e are interested in decomposition
of a payoff function
,
with respect to a tensor product of a wavelet basis
.
The function
has the
form
for a given vector
and
and
for given parameters
.
We introduce the
notations
In order to perform decompositions with respect to basis
we need to evaluate scalar products of the form
.
Hence, we proceed with evaluation of the
integral
We introduce the index
sets
and the dimensionality parameter
:
We assume without loss of generality
that
We
have
where
The
may be evaluated with already developed means, see the section
(
Piecewise polynomials in
parallel
).
The set
has the following
representation:
We introduce an alternative notation for
:
where
For every dimension
we have a subdivision
of the interval of integration
along the variable
.
The entire
-dimensional
domain of the integral
has
subdivision
The
restriction
is a multidimensional polynomial for each
.
For every subdomain
that does not contain the boundary
the integral
decomposes into a product of integrals along every dimension. Such calculation
is covered in the section (
Piecewise
polynomials
).
We now calculate the integral
for a subdomain
that intersects the boundary
.
We use the convenience notation
.
If
then
for some numbers
,
.
We arrive to the following
representation:
Note that the integrals are of the same form as the original integral but in
dimensions. However, in order to arrive to a recursive recipe, we need to
consider a slightly more complicated area of integration
.
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