Uniform convergence for cauchy principal value integrals of modified quasi-interpolatory splines^∗

Quasi-interpolatory splines based on Schoenberg points

“…Convergence results were already proved for product quadrature rules based on QI splines using parameters T 1 − T 5 in [1,[3][4][5]. These convergence results are both for bounded [1,3,4] and unbounded integrands [4].…”

“…These convergence results are both for bounded [1,3,4] and unbounded integrands [4]. Pointwise and uniform convergence results are proved in [1,3,5] for sequences of CPV integrals of these splines. We apply these convergence results to the QI splines based on T 6 .…”

“…The number of function evaluations is n + p − 2 for T 3 , T 4 , and n + p + 1 for T 5 . In general, the sets T 3 − T 5 contain much fewer points, and hence it is more economical to use them.…”

“…In several recent papers [1][2][3][4][5] Dagnino et al studied the application of quasiinterpolatory (QI) splines to numerical integration. This class of QI splines, introduced in [6], can be defined, starting with a positive integer p ≥ 2, the order of the spline space, a partition We shall assume throughout this paper that H n → 0 as n → ∞.…”

“…which can be defined using parameters T 6 since τ 11 := ζ 1 = −1 and τ 1n := ζ n = 1, we can prove as in [5] that J (wQ n f ; λ) → J (wf ; λ) as n → ∞, uniformly with respect to λ in …”