Advertisement

Least Squares and Chebyshev Systems

  • Alexander A. Roytvarf
Chapter

Abstract

As readers know, polynomials of degree n, in other words linear combinations of n + 1 monomials 1,…, t n , may have at most n real zeros. A far-reaching generalization of this fact raises a fundamental concept of Chebyshev systems, briefly, T-systems. Those systems are defined as follows. For a set (or system) of functions F = {f 0,…}, a linear combination of a finite number of elements, f = ∑c i f i , is called a polynomial on F (considered nonzero when ∃i: c i ≠ 0). A system of n + 1 function F = {f 0,…,f n } on an interval (or a half-interval, or a non-one-point segment) I ⊆ ℝ is referred to as T-system, when nonzero polynomials on F may have at most n distinct zeros in I (zeros of extensions outside I are not counted). The basic ideas of the theory of T-systems may be understood by readers with relatively limited experience. In this chapter we focus both on these ideas and on some nice applications of this theory such as estimation of numbers of zeros and critical points of functions (in analysis and geometry), a real-life problem in tomography, interpolation theory, approximation theory, and least squares.

We provide a comprehensive discussion of the linear least-squares technique and present its analytic, algebraic, and geometric versions. We examine its links to linear algebra and combinatorial analysis (in this connection we discuss some problems, previously considered in “ A Combinatorial Algorithm in Multiexponential Analysis” and “ Convexity and Related Classic Inequalities” chapters, from a different point of view). We also discuss some features of the least-squares solutions (e.g., passing through determinate points, asymptotic properties). In addition, we outline the connections between least squares and probability theory and statistics. Finally, we discuss real-life applications to polynomial interpolation (such as finding the best polynomial fitting for two-dimensional surfaces) and signal processing in nuclear magnetic resonance (NMR) technology (such as finding covariance matrices for maximal likelihood estimating parameters).

