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5
FUNCTIONAL SPACES
Before studying the convergence properties of orthogonal polynomials, it is necessary
to specify the classes of functions to be approximated. This leads to the definition of
new functional spaces. An elementary introduction of the so called Sobolev spaces is
given in this chapter. Beginners, with a weak mathematical background, can skip over
this part in the first reading.
5.1 The Lebesgue integral
The Lebesgue integral extends the classical Riemann integral presented in the intro-
ductory courses of calculus. This generalization allows the definition of integrals of
functions that belong to very large classes. This improvement does not only offer the
possibility of dealing with pathological functions (most of which are not of interest to
us), but provides new tools to help prove sophisticated results within the framework of
a simple and elegant theory.
We start by reviewing the main properties of the Lebesgue measure theory. For a
deeper analysis, we refer to the books by halmos(1971), kolmogorov and fomin(1961),
hartman and mikusiński(1961), hildebrandt(1963), temple(1971), rao(1987), etc..