Cardinal interpolation and spline functions: II interpolation of data of power growth

Abstract

Let m be a natural number and let S m denote the class of functions S( x) of the following nature: If m is even, then S( x) is a polynomial of degree m − 1 in each unit interval ( v, v + 1) for all integer values of v, while S( x) ϵ C m − 2 on the entire real axis. If m is odd, then the conditions are the same except that the intervals ( v, v + 1) are replaced by ( v − 1 2 , v + 1 2 ). The main result is as follows: If a sequence ( y v )(−∞ < v < ∞) of numbers is preassigned such that y v = O(¦ v ¦ 8) as v → ±∞, with s ⩾ 0 , then there exists a unique S( x) ϵ S m satisfying the relations S( v) = y v , for all integer v, and the growth condition S(x) = O(¦ x ¦ 8) as x → ±∞ .