Innovation in Engineering Computational Technology
Chapter
17 R. Cruz-Barroso, P. González-Vera and F. Perdomo-Pío
Department of Mathematical Analysis, La Laguna University, Tenerife, Canary Islands, Spain Keywords: orthogonal Laurent polynomials on the unit circle, Fourier expansion, Szego quadratures, interpolation process.
Let One of the most usual techniques with excellent results in the estimation of integrals (1) is the Trapezoidal rule: where  are the  -th roots of the unit. The
efficient computation of the coefficients
 has given rise to the well known algorithm
of the Fast Fourier Transform (FFT) whose basic theoretical
background is the following result: "Let
 be the unique
polynomial of degree  at most, interpolating  at the
-th roots of unity. Then,
 with
 given by 2 for
 ".
We start by showing that the above result and the relations
(2) arise in a more general context, considering a
weight function
 ,
 ). Now, the
calculation of the coefficients  implies the computation of
weighted integrals on the unit circle, and since the weight function
can exhibit singularities near  , the Trapezoidal rule seems to
be unadvisable. In this work, more general procedures than the
Trapezoidal rule are proposed in connection with the so called
"Szego quadrature formulas" [2]. Such formulas are
revisited, considering characterization results (for the nodes,
weights and domains of exactness), error estimations and a
convergence result.
The next purpose is concerned with the calculation of the Fourier
coefficients of functions with polar singularities, that is, with
functions of the form
Finally, the concepts of Fourier expansion and interpolation are
related and a connection with the Gauss-Christoffel, Gauss-Radau and
Gauss-Lobatto formulae are established. More precisely, we shall be
concerned with interpolation properties for real functions either
References
| ||||||||||
be a complex function defined on the unit circle
and consider its Fourier series
, where
on 
of orthogonal Laurent polynomials on
[
where
, 
which is a polynomial
with all its zeros not on 
we need to compute
,
where 
. In this respect, two
situations are particulary analysed (a symmetric and a non-symmetric
distribution of poles) and the Szego, interpolatory and
Trapezoidal rules computed, showing with some numerical experiments
that, as expected, both Szego and interpolatory rules give better
results than the Trapezoidal one and that the results strongly
depend on the "soft" part of the function whose Fourier
coefficients we need to compute. From the computations it seems it
can be also deduced that the number of coefficients 
which can
be effectively computed is approximately 
-periodic by means of trigonometric polynomials or not
periodic by means of algebraic polynomials taking as interpolation
nodes the roots of
. Moreover, some results of
interpolation on intervals 
of the real line are deduced and
a convergence result established. Here, the interpolation nodes are
zeros of orthogonal polynomials with respect to the Chebyshev-type
weight functions, namely
where
.