Wall Polynomials on the Real Line: A Classical Approach to OPRL Khrushchev’s FormulaSpringer Science and Business Media LLC - - 2023
Maŕıa José Cantero, Leandro del Moral Ituarte, L. Velázquez
AbstractThe standard proof of Khrushchev’s formula for orthogonal polynomials on the unit circle given in Khrushchev (J Approx Theory 108:161–248, 2001, J Approx Theory 116:268–342, 2002) combines ideas from continued fractions and complex analysis, depending heavily on the theory of Wall polynomials. Using operator theoretic tools instead, Khrushchev’s formula has...... hiện toàn bộ
The left-definite Legendre type boundary problemSpringer Science and Business Media LLC - Tập 7 - Trang 485-500 - 1991
W. N. Everitt, L. L. Littlejohn, S. C. Williams
The left-definite Legendre type boundary problem concerns the study of a fourth-order singular differential expressionM
k
[−] in a weighted Sobolev spaceH generated by a Dirichlet inner product. The fourth-order differential equation
$$M_k [y] = \lambda y$$
has orthogo...... hiện toàn bộ
Simple Cubature Formulas with High Polynomial ExactnessSpringer Science and Business Media LLC - Tập 15 - Trang 499-522 - 1999
Erich Novak, Klaus Ritter
We study cubature formulas for d -dimensional integrals with arbitrary weight function of tensor product form. We present a construction that yields a high polynomial exactness: for fixed degree, the number of knots depends on the dimension in an order-optimal way. The cubature formulas are universal: the order of convergence is almost optimal for two different scales of function spaces. The cons...... hiện toàn bộ
Splines on Triangulations with Hanging VerticesSpringer Science and Business Media LLC - Tập 36 - Trang 487-511 - 2012
Larry L. Schumaker, Lujun Wang
Polynomial spline spaces defined on triangulations with hanging vertices are studied. In addition to dimension formulae, explicit basis functions are constructed, and their supports and stability are discussed. The approximation power of the spaces is also treated.
Quadratic Convergence of Newton's Method for Convex Interpolation and SmoothingSpringer Science and Business Media LLC - Tập 19 - Trang 123-143 - 2003
Dontchev, Qi
Abstract. In this paper, we prove that Newton's method for convex best interpolation is locally quadratically convergent, giving an answer to a question of Irvine, Marin, and Smith [7] and strengthening a result of Andersson and Elfving [1] and our previous work [5]. A damped Newton-type method is presented which has global quadratic convergence. Analogous results are obtained for the convex smoo...... hiện toàn bộ
Hermite Interpolation of Data Sets by Minimal Spline SpacesSpringer Science and Business Media LLC - Tập 23 - Trang 211-227 - 2005
Manfred Sommer
Hermite interpolation of 2n + k data by spline spaces of order k with n variable knots counting multiplicities is studied. A characterization of the minimal spline spaces which admit a solution of the interpolation problem is obtained. A sufficient condition on uniqueness of interpolating spline functions is given.