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1
SPECIAL FAMILIES
OF POLYNOMIALS
Approximating functions in spectral methods are related to polynomial solutions of
eigenvalue problems in ordinary differential equations, known as Sturm-Liouville prob-
lems. These originate from applying the method of separation of variables in the analysis
of boundary-value problems. We outline both basic and remarkable properties of the
most commonly used families of polynomials of this kind.
1.1 Sturm-Liouville problems
Let I denote an open interval in R. We consider the continuous functions a : Ī → R,
b : I → R, w : I → R, satisfying a ≥ 0 in Ī and w > 0 in I. We are concerned with
the solutions (λ,u) to the following eigenvalue problem:
(1.1.1) −(au ′ ) ′ + bu = λw u in I , λ ∈ R.
A large variety of boundary conditions can be assumed in order to determine uniquely,
up to a constant factor, the eigenfunction u. Problem (1.1.1) is said to be regular if
a > 0 in Ī. When a vanishes at least at one point in Ī, then we have a singular problem.
The literature about this subject gives plenty of results. For a general theory, we can
refer for instance to courant and hilbert(1953), yosida(1960), brauer and nohel