Histospline projections on a uniform partition
Let \(x_ i=a+ih\), \(i=0,...,N\), \(h=(b-a)/N\) be a uniform partition of the interval [a,b]. The paper is concerned with the existence and unicity of periodic histospline projections s determined by the conditions \(\int_{\Delta_ i}s(x)dx=\int_{\Delta_ i}f(c)dx\), \(\Delta_ i=[x_ i+vh,\quad x_ i+(v+1)h]\), \(i=0,...,N-1\), where \(v\in [0,1]\) is fixed and f is a function with continuous and periodic derivatives up to order \(n+1\) on [a,b]. The proofs are based on a linear dependence formula relating the quantities \(s^{(k)}(x_ i+uh)\) and the integrals of s over \([x_ i+vh\), \(x_ i+(v+l)h]\), \(u,v\in [0,1]\), proved by the authors in an earlier paper [Ann. Sci. Math. Que. 10, 5-15 (1986)]. Error bounds of the form \(\| f^{(k)}-s^{(k)}\|_{\infty}\simeq O(h^{n+1-k})\) are also obtained.
scientific article; zbMATH DE number 4084249
scientific article