Recurrence relations and fast algorithms

Abstract

We construct fast algorithms for evaluating transforms associated with families of functions which satisfy recurrence relations. These include algorithms both for computing the coefficients in linear combinations of the functions, given the values of these linear combinations at certain points, and, vice versa, for evaluating such linear combinations at those points, given the coefficients in the linear combinations; such procedures are also known as analysis and synthesis of series of certain special functions. The algorithms of the present paper are efficient in the sense that their computational costs are proportional to n ln n at any fixed precision of computations, where n is the amount of input and output data. Stated somewhat more precisely, we find a positive real number C such that, for any positive integer n ⩾ 10 and positive real number ε ⩽ 1 / 10 , the algorithms require at most C n ( ln n ) ( ln ( 1 / ε ) ) 3 floating-point operations to evaluate at n appropriately chosen points any linear combination of n special functions, given the coefficients in the linear combination, where ε is the precision of computations.