A function, continuous in a finite close interval, can be approximated with a preassigned accuracy by Polynomials. A function of a Real variable which is continuous and has period can be approximated by trigonometric Polynomials.
References
Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 5, 1975.