Splines (with optimal knots) are better

Burchard, Hermann G. W.

doi:10.1080/00036817408839073

Cited by 76 publications

(43 citation statements)

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“…Examples of functions in Xp\Np,n axe most easily constructed using (3.6), since it is shown in [2] that / G Np,n iff /EX' and BpAf) < °°-Thus> an example in [1] which is in W1>Uoc n L1 but has Il ¿/Il 1/2 = oo is in L1 \ N1'1. As shown in [7], /(/) = |log(0|_I, belonging to X°°(0, i), has ||Z)2/||I/2 = oo hence is in Xo0 \ N00,2.…”

Section: Definitionmentioning

confidence: 99%

On the degree of convergence of piecewise polynomial approximation on optimal meshes

Burchard¹

1977

Trans. Amer. Math. Soc.

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Abstract. The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain /"-spaces Ng'" (introduced for this purpose in [2]). We obtain two o-results and use pairs of inequalities of Bernstein-and Jackson-type to prove several direct and converse theorems. For/in Ng'" we define a derivative D"-'f in V, a = (n + p~l)~x, which agrees with D"ffot smooth/, and prove several properties of D"'".1. Introduction and summary of results. In this paper we prove direct and converse theorems for piecewise polynomial and spline approximation, of fixed degree, on optimal meshes in the Lp-norm, 1 < p < oo. There exists an intimate relationship between this problem and certain /"-spaces Ng,n. These were first introduced, in this connection, by Burchard and Hale [2], who established direct theorems of the 0(k~")-)and. Here, k is the order of the mesh, and n -1 the degree of the polynomial segments. In the present paper, we further analyze the spaces Ng,n. We obtain two direct "o" theorems (o(k~n) and o(k~n+x) resp.), as well as other direct and converse theorems, some of which give necessary and sufficient conditions for the degree of convergence 0(k~e) with 0 < 0 < n (the suboptimal case) or n -1 < 0 (n > 2). For 9 = n = 1 (the optimal case) direct and converse theorems do not yet match up, except forp = oo, a problem solved by J. P. Kahane.We mention that the o(fc-n+I)-result amounts to a significant weakening of the hypothesis in a theorem of Freud and Popov, which in turn is an improved extension of a theorem of Korneicuk [9]. Our o(k~")-theoxem shows, roughly speaking, if D"~xf EB\ and D"~xf is a singular function, then/ can be approximated more rapidly by piecewise solutions s of D"s = 0

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Section: Definitionmentioning

confidence: 99%

On the degree of convergence of piecewise polynomial approximation on optimal meshes

Burchard¹

1977

Trans. Amer. Math. Soc.

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“…5. Burchard [3] has obtained useful error bounds for best Lv approximations by variable-knot polynomial splines; these are similar in form to but less sharp than the asymptotic estimates in Sec. 6.…”

Section: Introductionmentioning

confidence: 91%

“…(IX) Burchard [3] has obtained an upper bound similar in form to (6.18) for Lr approximation by polynomial splines, i.e. the case L = Dn.…”

Section: T That Is If T Is Finer Than S Then E(/ T) < E(j S)mentioning

confidence: 99%

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Untitled

1975

Quart. Appl. Math.

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Introduction.The feature-selection problem in pattern analysis is concerned with the assignment of analytically or computationally tractable representations to patterns. Two common goals in the mathematical analysis of this problem are to make the representations precise in preserving salient characteristics of patterns, and concise in eliminating redundancy in the representations.When the objects of the pattern analysis are curves, or line patterns, defined by functions on a subset of the real line, then the feature selection procedure is conveniently formulated in one way as an approximation problem. "Optimal" representations of patterns are derived as best approximations of the patterns by members of a suitablychosen class of approximating functions. The motivation for the present study is the feature-selection problem for line patterns that exhibit a characteristic quality of piecewise regularity. We are led to the analysis of problems of nonlinear approximation of real-valued functions on the line. The connections between line-pattern analysis and the approximation-theoretic problems will be drawn in greater detail in the next section.The principal mathematical results of this paper belong to the area of approximation theory. The classes of approximating functions that we consider are closely related to extended Chebyshevian splines with variable knots on a finite interval of the real line. They are defined by a linear differential operator L with constant coefficients. An approximating function will satisfy L = 0 except at a finite number of points h , • • • , tv in the domain of interest, and at the points t, continuity constraints may be imposed on 4>. The points t, are referred to as knots of . The approximating classes include, in particular, finite-order algebraic and trigonometric polynomials and splines.For a line pattern identified with a function / we consider the problem of optimal approximation of / by such a function with respect to a norm on the space of line patterns. Approximations are optimized with respect to dependence on knots and this makes the optimization problem distinctly nonlinear. We concentrate on mean-square

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“…There are various analyses of the approximation error of a given function by piecewise polynomials with a fixed number of pieces of fixed order (see, e.g., [7], [8], [10], [12], [21] and the references cited there). In principle, such studies may relate to the finite-element method since that method leads to optimal approximations under the energy norm.…”

Section: Introduction For the Numerical Solution Of Boundary Value Pmentioning

confidence: 99%

Untitled

1979

Math. Comp.

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Abstract. A theory of a posteriori estimates for the finite-element method was developed earlier by the authors.Based on this theory, for a two-point boundary value problem the existence of a unique optimal mesh distribution is proved and its properties analyzed. This mesh is characterized in terms of certain, easily computable local error indicators which in turn allow for a simple adaptive construction of the mesh and also permit the computation of a very effective a posteriori error bound. While the error estimates are asymptotic in nature, numerical experimentsshow the results to be excellent already for 10% accuracy. The approaches are not restricted to the model problem considered here only for clarity; in fact, they allow for rather straightforward extensions to more general problems in one dimension, as well as to higher-order elements.

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Splines (with optimal knots) are better

Cited by 76 publications

References 1 publication

On the degree of convergence of piecewise polynomial approximation on optimal meshes

On the degree of convergence of piecewise polynomial approximation on optimal meshes

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