A unified approach to certain counterexamples in approximation theory in connection with a uniform boundedness principle with rates

Dickmeis, W.; Nessel, R. J.

doi:10.1016/0021-9045(81)90041-1

Cited by 13 publications

(5 citation statements)

References 13 publications

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“…This is done in Section 2 via a rather abstract theorem, given in terms of sublinear operators in Banach spaces. In fact, these considerations continue our previous investigations [8][9][10] on quantitative uniform boundedness and condensation principles. Accordingly, the method of proof of Theorem 2.1 essentially consists in appropriate quantitative extensions of the familiar gliding hump method.…”

Section: Irt*f] = (Gs(lrs If ] )supporting

confidence: 80%

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A general approach to counterexamples in numerical analysis

1984

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Summary. The purpose of this paper is to indicate a unified approach to quantitative negative results in numerical analysis. This is done via a rather general theorem which in fact subsumes our previous quantitative uniform boundedness principles. The proof is based upon a gliding hump method. The general theorem is exemplarily applied to discuss the sharpness of various direct and inverse approximation results, known for the compound trapezoidal rule and for the approximate solution of the heat equation. The treatment outlines a program which may also be worked out for other procedures.

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Section: Irt*f] = (Gs(lrs If ] )supporting

confidence: 80%

“…[8][9][10]) which essentially correspond to (2.4-7)o(2.12,13) (cf. Corollary 3.1), the present resonance Theorem2.1 additionally yields comparison results of type (2.14, 15).…”

Section: [T~li> O~(z~)it~g~i-it~l_ I-it~(f~-l)imentioning

confidence: 99%

A general approach to counterexamples in numerical analysis

1984

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“…For this theorem, first equipped with rates by Butzer-Scherer-Westphal (1973) [24], the necessary and sufficient conditions for convergence are determined by a Jacksontype inequality for the operators as well as a condition on the operator norms, which may now be unbounded. This theorem is connected with the towering, uniform boundedness principle of functional analysis, which was generalized to a form equipped with rates by Rolf Nessel and his research team (1981)(1982)(1983) [33], to a powerful theorem covering the sharpness of error bounds. This ensures that our results are the best possible.…”

Section: Applications Of Functional Analysis To Numerical Analysis Mmentioning

confidence: 99%

A retrospective on 60 years of approximation theory and associated fields

Butzer

2009

Journal of Approximation Theory

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In collecting material for an autobiographical statement, I realized that I had been privileged to experience and participate in the emergence and evolution of approximation theory across some 60 years. As a student and then as a professional, I had observed the several strands of development within this exciting enterprise. As the son of a German emigré family, in England and Canada, I was schooled on two continents, comfortable with three languages, and exposed to a world of cultural contrasts. In my studies at Toronto (1948Toronto ( -1951, I was exposed to expatriate lecturers who had worked with the British and Russian pioneers in Fourier analysis and approximation. The Cambridge lecturing and tutoring style that permeated Toronto at the time also had a lifelong impact on my classroom presence and above all, my challenging and nurturing relationship with several generations of students.After a first position at McGill University, I spent some time in Paris and then 2 years in Mainz, where I had very productive interactions with German and US mathematicians. After semester stays in Freiburg and Würzburg, I took up a position at the Aachen University of Technology in 1958, at a time when the mathematics department was expanding its program and mission. After my promotion to chair professor in 1962, my research was diversifying and I was training my first doctoral students.Convinced of the importance of international and collaborative scholarship, I organized a first international symposium at the Oberwolfach Conference Center in August 1963. It was, perhaps unexpectedly, successful and was followed by seven further conferences across 20 years. In all, these Oberwolfach symposia-with Béla Szőkefalvi-Nagy as co-organizer from the fourth onwards-drew about 250 different experts from 24 countries including Hungary, Bulgaria, Poland, Roumania and eventually Russia, the roster of participants almost representing a Who's Who in approximation theory and associated fields such as harmonic analysis, functional analysis and operator theory, integral transform theory, orthogonal polynomials, interpolation, special functions, divergent series. The scope of the contacts so facilitated served to weaken national controls over research, favoring the emergence of new clusters of specialists, with partially overlapping interests, all thriving with the heightened interchange of ideas, methods and goals. The symposia and new research

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“…The corresponding weak problem is to find a solution u E W'2(0, 1) satisfying a(u, v) (f, v)L2(o,, for all v 6 Wo'2(0, 1),…”

Section: Introductionmentioning

confidence: 99%

On the sharpness of a superconvergence estimate in connection with one-dimensional Galerkin methods

Goebbels¹

1999

J Inequal Appl

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The present paper studies some aspects of approximation theory in the context of onedimensional Galerkin methods. The phenomenon of superconvergence at the knots is well-known. Indeed, for smooth solutions the rate of convergence at these points is O(h2r) instead of O(h +), where is the degree of the finite element space. In order to achieve a corresponding result for less smooth functions, we apply K-functional techniques to a Jackson-type inequality and estimate the relevant error by a modulus of continuity. Furthermore, this error estimate requires no additional assumptions on the solution, and it turns out that it is sharp in connection with general Lipschitz classes. The proof of the sharpness is based upon a quantitative extension of the uniform boundedness principle in connection with some ideas of Douglas and Dupont [Numer. Math. 22] (1974) 99-109. Here it is crucial to design a sequence of test functions such that a Jackson-Bernstein-type inequality and a resonance condition are satisfied simultaneously.

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A unified approach to certain counterexamples in approximation theory in connection with a uniform boundedness principle with rates

Cited by 13 publications

References 13 publications

A general approach to counterexamples in numerical analysis

A general approach to counterexamples in numerical analysis

A retrospective on 60 years of approximation theory and associated fields

On the sharpness of a superconvergence estimate in connection with one-dimensional Galerkin methods

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