Norms on L of Periodic Interpolation Splines with Equidistant Nodes

Abstract

We consider the set S r,n of periodic (with period 1) splines of degree r with deficiency 1 whose nodes are at n equidistant points x i=i / n. For n-tuples y = (y 0, ... , y n-1), we take splines s r,n (y, x) from S r,n solving the interpolation problem $$s_{r,n} (y,t_i ) = y_i,$$ where t i = x i if r is odd and t i is the middle of the closed interval [x i , x i+1 ] if r is even. For the norms L * of the operator y → s r,n (y, x) treated as an operator from l 1 to L 1 [0, 1] we establish the estimate $$L_{r,n}^ * = \frac{4}{{\pi ^2 n}}log min(r,n) + O\left( {\frac{1}{n}} \right)$$ with an absolute constant in the remainder. We study the relationship between the norms L * and the norms of similar operators for nonperiodic splines.