Schoenberg, I. J. Spline interpolation and best quadrature formulae. (English) Zbl 0136.36202 Bull. Am. Math. Soc. 70, 143-148 (1964). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 24 Documents Keywords:approximation and series expansion × Cite Format Result Cite Review PDF Full Text: DOI Online Encyclopedia of Integer Sequences: T(n,k) = A321119(n) - (-1)^k*A321119(n-2*k)/2 for 0 < k < n, with T(0,0) = 0 and T(n,0) = T(n,n) = A002530(n+1) for n > 0, triangle read by rows; unreduced numerator of the weights of Holladay-Sard’s quadrature formula. a(n) = ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2); n-th row common denominator of A321118. References: [1] Carl de Boor, Best approximation properties of spline functions of odd degree, J. Math. Mech. 12 (1963), 747 – 749. · Zbl 0116.27601 [2] John C. Holladay, A smoothest curve approximation, Math. Tables Aids Comput 11 (1957), 233 – 243. · Zbl 0084.34904 [3] G. Peano, Residuo in formulas de quadratura, Mathesis 34 (1914), 1-10. · JFM 45.0451.04 [4] Arthur Sard, Best approximate integration formulas; best approximation formulas, Amer. J. Math. 71 (1949), 80 – 91. · Zbl 0039.34104 · doi:10.2307/2372095 [5] I. J. Schoenberg and Anne Whitney, On Pólya frequence functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves, Trans. Amer. Math. Soc. 74 (1953), 246 – 259. · Zbl 0051.33606 [6] I. J. Schoenberg, Spline functions, convex curves and mechanical quadrature, Bull. Amer. Math. Soc. 64 (1958), 352 – 357. · Zbl 0085.33701 [7] I. J. Schoenberg, On interpolation by spline functions and its minimal properties, On Approximation Theory (Proceedings of Conference in Oberwolfach, 1963), Birkhäuser, Basel, 1964, pp. 109 – 129. [8] J. L. Walsh, J. H. Ahlberg, and E. N. Nilson, Best approximation properties of the spline fit, J. Math. Mech. 11 (1962), 225 – 234. · Zbl 0196.48603 [9] J. L. Walsh, J. H. Ahlberg and E. N. Nilson, Best approximation and convergence properties of higher-order spline fits, Abstract 63t-103, Notices Amer. Math. Soc. 10 (1963), 202. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.