Condition
(Sparse tensor product setup) We
introduce the mesh
(see the section
Wavelet analysis
) on
.
We assume that
and
are dual GMRAs, scaling functions and wavelets on
,
see the sections
(
Compactly
supported smooth biorthogonal wavelets
) and
(
Construction
of MRA and wavelets on half line or an interval
). We assume that
satisfies condition (
Frame formula 1
) of the
proposition (
Frame property 2
) and we showed
how to achieve that in the section
(
Spline functions
).

Note that we denote
the entire
-basis
of semi-orthogonal functions constructed in the section
(
Construction
of MRA and wavelets on half line or an interval
). In particular, the
statement of the proposition
(
Reproduction of polynomials 4
)
adapted to GMRA would read
for some
,
(thus, there are
vanishing moments).

We keep the
notation
but it now includes the space
so that
is in the closure of the linear span of
with
being what is
in the notation of the definitions
(
Generalized multiresolution
analysis
) and (
Dual GMRA
).

Therefore, the index
in the notation
refers to the inclusion
rather than scale operation and the index
refers to the numbering of functions produced by the procedure of the section
(
Construction
of MRA and wavelets on half line or an interval
) rather than shift
operation. However, the old and new meaning of these indexes is close, in
particular, there is
-norm
stability when changing these indexes, see the formulas
(
Property of scale and transport
2
),(
Property of scale and
transport 3
).

We keep the notation
for the
spaces
with the
property

Moreover,
for some
.

We use the
notation
which is finite because we restrict our attention to bounded sets.
Furthermore,

We will be extending wavelet approximation to the domain
.