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.

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.

scholarly journals Higher Rank Wavelets

2011 ◽  
Vol 63 (3) ◽  
pp. 689-720
Author(s):  
Sean Olphert ◽  
Stephen C. Power
Keyword(s):  
Orthonormal Basis ◽  
Fourier Transforms ◽  
Latin Square ◽  
Haar Wavelets ◽  
Scaling Functions ◽  
Wavelet Sets ◽  
Compactly Supported ◽  
Rank 2 ◽  
Higher Rank ◽  
Meyer Wavelet

Abstract A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in L2(ℝd). While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.

ORTHONORMAL MRA WAVELETS: SPECTRAL FORMULAS AND ALGORITHMS

2012 ◽  
Vol 10 (01) ◽  
pp. 1250008 ◽  
Author(s):  
F. GÓMEZ-CUBILLO ◽  
Z. SUCHANECKI ◽  
S. VILLULLAS
Keyword(s):  
Multiresolution Analysis ◽  
Matrix Representation ◽  
Infinite Product ◽  
Scaling Functions ◽  
Compactly Supported ◽  
Orthonormal Bases ◽  
Spectral Decompositions ◽  
Product Matrix ◽  
Orthonormal Wavelets

Spectral decompositions of translation and dilation operators are built in terms of suitable orthonormal bases of L2(ℝ), leading to spectral formulas for scaling functions and orthonormal wavelets associated with multiresolution analysis (MRA). The spectral formulas are useful to compute compactly supported scaling functions and wavelets. It is illustrated with a particular choice of the orthonormal bases, the so-called Haar bases, which yield a new algorithm related to the infinite product matrix representation of Daubechies and Lagarias.

STABILITY OF BIORTHOGONAL WAVELET BASES

2003 ◽  
Vol 01 (01) ◽  
pp. 75-92
Author(s):  
PAUL F. CURRAN ◽  
GARY McDARBY
Keyword(s):  
Linear Operator ◽  
Lifting Scheme ◽  
Simple Eigenvalue ◽  
Single Parameter ◽  
Multiple Eigenvalue ◽  
Scaling Functions ◽  
Biorthogonal Wavelet ◽  
Wavelet Bases ◽  
Compactly Supported

We investigate the lifting scheme as a method for constructing compactly supported biorthogonal scaling functions and wavelets. A well-known issue arising with the use of this scheme is that the resulting functions are only formally biorthogonal. It is not guaranteed that the new wavelet bases actually exist in an acceptable sense. To verify that these bases do exist one must test an associated linear operator to ensure that it has a simple eigenvalue at one and that all its remaining eigenvalues have modulus less than one, a task which becomes numerically intensive if undertaken repeatedly. We simplify this verification procedure in two ways: (i) we show that one need only test an identifiable half of the eigenvalues of the operator, (ii) we show that when the operator depends upon a single parameter, the test first fails for values of that parameter at which the eigenvalue at one becomes a multiple eigenvalue. We propose that this new verification procedure comprises a first step towards determining simple conditions, supplementary to the lifting scheme, to ensure existence of the new wavelets generated by the scheme and develop an algorithm to this effect.

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