Construction of D m -splines in L ðmÞ 2 ð0; 1Þ space by Sobolev method A. Cabada a , A.R. Hayotov a,b,⇑ , Kh.M. Shadimetov b a University of Santiago de Compostela, Department of Mathematical Analysis, Campus Vida, 15782 Santiago de Compostela, Galicia, Spain b Institute of Mathematics, National University of Uzbekistan, Do‘rmon yo‘li str., 29, Tashkent 100125, Uzbekistan article info Keywords: Polynomial spline Hilbert space The norm minimizing property S.L. Sobolev’s method Discrete argument function abstract In the present paper, using S.L. Sobolev’s method, interpolation D m -splines that minimizes the expression R 1 0 ðu ðmÞ ðxÞÞ 2 dx in the L ðmÞ 2 ð0; 1Þ space are constructed. Explicit formulas for the coefficients of the interpolation splines are obtained. The obtained interpolation spline is exact for polynomials of degree m  1. Some numerical experiments are presented. Moreover the connection between the obtained interpolation splines and the optimal quadrature formulas are shown. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction. Statement of the problem In order to find an approximate representation of a function u by elements of a certain finite dimensional space, it is possible to use values of this function at some finite set of points x b , b ¼ 0; 1; ... ; N. The corresponding problem is called the interpolation problem, and the points x b the interpolation nodes. There are polynomial and spline interpolations. It is known that the polynomial approximation is non-practical for approximation of functions with finite and small smoothness, which often occurs in applications. This circumstance makes necessary to work with the splines. Spline functions are very useful in applications. Classes of spline functions possess many nice structural properties as well as excellent approximation powers. They are used, for example, in data fitting, function approximation, numerical quadrature, and the numerical solution of ordinary and partial differential equations, integral equations, and so on. Many books are devoted to the theory of splines, for example, Ahlberg et al. [1], Arcangeli et al. [2], Attea [3], Berlinet and Thomas-Agnan [4], Bojanov et al. [5], de Boor [7], Eubank [9], Green and Silverman [12], Ignatov and Pevniy [17], Korneichuk et al. [19], Laurent [20], Mastroianni and Milovanovic ´ [21], Nürnberger [23], Schumaker [26], Stechkin and Subbotin [35], Vasilenko [36], Wahba [37] and others. If the exact values uðx b Þ of an unknown smooth function uðxÞ at the set of points fx b ; b ¼ 0; 1; ... ; Ng in an interval ½a; b are known, it is usual to approximate u by minimizing Z b a ðg ðmÞ ðxÞÞ 2 dx; ð1:1Þ in the set of interpolating functions (i.e., gðx b Þ¼ uðx b Þ; b ¼ 0; 1; ... ; N) of the Sobolev space L ðmÞ 2 ða; bÞ. Here L ðmÞ 2 ða; bÞ is the Sobolev space of functions with a square integrable mth generalized derivative. It turns out that the solution is a natural polynomial spline of degree 2m  1 with knots x 0 ; x 1 ; ... ; x N called the interpolating D m -spline for the points ðx b ; uðx b ÞÞ. In the non periodic case this problem has been investigated, at the first time, by Holladay [16] for m ¼ 2. His results have been http://dx.doi.org/10.1016/j.amc.2014.07.033 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author at: Institute of Mathematics, National University of Uzbekistan, Do‘rmon yo‘li str., 29, Tashkent 100125, Uzbekistan. E-mail addresses: alberto.cabada@usc.es (A. Cabada), abdullo_hayotov@mail.ru (A.R. Hayotov). Applied Mathematics and Computation 244 (2014) 542–551 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc