Riassunto
for example, the so-called Lp approximation, the Bernstein approxima tion problem (approximation on the real line by certain entire functions), and the highly interesting studies of J. L. WALSH on approximation in the complex plane. I would like to extend sincere thanks to Professor L. COLLATZ for his many encouragements for the writing of this book. Thanks are equally due to Springer-Verlag for their ready agreement to my wishes, and for the excellent and competent composition of the book. In addition, I would like to thank Dr. W. KRABS, Dr. A. -G. MEYER and D. SCHWEDT for their very careful reading of the manuscript. Hamburg, March 1964 GUNTER MEINARDUS Preface to the English Edition This English edition was translated by Dr. LARRY SCHUMAKER, Mathematics Research Center, United States Army, The University of Wisconsin, Madison, from a supplemented version of the German edition. Apart from a number of minor additions and corrections and a few new proofs (e. g. , the new proof of JACKSON'S Theorem), it differs in detail from the first edition by the inclusion of a discussion of new work on comparison theorems in the case of so-called regular Haar systems (§ 6) and on Segment Approximation (§ 11). I want to thank the many readers who provided comments and helpful suggestions. My special thanks are due to the translator, to Springer-Verlag for their ready compliance with all my wishes, to Mr.
Contenuti
I. Linear Approximation.- § 1. The General Linear Approximation Problem.- 1.1. Statement of the Problem. Existence Theorem.- 1.2. Strictly Convex Spaces. Hilbert Space.- 1.3. Maximal Linear Functionals.- § 2. Dense Systems.- 2.1. A General Criterion of Banach.- 2.2. Approximation Theorems of Weierstrass and Müntz.- 2.3. Approximation Theorems in the Complex Plane.- § 3. General Theory of Linear Tchebycheff Approximation.- 3.1. Fundamentals. The Theorem of Kolmogoroff.- 3.2. The Haar Uniqueness Theorem. Linear Functionals and Alternants.- 3.3. Further Uniqueness Results.- 3.4. Invariants.- 3.5. Vector-valued Functions.- § 4. Special Tchebycheff Approximations.- 4.1. Tchebycheff Systems.- 4.2. Tchebycheff Polynomials.- 4.3. The Function (x ? a)?1.- 4.4. A Problem of Bernstein and Achieser.- 4.5. Zolotareff’s Problem.- § 5. Estimating the Magnitude of Error in Trigonometric and Polynomial Approximation.- 5.1. Projection Operators. Linear Polynomial Operators.- 5.2. The Connection between Trigonometric and Polynomial Approximation.- 5.3. The Fejér Operator.- 5.4. The Korovkin Operators.- 5.5. The Theorems of D. Jackson.- 5.6. The Theorems of Bernstein and Zygmund.- 5.7. Supplements.- § 6. Approximation by Polynomials and Related Functions.- 6.1. Foundations.- 6.2. Upper Bounds for En (f).- 6.3. Lower Bounds for En (f).- 6.4. Dependence of the Approximation on the Interval.- 6.5. Regular Haar Systems.- 6.6. Asymptotic Results.- 6.7. Results for the Alternants.- § 7. Numerical Methods for Linear Tchebycheff Approximation.- 7.1. The Iterative Methods of Remez.- 7.2. Initial Approximations.- 7.3. Direct Methods.- 7.4. Discretization. Other Methods.- II. Non-linear Approximation.- § 8. General Theory of Non-linear Tchebycheff Approximation.- 8.1. Survey of the Problem. A Generalization of the Kolmogoroff Theorem.- 8.2. The Haar Uniqueness Theorem. Alternants.- 8.3. The Investigations of Rice.- 8.4. The Newton Iteration Method.- 8.5. H-Sets.- § 9. Rational Approximation.- 9.1. Existence. Invariants. A Theorem of Walsh.- 9.2. Theorems on Alternants. Anomalies. Continuity. Examples.- 9.3. Asymptotic Results. Small Intervals.- 9.4. Numerical Methods.- § 10. Exponential Approximation.- 10.1. The Results of Rice.- 10.2. An Anomaly Theorem. Constructive Methods.- § 11. Segment Approximation.- 11.1. Statement of the Problem. Hypotheses.- 11.2. The principle of Lawson.- 11.3. Equi-degree Polynomial Approximation.
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