which converges uniformly f or all values of 6. This is of course a general fact, tha t if a given function can be uniformly approximated as closely as desired by a linear combination of other functions, then that function can be expanded in a uniformly convergent series of which each term is a linear combination of those other functions, and conversely. This fact is easy to prove and will be frequently used in the sequel. If in the (x, ;y)-plane we introduce polar coordinates (r, 6) and consider the function ƒ(#) defined on the unit circle C, Weierstrass's theorem refers to the approximation of a function f(0) continuous on C by trigonometric polynomials, or what is the same thing, by polynomials of the form