Given any collection of real numbers $\{ \mu _i \} _{i = 0}^\infty $, called moments, satisfying a Hamburger-like condition $\Delta _n = \det [\mu _{i + j} ]_{i,j = 0}^n \ne 0$ and a growth condition $| {\mu _n } | < cM^n n!$, where c, M are constant, $n = 0,1, \cdots $, the Chebyshev polynomials $p_0 = 1$, \[p_n (x) = \left[ {{1 / {\Delta _{n - 1} }}} \right]\left| {\begin{array}{*{20}c} {\mu _0 } & {\mu _1 } & \cdots & {\mu _n } \\ \vdots & {} & {} & \vdots \\ {\mu _{n - 1} } & {\mu _n } & \cdots & {\mu _{2n - 1} } \\ 1 & x & \cdots & {x^n } \\ \end{array} } \right|,\]$n = 1,2, \cdots $ are shown to be orthogonal with respect to the linear functional \[w(x) = \sum_{n = 0}^\infty {( - 1)^n \mu _n \delta ^{(n)} } {{(x)} / {n!}}.\] The problem of the existence of extensions of w to a space of test functions which includes polynomials is also discussed. It is shown that if $F^{ - 1} w(t)$ has an analytic continuation which has a classical Fourier transform, then that transform is the desired extension. If t...