Generalized Fourier series

In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we consider that of square-integrable functions defined on an interval of the real line, which is important, among others, for interpolation theory.

Consider a set of square-integrable functions with values in ${\displaystyle \mathbb {F} =\mathbb {C} {\mbox{ or }}\mathbb {R} }$,

which are pairwise orthogonal for the inner product

where w(x) is a weight function, and ${\displaystyle {\overline {\cdot }}}$ represents complex conjugation, i.e. ${\displaystyle {\overline {g}}(x)=g(x)}$ for ${\displaystyle \mathbb {F} =\mathbb {R} }$.

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