Abstract:
We establish some theorems on the representation of
functions $f\in L^2(\mathbb R^d)$ by series of the form
$f=\sum_{j\in\mathbb N}\sum_{k\in\mathbb Z^d}c_{j,k}\psi_{j,k}$
that are absolutely convergent with respect to the index $j$ (that is,
$\sum_{j\in\mathbb N}\bigl\|\sum_{k\in\mathbb Z^d}c_{j,k}\psi_{j,k}\bigr\|_2<\infty$),
where $\psi_{j,k}(x)=|{\det a_j}|^{1/2}\psi(a_jx-bk)$, $j\in\mathbb N$, $k\in\mathbb Z^d$,
is an affine system of functions. We prove the validity of the Bui–Laugesen
conjecture on the sufficiency of the Daubechies conditions for a positive
solution of the affine synthesis problem in the space $L^2(\mathbb R^d)$.
A constructive solution is given for this problem under a localization
of the Daubechies conditions.
Keywords:representation of functions by series, affine system, affine synthesis.
Fulltext:
PDF file (488 kB)