Keywords

Chebyshev Polynomial Trigonometric Polynomial Distinct Zero Unique Existence Geometric Version 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Akhiezer, N.I.: Классическая проблема моментов и некоторые вопросы анализа, связанные с ней. “Физматгиз” Press, Moscow (1961). [English transl. The Classical Moment Problem and Some Related Questions in Analysis. Oliver and Boyd Press, Edinburgh/London (1965)]Google Scholar
  2. Arnol’d, V.I.: Теорема Штурма и симплектическая геометрия. Функц. анализ и его прилож. 19(4), 1–10 (1985). [English transl. The Sturm theorem and symplectic geometry. Functional Anal. App. 19(4)]Google Scholar
  3. Arnol’d, V.I.: Сто задач (One Hundred Problems). МФТИ Press, Moscow (1989). RussianGoogle Scholar
  4. Arnol’d, V.I.: Topological Invariants of Plane Curves and Caustics. University Lecture Series, vol. 5. American Mathematical Society, Providence (1994)MATHGoogle Scholar
  5. Arnol’d, V.I.: Лекции об уравнениях с частными производными (Lectures on Partial Differential Equations). “Фазис” Press, Moscow (1997). [English transl. Lectures on Partial Differential Equations (Universitext). 1st edn. Springer (2004)]Google Scholar
  6. Arnol’d, V.I.: Что такое математика (What is Mathematics)? МЦНМО Press, Moscow (2002). RussianGoogle Scholar
  7. Banach, S.: Théorie des operations linéaries. Paris (1932). [English transl. Theory of Linear Operations (Dover Books on Mathematics). Dover (2009)]Google Scholar
  8. Berezin, I.S., Zhidkov, N.P.: Методы вычислений, ТТ. 1–2. “Физматгиз” Press, Moscow (1959–1960). [English transl. Computing Methods. Franklin Book Company (1965)]Google Scholar
  9. Bernshtein, D.N.: Число корней системы уравнений. Функц. анализ и его прилож. 9(3), 1–4 (1975). [English transl. The number of roots of a system of equations. Functional Anal. App. 9(3), 183–185]Google Scholar
  10. Briskin, M., Elichai, Y., Yomdin, Y.: How can singularity theory help in image processing. In: Gromov, M., Carbone, A. (eds.) Pattern Formation in Biology, Vision and Dynamics, pp. 392–423. World Scientific, Singapore (2000)CrossRefGoogle Scholar
  11. Brudnyi, A., Yomdin, Y. Remez Sets (preprint)Google Scholar
  12. Courant, R., Hilbert, D.: Methods of Mathematical Physics. Wiley, New York (1953–1962)Google Scholar
  13. Cramér, H.: Mathematical Methods of Statistics. Princeton University Press, Princeton (1946)MATHGoogle Scholar
  14. Dieudonné, J.: Foundation of Modern Analysis. Academic, New York/London (1960)Google Scholar
  15. Elichai, Y., Yomdin, Y.: Normal forms representation: A technology for image compression. SPIE. 1903, Image and Video Processing, 204–214 (1993)Google Scholar
  16. Erdelyi, A. (ed.): Higher Transcendental Functions, vol. 1–3. McGraw-Hill, New York/Toronto/London (1953)Google Scholar
  17. Gantmacher, F.R., Krein, M.G. Осцилляционные матрицы и ядра, и малые колебания механических систем. “Гостехиздат” Press, Moscow-Leningrad (1950). [English transl. Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems. US Atomic Energy Commission, Washington (1961)]Google Scholar
  18. Gelfand, I.M., Shilov, G.E., Vilenkin, N.Ya., Graev, N.I.: Обобщённые функции, ТТ. 1–5. “Наука” Press, Moscow (1959–1962). [English transl. Generalized Functions. V.’s 1–5. Academic Press (1964)]Google Scholar
  19. Haviv, D., Yomdin, Y.: Model based representation of surfaces (preprint)Google Scholar
  20. Helgason, S.: The Radon Transform. Progress in Mathematics, vol. 5. Birkhaüsser, Boston/Basel/Stuttgart (1980)MATHGoogle Scholar
  21. Helgason, S.: Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions. Academic (Harcourt Brace Jovanovich), Orlando/San Diego/San Francisco/New York/London/Toronto/Montreal/Tokyo/São Paulo (1984)MATHGoogle Scholar
  22. Herman, G.T.: Image Reconstruction from Projections. The Fundamentals of Computerized Tomography. Academic, New York/London/Toronto/Sydney/San Francisco (1980)MATHGoogle Scholar
  23. Karlin, S., Studden, W.J.: Tchebycheff Systems: With Application in Analysis and Statistics. Interscience Publishers A. Divison of Willey, New York/London/Sydney (1966)Google Scholar
  24. Khovanskii, A.G.: Малочлены. “Фазис” Press, Moscow (1997). [English transl. Fewnomials. Translations of Mathematical Monographs 88, AMS, Providence/Rhode Island (1991)]Google Scholar
  25. Klein, F.: Vorlesungen über die Entwicklung der Mathematic im 19. Jahrhundert. Teil 1. Für den Druck bearbeitet von Courant, R., Neugebauer, O. Springer, Berlin (1926). [English transl. Development of Mathematics in the Nineteenth Century (Lie Groups Series, No 9). Applied Mathematics Group publishing (1979)]Google Scholar
  26. Krein, M.G., Nudelman, A.A.: Проблема моментов Маркова и экстремальные задачи. “Наука” Press, Moscow (1973). [English transl. The Markov Moment Problem and Extremal Problems, Translations of Math. Monographs, V.50. American Mathematical Society, Providence/Rhode Island (1977)]Google Scholar
  27. Lang, S.: Algebra. Addison-Wesley, Reading/London/Amsterdam/Don Mills/Sydney/Tokyo (1965)MATHGoogle Scholar
  28. Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston (1964)MATHGoogle Scholar
  29. McLachlan, N.W.: Theory and Application of Mathieu Functions. Dover, New York (1964)MATHGoogle Scholar
  30. Polya, G., & Szegö, G.: Aufgaben und Lehrsätze aus der Analysis. Springer, Göttingen/Heidelberg/New York (1964). [English transl. Problems and Theorems in Analysis I, II. Springer, Reprint edition (1998)]Google Scholar
  31. Prasolov, V.V., Soloviev, Y.P.: Эллиптические функции и алгебраические уравнения (Elliptic Functions and Algebraic Equations). “Факториал” Press, Moscow (1997). RussianGoogle Scholar
  32. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in Fortran 77. The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge/New York/Melbourne (1992)Google Scholar
  33. Riesz, F. & Sz.-Nagy, B.: Leçons d’analyse fonctionnnelle. Sixtème edition, Budapest (1972). [English transl. Functional Analysis. Dover (1990)]Google Scholar
  34. Rudin, W.: Functional Analysis. McGraw-Hill, New York/St. Louis/San Francisco/Düsseldorf/Johannesburg/London/Mexico/Montreal/New Delhi/Panama/Rio de Janeiro/Singapore/Sydney/Toronto (1973)MATHGoogle Scholar
  35. Smith, K.T., Solmon, D.C., Wagner, S.L.: Practical and mathematical aspects of the problem of reconstructing objects from radiographs. Bull. AMS 83(6), 1227–1270 (1977)MathSciNetMATHCrossRefGoogle Scholar
  36. Stoker, J.J.: Nonlinear Vibrations in Mechanical and Electrical Systems. Interscience, New York (1950)MATHGoogle Scholar
  37. Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence/Rhode Island (1981)Google Scholar
  38. Van der Waerden, B.L.: Mathematische statistik. Springer, Berlin/Göttingen/Heidelberg (1957). [English transl. Mathematical Statistics. Springer (1969)]Google Scholar
  39. Van der Waerden, B.L.: Algebra II. Springer, Berlin/Heidelberg/New York (1967)CrossRefGoogle Scholar
  40. Van der Waerden, B.L.: Algebra I. Springer, Berlin/Heidelberg/New York (1971)MATHGoogle Scholar
  41. Vilenkin, N.Ya.: Специальные функции и теория представлений групп. “Наука” Press, Moscow (1965). [English transl. Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs 22, American Mathematical Society, Providence/Rhode Island (1968)]Google Scholar
  42. Walker, R.J.: Algebraic Curves. Princeton University Press, Princeton/New Jersey (1950)MATHGoogle Scholar
  43. Wiener, Z., Yomdin, Y.: From formal numerical solutions of elliptic PDE's to the true ones. Math. Comput. 69(229), 197–235 (2000)MathSciNetMATHGoogle Scholar
  44. Yomdin, Y.: Discrete Remez inequality. Isr. J. of Math. (submitted)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Alexander A. Roytvarf
    • 1
  1. 1.Rishon LeZionIsrael

Personalised recommendations