly supported windows g ∈ Ck(ℝ) associated with a family of smooth compactly supported dual windows [Formula: see text]. For any n ∈ ℕ the pair of dual windows g, hn ∈ Ck(ℝ) have compact support in the interval [-2/3, 2/3] and share the property of being constant on half the length of their support. We therefore obtain arbitrary smoothness of the dual pair of windows g, hn without increasing their support.

Riesz basis of wavelets constructed from trigonometric B-splines

2015 ◽  
Vol 13 (04) ◽  
pp. 419-436
Author(s):  
Mahendra Kumar Jena
Keyword(s):  
Differential Operator ◽  
Sobolev Space ◽  
Sobolev Spaces ◽  
Riesz Basis ◽  
Scaling Function ◽  
Wavelet Bases ◽  
B Splines ◽  
Compactly Supported ◽  
The Scaling Function

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].

Reproducing solutions to PDEs by scaling functions

2020 ◽  
Vol 18 (03) ◽  
pp. 2050017
Author(s):  
Victor G. Zakharov
Keyword(s):  
Constant Coefficient ◽  
Differential Operators ◽  
Invariant Polynomial ◽  
Scaling Functions ◽  
Scale Invariant ◽  
Compactly Supported

A generalization of the multivariate Strang–Fix conditions to no scale-invariant (only shift-invariant) polynomial spaces multiplied by exponents is introduced. A method to construct nonstationary compactly supported interpolating scaling functions that the scaling functions reproduce polynomials multiplied by exponents is presented. The polynomials (multiplied by exponents) are solutions to systems of linear constant coefficient PDEs, where the symbols of the differential operators that define PDEs can be no scale-invariant and can contain constant terms. Analytically calculated graphs of the scaling functions, including nonstationary, are presented. A concept of the so-called [Formula: see text]-separate MRAs is considered; and it is shown that, in the case of isotropic dilation matrices, the [Formula: see text]-separate scaling functions appear naturally.

scholarly journals Polynomial spaces reproduced by elliptic scaling functions

2015 ◽  
Vol 13 (06) ◽  
pp. 1550042 ◽  
Author(s):  
Victor G. Zakharov
Keyword(s):  
Differential Operator ◽  
Quadratic Form ◽  
Scaling Function ◽  
Polynomial Solution ◽  
Invariant Polynomial ◽  
Scaling Functions ◽  
Polynomial Space ◽  
Scale Invariant ◽  
Compactly Supported ◽  
Dilation Matrix

In this paper, we consider the so-called elliptic scaling functions [V. G. Zakharov, Elliptic scaling functions as compactly supported multivariate analogs of the B-splines, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014) 1450018]. Any elliptic scaling function satisfies the refinement relation with a real isotropic dilation matrix; and, in the paper, we prove that any real isotropic matrix is similar to an orthogonal matrix and the similarity transformation matrix determines a positive-definite quadratic form. We prove that the polynomial space reproduced by integer shifts of a compactly supported function can be usually considered as a polynomial solution to a system of constant coefficient PDE’s. We show that the algebraic polynomials reproduced by a compactly supported elliptic scaling function belong to the kernel of a homogeneous elliptic differential operator that the differential operator corresponds to the quadratic form; and thus any elliptic scaling function reproduces only affinely-invariant polynomial spaces. However, in the paper, we present nonstationary elliptic scaling functions such that the scaling functions can reproduce no scale-invariant (only shift-invariant) polynomial spaces.

Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

2021 ◽  
pp. 1-16
Author(s):  
Alexander Dabrowski
Keyword(s):  
General Type ◽  
Differential Operators ◽  
Laplacian Operator ◽  
Variational Characterization ◽  
Repeated Eigenvalues ◽  
Direct Extension ◽  
Asymptotic Formulae ◽  
Elliptic Differential Operators ◽  
Domain Perturbations ◽  
Boundary Deformation

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

scholarly journals Pseudo-monotonicity and degenerated or singular elliptic operators

1998 ◽  
Vol 58 (2) ◽  
pp. 213-221 ◽  
Author(s):  
P. Drábek ◽  
A. Kufner ◽  
V. Mustonen
Keyword(s):  
Sobolev Spaces ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Elliptic Operators ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

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