Approximation with interpolatory constraints

2016

Journal of Approximation Theory, 1996

A theory of best approximation with interpolatory contraints from a finitedimensional subspace M of a normed linear space X is developed. In particular, to each x # X, best approximations are sought from a subset M(x) of M which depends on the element x being approximated. It is shown that this``parametric approximation'' problem can be essentially reduced to the``usual'' one involving a certain fixed subspace M 0 of M. More detailed results can be obtained when (1) X is a Hilbert space, or (2) M is an``interpolating subspace'' of X (in the sense of [1]).

Journal of Approximation Theory, 2001

The continuity conditions at the endpoints of interpolation theorems, &Ta& B j M j &a& A j for j=0, 1, can be written with the help of the approximation functional:

2001

The problem of uniform approximants subject to Hermite interpolatory constraints is considered with an alternate approach. The uniqueness and the convergence aspects of this problem are also discussed. Our approach is based on the work of P. Kirchberger (1903) and a generalization of Weierstrass approximation theorem. 2000 Mathematics Subject Classification: 41A05, 41A10, 41A29, 41A52. 1. Introduction. Let

Constructive Approximation, 2000

We discuss whether or not it is possible to have interpolatory pointwise estimates in the approximation of a function f 2 C 0; 1], by polynomials. For the sake of completeness as well as in order to strengthen some existing results, we discuss brie y the situation in unconstrained approximation. Then we deal with positive and monotone constraints where we show exactly when such interpolatory estimates are achievable by proving a rmative results and by providing the necessary counterexamples in all other cases.

2007

We discuss whether or not it is possible to have interpolatory pointwise estimates in the approximation of a function f 2 C 0; 1], by polynomials. For the sake of completeness as well as in order to strengthen some existing results, we discuss brieey the situation in unconstrained approximation. Then we deal with positive and monotone constraints where we show exactly when such interpolatory estimates are achievable by proving aarmative results and by providing the necessary counterexamples in all other cases. The eeect of the endpoints of the nite interval on the quality of approximation of continuous functions by algebraic polynomials, was rst observed by Nikolski Nik46]. Later pointwise estimates of this phenomenon were given by Timan Tim51] (k = 1), Dzjadyk Dzj58, Dzj77] (k = 2), Freud Fre59] (k = 2), and Brudny Bru63] (k 2), who proved that if f 2 C r 0; 1], then for each n N = r + k ? 1, a polynomial p n 2 n exists, such that (1.1) jf(x) ? p n (x)j c(r; k) r n (x)! k (f (r) ; ...

Annali di Matematica Pura ed Applicata, 1982

In the present paper we study a general ]orm of Peetre's J-and K-methods o] interpolation. Special emphasis is given to the equivalence theorem for J-and K-spaces and to reiteration theorems.

Journal of Approximation Theory, 1987

Arkiv för matematik, 1984

Given an interpolation couple (A0, A~), the approximation functional is dcfined

Mathematics of Computation, 2003

In the first part of this paper we apply a saddle point theorem from convex analysis to show that various constrained minimization problems are equivalent to the problem of smoothing by spline functions. In particular, we show that near-interpolants are smoothing splines with weights that arise as Lagrange multipliers corresponding to the constraints in the problem of near-interpolation. In the second part of this paper we apply certain fixed point iterations to compute these weights. A similar iteration is applied to the computation of the smoothing parameter in the problem of smoothing.