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 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 → ±∞ .