  We give necessary and sufficient conditions for an orthogonal series to
converge in the mean-squares to a nonnegative function. We present many
examples and applications, in analysis and probability. In particular, we give
necessary and sufficient conditions for a Lancaster-type of expansion $%
\sum_{n\geq 0}c_{n}\alpha _{n}(x)\beta _{n}(y)$ with two sets of orthogonal
polynomials $\left\{ \alpha _{n}\right\} $ and $\left\{ \beta _{n}\right\} $ to
converge in means-squares to a nonnegative bivariate function. In particular,
we study the properties of the set $C(\alpha ,\beta )$ of the sequences
$\left\{ c_{n}\right\} ,$ for which the above-mentioned series converge to a
nonnegative function and give conditions for the membership to it. Further we
show that the class of bivariate distributions for which a Lancaster type
expansion can be found, is the same as the class of distributions having all
conditional moments in the form of polynomials in the conditioning random
variable.
