The characterization of continuous, four-coefficient scaling functions and wavelets (original) (raw)

Approximation of functions by wavelet expansions with dilation matrix

Filomat, 2023

In this paper, we obtain the degree of approximation of a function f in L p (1 ≤ p ≤ ∞) norm under general conditions of the pointwise and uniform convergence of wavelet expansions associated with the multiresolution analysis with dilation matrix. Our results show that the degree has the exponential decay (faster than any polynomial) for the function f in L p (R) on a finite interval (a, b).

On multiresolution analysis (MRA) wavelets in ℝ n

The Journal of Fourier Analysis and Applications, 2000

We prove that for any expansive n x n integral matrix A with l det A [ = 2, there exist A-dilation minimally supported frequency (MSF) wavelets that are associated with a multiresolution analysis (MRA). The condition I det A I = 2 was known to be necessary, and we prove that it is sufficient. A wavelet set is the support set of the Fourier transform of an MSF wavelet. We give some concrete examples of MRA wavelet sets in the plane. The same technique of proof is also applied to yield an existence result for A-dilation MRA subspace wavelets. An orthonormal wavelet for a dilation factor a > 0 in R is a single function ~p E L2(R) with the property that {a~r n, lEZ} is an orthonormal basis for L2(R). The proof of the existence of wavelets for any dilation factor a > 1 can be found in [6]. Similarly, one can consider wavelets in R n. If A is a real expansive matrix (equivalently, all the eigenvalues of A are required to have absolute value greater than 1), an A-dilation wavelet is a single function ~p E L2(R n) (product Lebesgue measure) with the property that {[detAl~r~(Amt-k) : m EZ, k EZ n} is an orthonormal basis for L2(Rn). In the article [7], Dai, Larson and Speegle proved the existence of wavelets for any expansive dilation matrix A. This was surprising since prior to this, several researchers had suspected that single function wavelets did not exist for A = 21 in the case n > 1. The method used in [7] was the construction of special wavelets of the form 1 ff~-I (~XE) (*) Math Subject Classifications. 42C15, 46E15.

Characterizations of Scaling Functions: Continuous Solutions

SIAM Journal on Matrix Analysis and Applications, 1994

A dilation equation is a functional equation of the form f (t) = N k=0 c k f (2t − k), and any nonzero solution of such an equation is called a scaling function. Dilation equations play an important role in several fields, including interpolating subdivision schemes and wavelet theory. This paper obtains sharp bounds for the Hölder exponent of continuity of any continuous, compactly supported scaling function in terms of the joint spectral radius of two matrices determined by the coefficients {c 0 ,. .. , c N }. The arguments lead directly to a characterization of all dilation equations that have continuous, compactly supported solutions.

Estimates of the smoothness of dyadic orthogonal wavelets of Daubechies type

Mathematical Notes, 2009

Suppose that ω(ϕ, · ) is the dyadic modulus of continuity of a compactly supported function ϕ in L 2 (R + ) satisfying a scaling equation with 2 n coefficients. Denote by α ϕ the supremum for values of α > 0 such that the inequality ω(ϕ, 2 −j ) ≤ C2 −αj holds for all j ∈ N. For the cases n = 3 and n = 4, we study the scaling functions ϕ generating multiresolution analyses in L 2 (R + ) and the exact values of α ϕ are calculated for these functions. It is noted that the smoothness of the dyadic orthogonal wavelet in L 2 (R + ) corresponding to the scaling function ϕ coincides with α ϕ .

Symmetric orthonormal scaling functions and wavelets with dilation factor 4

Advances in Computational Mathematics, 1998

Abstract: It is well known that in the univariate case, up to an integer shift andpossible sign change, there is no dyadic compactly supported symmetric orthonormalscaling function except for the Haar function. In this paper we are concerned with theconstruction of symmetric orthonormal scaling functions with dilation factor d = 4.Several examples of such scaling functions are provided in this

An algebraic approach to discrete dilations. Application to discrete wavelet transforms

The Journal of Fourier Analysis and Applications, 2000

We investigate the connections between continuous and discrete wavelet transforms on the basis of algebraic arguments. The discrete approach is formulated abstractly in terms of the action of a semidirect product A × Γ on ℓ 2 (Γ), with Γ a lattice and A an abelian semigroup acting on Γ. We show that several such actions may be considered, and investigate those which may be written as deformations of the canonical one. The corresponding deformed dilations (the pseudodilations) turn out to be characterized by compatibility relations of a cohomological nature. The connection with multiresolution wavelet analysis is based on families of pseudodilations of a different type.

The Theory of Wavelets with Composite Dilations

Applied and Numerical Harmonic Analysis

A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L 2 (R n) under the action of lattice translations and dilations by products of elements drawn from non-commuting matrix sets A and B. Typically, the members of B are shear matrices (all eigenvalues are one) while the members of A are matrices expanding or contracting on a proper subspace of R n. These wavelets are of interest in applications because of their tendency to produce "long, narrow" window functions well suited to edge detection. In this paper, we discuss the remarkable extent to which the theory of wavelets with composite dilations parallels the theory of classical wavelets, and present several examples of such systems.

L 2 (R) Solutions of Dilation Equations and Fourier-Like Transforms

2002

We state a novel construction of the Fourier transform on L 2 (R) based on translation and dilation properties which makes use of the multiresolution analysis structure commonly used in the design of wavelets. We examine the conditions imposed by variants of these translation and dilation properties. This allows other characterizations of the Fourier transform to be given, and operators which have similar properties are classified. This is achieved by examining the solution space of various dilation equations, in particular we show that the L 2 (R) solutions of f (x) = f (2x) + f (2x − 1) are in direct correspondence with L 2 ).