Biorthogonal systems of functionals and decomposition matrices for minimal splines

For the minimal splines of arbitrary order on a nonuniform grid, a system of linear functionals biorthogonal to the system of coordinate splines is constructed. The matrices of refining and sparsing decompositions are obtained for the spaces of splines of arbitrary order associated with infinite and finite nonuniform grids on an interval and on a segment, respectively.     

Calibration relations for nonpolynomial splines

Nonpolynomial (X, A, ϕ)-splines of the third order and the special case of B _ϕ-splines of class C² are studied. For such splines calibration relations are obtained, owing to which the coordinate splines on the original grid is represented in terms of the coordinate splines on a refined grid. A nonlinear mapping (ℝ⁴)⁹ ↦ ℝ⁴ and locally orthogonal chains of vectors are used for this purpose. Bibliography: 22 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 39–54.     

Calibration relations for nonpolynomial splines

We construct wavelet decompositions and the corresponding decomposition – reconstruction algorithms in the case of an infinite flow (a grid on an open interval) and a finite flow (a grid on a segment) for a space of Lagrange type splines (in general, not polynomial). Bibliography: 11 titles.     

Cones and recurrence relations for simplex splines

We prove some new relations between functions defined as shadows of cones (cone splines) and simplices (simplex splines). We use them to show how ans-variate simplex spline of some orderk can be written as a sum ofk+1 (s-l)-variate simplex splines of orderk-1. A recurrence relation on the spatial dimension of the simplex spline,s, is proposed as an interesting alternative to the recurrence relation in [17], where one uses the orderk for recursion, but not the spatial dimensions.     

Some duality relations for local splines

We obtain duality relations for local periodic cubic and parabolic splines of minimal defect and establish some of their corollaries.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 12–19, January, 1995.This research was supported by the Ukrainian State Committee on Science and Technology.     

Error bounds for minimal energy bivariate polynomial splines

Summary. We derive error bounds for bivariate spline interpolants which are calculated by minimizing certain natural energy norms. Received March 28, 2000 / Revised version received June 23, 2000 / Published online March 8, 2002 RID="*" ID="*" Supported by the National Science Foundation under grant DMS-9870187 RID="**" ID="**" Supported by the National Science Foundation under grant DMS-9803340 and by the Army Research Office under grant DAAD-19-99-1-0160     

Permutohedra and minimal matrices

The notions of minimality, π-uniqueness and additivity originated in discrete tomography. They have applications to Kronecker products of characters of the symmetric group and arise as the optimal solutions of quadratic transportation problems. Here, we introduce the notion of real-minimality and give geometric characterizations of all these notions for a matrix A, by considering the intersection of the permutohedron determined by A with the transportation polytope in which A lies. We also study the computational complexity of deciding if the properties of being additive, real-minimal, π-unique and minimal hold for a given matrix, and show how to efficiently construct some matrix with any of these properties.     

Linear dependence relations for polynomial splines at midknots

Linear relations between midknot values of a smooth polynomial spline and its derivatives are derived for the case of a uniformly spaced set of knots. The leading term of the truncation error for midknot interpolating splines is determined for these relations. Comparisons are made with the corresponding relations and truncation errors associated with knot-interpolating splines.     

Embedding and wavelet decomposition of spaces of minimal splines

On a sequence of embedded nonuniform grids, chains of embedded spaces of minimal splines (not necessarily polynomial) are constructed. The wavelet decomposition is given. The basis wavelets are compactly supported and admit simple analytic representation. The corresponding decomposition and reconstruction formulas are derived. The variety of spaces under consideration is identified with the variety of complete sequences of points of the direct product of an interval and a projective plane. Bibliography: 20 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 15–31     

On consistency relations for polynomial splines on a uniform partition

In this paper we present linear dependence relations connecting spline values, derivative values and integral values of the spline. These relations are useful when spline interpolants or histospline projections of a function are considered.This work was supported in part by the Ministère de l'Éducation du Québec and by the Department of the National Defence of Canada.     

Reconstruction of splines from local average samples

We study the reconstruction of cardinal splines $f\left(t\right)$ from their average samples $y_n=f1h\left(n\right)$, $n\in \mathbb{Z}$, when the average function $h\left(t\right)$ has support in $[?1/2,1/2]$. We investigate the existence and uniqueness of the solution of the following problem: For given dates $y_n$, find a cardinal spline $f\left(t\right)$, of a given degree, satisfying  $y_n=f1h\left(n\right)$, $n\in \mathbb{Z}$.     

Some limit relations for multidimensional positive-definite toeplitz matrices

Odessa State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 1, pp. 55–56, January–March, 1988.     

Recurrence relations for polynomial sequences via Riordan matrices

We give recurrence relations for any family of generalized Appell polynomials unifying so some known recurrences for many classical sequences of polynomials. Our main tool to get our goal is the Riordan group. We use the product of Riordan matrices to interpret some relationships between different polynomial families. Moreover using the Hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus.     

A fast minimal residual algorithm for shifted unitary matrices

A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = ρU + ζI, where ρ and ζ are constants, U is a unitary matrix and I is the identity matrix. We show that for such matrices a Krylov subspace basis can be generated by recursion formulas with few terms. This leads to a minimal residual algorithm that requires little storage and makes it possible to determine each iterate with fairly little arithmetic work. This algorithm provides a model for iterative methods for non-Hermitian linear systems of equations, in a similar way to the conjugate gradient and conjugate residual algorithms. Our iterative scheme illustrates that results by Faber and Manteuffel [3,4] on the existence of conjugate gradient algorithms with short recurrence relations, and related results by Joubert and Young [13], can be extended.