Two-Scale Difference Equations. I. Existence and Global Regularity of Solutions

Refinement equations and spline functions

“…First, when λ and all the d j are integers, the regularity of refinable functions is related to the spectrum of a certain matrix that is constructed by the refinement coefficients c j . In this case, a negative answer to this question was given in [6]. However, the matrix can only be constructed in the case when λ and all the d j are integers.…”

“…In [6], Daubechies and Lagarias showed that up to a scalar multiple the refinement equation (1) Moreover, they also showed the nonexistence of a C ∞ -refinable function with compact support in one dimension when the number λ and all the d j are integers. In [2], Cavaretta et al extended this result to higher dimensions by a matrix method.…”

On dilation equations and the Hölder continuity of the de Rham functions

“…where /(£) = /e' xi f{x) dx is the Fourier transform of/, and It is stated in [1] that it is not known how to prove the continuity of the de Rham function directly from (3). The purpose of this note is to provide a proof of their result using precisely (3).…”

“…The purpose of this note is to provide a proof of their result using precisely (3). For results on more general dilation equations, we refer the reader to [1] and [2]. …”

Wavelet‐based estimation of a discriminant function

“…We give the matlab program based on Daubechies-Lagarias algorithm (Daubechies and Lagarias, 1991;1992), that calculates a value of scaling function at an arbitrary design point with a prescribed precision.…”