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
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 * r,n 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
with an absolute constant in the remainder. We study the relationship between the norms L * r,n and the norms of similar operators for nonperiodic splines.
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Subbotin, Y.N., Telyakovskii, S.A. Norms on L of Periodic Interpolation Splines with Equidistant Nodes. Mathematical Notes 74, 100–109 (2003). https://doi.org/10.1023/A:1025075301686
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DOI: https://doi.org/10.1023/A:1025075301686