Abstract. Some algorithms are introduced, whereby a function defined on an arbitrarily spaced set of abscissas may be interpolated or approximated by trigonometric or hyperbolic polynomials. The interpolation may be ordinary or osculatory. Least squares approximation is included; the approximant may be a pure sine series or a cosine series or a balanced trigonometric or hyperbolic polynomial. An application to a periodicity-search is described.An extensive set of algorithms is available for functional approximation and interpolation in terms of polynomials. The present article develops some corresponding algorithms for nonpolynomial approximants. The classes of approximant (interpolant) considered are sine polynomial, cosine polynomial, balanced trigonometric polynomial and their analogs in terms of hyperbolic functions. The classes of approximation considered are interpolation on ordinates, osculatory and hyperosculatory interpolation, weighted least-squares approximation, weighted least-squares approximation subject to some ordinate and derivative constraints. Let /' be the range of 0 induced by the requirement x£/ and let n,,(0) be a function of degree (j, j) such that n,,(0,) = 0 for / g 2j; moreover, these 2j zeros are the only zeros of IT,,-in /'. For consecutive j we can now construct the functions IT,-,-, which are unique to within a normalization factor. Starting with noo(0) = 1, we may define for y = 0,1 • ■ • ní+1,,+i = gX©)!!,-,, where g,(0) = a¡ sin 0 + /3, cos 0 -y¡.