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In-Depth Information
a
φ
φ
φ
φ
φ
2
3
4
5
1
x
max
x
min
b
ψ
ψ
ψ
ψ
ψ
1
2
3
4
5
x
max
x
min
Fig. 6.2 Nodal basis (a) and multilevel basis (b) at the interval [
x
min
,
x
max
]. For the nodal basis
the function
ϕ
2
is shown in bold; for the multilevel basis the function
ψ
3
is in bold
In practice of approximation theory, two types of bases are of special impor-
tance: the nodal basis and the multi-scale basis. The nodal basis has a local support
and is easy to use in practice. It can well approximate jumping functions (“well”
means it requires few coefficients), but poorly however smooth functions. An
example of a nodal basis in the one-dimensional case is shown in Fig.
6.2a
for
the linear case, where this basis is also called the Courant hat function. In its
assigned node this takes the value 1 and declines linearly to the two adjacent
nodes. Apart from this it is constantly 0. The Courant hat function can easily be
calculated and is often used in finite element method (FEM) approaches.
The multi-scale basis has in most cases a global support and employs basis
functions of different “frequencies.” It is usually less easy to use. In contrast to the
nodal bases, multi-scale bases well approximate smooth functions but poorly
however jumping functions. A classic example is the Fourier basis. An example
of a multi-scale basis in the same linear function space as in Fig.
6.2a
is shown in
Fig.
6.2b
. This type of basis is also called a multilevel basis, since its nodes are
distributed hierarchically in different levels. In the example, the basis function
nodes
ψ
5
form the coarse grid and are called coarse grid functions. The
basis function nodes
ψ
1
,
ψ
3
, and
ψ
4
form the fine grid and are called fine grid functions
(in our example they correspond to the basis functions
ψ
2
and
ϕ
4
of the nodal basis).
Of course also more than two levels can be used. (Note: In contrast to the Fourier
basis, in the multilevel basis the support of the functions is global only to a certain
degree, and in fact the multilevel basis considered here exhibits poorer approxima-
tion properties for smooth functions than does the Fourier basis. We use it however
in the interests of a better illustration, since its function space is effectively identical
to that of the nodal basis under consideration.)
ϕ
2
and
