Untitled - Cdm.unimo.it

68 Polynomial Approximation of Differential Equations
Henceforth, depending on the basis used for the representation of a polynomial, we
distinguish among the two cases with the help of the following terminology. When the
space of polynomials is intended to be the span of the uk’s, then it is isomorph to the
set of Fourier coefficients. In this situation we call it the frequency space. Otherwise,
when the isomorphism is naturally established with the set of the point values, it will
be denoted by physical space. There are situations in which it is more convenient to
work with one of the two representations. The discrete Fourier transform allows passage
from the physical to the frequency space, thus both representations can be used to their
fullest advantage.
The cost of implementing the transform or its inverse is proportional to n 2 , since
it corresponds to a matrix-vector multiplication. Most of this chapter is dedicated to
the analysis of a special case which occurs when Chebyshev polynomials are used. For
this choice of polynomials, fast algorithms have been developed to perform the discrete
Fourier transform, considerably reducing the computational expense.
4.2 Aliasing
Two operators Πw,n and Iw,n have been respectively introduced in sections 2.4 and
3.3. For a continuous function f, both the images Πw,n−1f and Iw,nf are in Pn−1,
which is interpreted as frequency or physical space respectively. Unless f is a polynomial
in Pn−1, the projection and the interpolant do not coincide. We show two examples.
In figures 4.2.1 and 4.2.2, we plot in [−1,1] the projection and the interpolant of the
same degree, corresponding to two different functions. In the first figure we have set
f(x) = 4|x| − 2 − 1, w(x) ≡ 1 (Legendre weight function) and n = 8. In the second
we have set f(x) = 1
2 − |sinπx|, w(x) = 1/ √ 1 − x 2 (Chebyshev weight function)
and n = 13.