Optimal Quadrature Formulas and Interpolation Splines Minimizing the Semi-Norm in the Hilbert Space $$K_{2}(P_{2})$$

Construction of Interpolation Splines Minimizing the Semi-norm in the Space K2(Pm)

“…where Remark . From Theorem 4.1, when m = 2, we get Theorem 7 of [17] and Theorem 3.1 of [19], and when m = 2, ω = 1 we get Theorem 3.1 of [18].…”

“…(0, 1) and K 2 (P 2 ) Hilbert spaces were constructed in works [8,17,18,19,31,32] by using Sobolev's method. Furthermore, the connection between interpolation spline and optimal quadrature formula in the sense of Sard in L (m) 2 (0, 1) and K 2 (P 2 ) spaces were shown in [8] and [18].…”

Construction of interpolation splines minimizing the semi-norm in the space $K_2(P_m)$

“…⊓ ⊔ Remark 4.1 From Theorem 4.1, when m = 2, we get Theorem 7 of [17] and Theorem 3.1 of [19], and when m = 2, ω = 1 we get Theorem 3.1 of the work [18].…”

On an optimal interpolation formula in $K_2(P_2)$ space

“…The equation (2.1) was solved in [5] and for the extremal function ψ ℓ was obtained the following expression…”

“…After that, this construction was modified, the degree of polynomials increased. The theory of splines based on variational methods studied and developed, for example, by J.H.Ahlberg et al [1], C. de Boor [3], A.R.Hayotov, G.V.Milovanivić and Kh.M.Shadimetov [5], I.J.Schoenberg [6], L.L.Schumaker [7], S.L.Sobolev [8], V.A.Vasilenko [12] and others.…”