On the convergence of odd-degree spline interpolation (original) (raw)

A remainder formula and limits of cardinal spline interpolants

Transactions of the American Mathematical Society, 1982

A Peano-type remainder formula \[ f ( x ) − S n ( f ; x ) = ∫ − ∞ ∞ K n ( x , t ) f ( n + 1 ) ( t ) d t f(x) - {S_n}(f;\,x) = \int _{ - \infty }^\infty {{K_n}(x,\,t){f^{(n + 1)}}(t)\,dt} \] for a class of symmetric cardinal interpolation problems C.I.P. ( E , F , x ) (E,\,F,\,{\mathbf {x}}) is obtained, from which we deduce the estimate | | f − S n , r ( f ; ) | | ∞ ⩽ K | | f ( n + 1 ) | | ∞ ||f - {S_{n,r}}(f;\,)|{|_\infty } \leqslant K||{f^{(n + 1)}}|{|_\infty } . It is found that the best constant K K is obtained when x {\mathbf {x}} comprises the zeros of the Euler-Chebyshev spline function. The remainder formula is also used to study the convergence of spline interpolants for a class of entire functions of exponential type and a class of almost periodic functions.

On the Interpolation by Discrete Splines with Equidistant Nodes

Journal of Approximation Theory, 2000

In this paper we consider discrete splines S(j), j ∈ Z, with equidistant nodes which may grow as O(|j| s) as |j| → ∞. Such splines are relevant for the purposes of digital signal processing. We give the definition of discrete B-splines and describe their properties. Discrete splines are defined as linear combinations of shifts of the B-splines. We present a solution to the problem of discrete spline cardinal interpolation of sequences of power growth and prove that the solution is unique within the class of discrete splines of a given order.

Optimal local spline interpolants

Journal of Computational and Applied Mathematics, 1987

A constructive proof is given of the existence of a local spline interpolant which also approximates optimally in the sense that its associated operator reproduces polynomials of maximal order. First, it is shown that such an interpolant does not exist for orders higher than the linear case if the partition points of the appropriate spline space coincide with the given interpolation points. Next, in the main result, the desired existence of an optimal local spline interpolant for all orders is proved by increasing, in a specified manner, the set of partition points. Although our interpolant reproduces a more restricted function space than its quasi-interpolant counterpart constructed by De Boor and Fix [l], it has the advantage of interpolating every real function at a given set of points. Finally, we do some explicit calculations in the quadratic case.

On two families of near-best spline quasi-interpolants on non-uniform partitions of the real line

Arxiv preprint math/ …, 2006

The univariate spline quasi-interpolants (abbr. QIs) studied in this paper are approximation operators using B-spline expansions with coefficients which are linear combinations of discrete or weighted mean values of the function to be approximated. When working with nonuniform partitions, the main challenge is to find QIs which have both good approximation orders and uniform norms which are bounded independently of the given partition. Near-best QIs are obtained by minimizing an upper bound of the infinity norm of QIs depending on a certain number of free parameters, thus reducing this norm. This paper is devoted to the study of two families of near-best QIs of approximation order 3.

On Optimal Error Bounds for Derivatives of Interpolating Splines on a Uniform Partition

Journal of Approximation Theory, 1999

Based on Peano kernel technique, explicit error bounds (optimal for the highest order derivative) are proved for the derivatives of cardinal spline interpolation (interpolating at the knots for odd degree splines and at the midpoints between two knots for even degree splines). The results are based on a new representation of the Peano kernels and on a thorough investigation of their zero distributions. The bounds are given in terms of Euler Frobenius polynomials and their zeros.