Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Müntz systems and orthogonal Müntz-Legendre polynomials


Authors: Peter Borwein, Tamás Erdélyi and John Zhang
Journal: Trans. Amer. Math. Soc. 342 (1994), 523-542
MSC: Primary 42C05; Secondary 39A10, 41A17
DOI: https://doi.org/10.1090/S0002-9947-1994-1227091-4
MathSciNet review: 1227091
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system $ \{ {x^{{\lambda _0}}},{x^{{\lambda _1}}}, \ldots \} $ with respect to Lebesgue measure on [0, 1]. In this paper, differential and integral recurrence formulae for the Müntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Laguerre polynomials. The uniform convergence of the Christoffel functions is proved equivalent to the nondenseness of the Müntz space on [0, 1], which implies that in this case the orthogonal Müntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1). Some inequalities for Müntz polynomials are also investigated, most notably, a sharp $ {L^2}$ Markov inequality is proved.


References [Enhancements On Off] (What's this?)

  • [1] J. M. Anderson, Müntz-Szász type approximation and the angular growth of L integral functions, Trans. Amer. Math. Soc. 169 (1972), 237-248. MR 0310259 (46:9360)
  • [2] P. B. Borwein, Zeros of Chebyshev polynomials in Markov systems, J. Approx. Theory 63 (1990), 56-64. MR 1074081 (92a:41002)
  • [3] P. B. Borwein and T. Erdélyi, Lacunary Müntz systems, J. Edinburgh Math. Soc. (to appear).
  • [4] -, Notes on lacunary Müntz polynomials, Israel J. Math. 76 (1991), 183-192. MR 1177339 (93h:41014)
  • [5] P. B. Borwein and E. B. Saff, On the denseness of weighted incomplete approximations, Progress in Approximation Theory (A. Gonchar and E. B. Saff, eds.), Springer-Verlag, 1992, pp. 419-429. MR 1240793 (95d:41012)
  • [6] E. W. Cheney, Introduction to approximation theory, McGraw-Hill, New York, 1966. MR 0222517 (36:5568)
  • [7] J. A. Clarkson and P. Erdös, Approximation by polynomials, Duke Math. J. 10 (1943), 5-11. MR 0007813 (4:196e)
  • [8] R. P. Feinerman and D. J. Newman, Polynomial approximation, Williams and Wilkins, Baltimore, Md., 1976. MR 0499910 (58:17657)
  • [9] G. Freud, Orthogonal polynomials, Pergamon Press, Oxford, 1971. MR 0284759 (44:1983)
  • [10] M. von Golitschek, A short proof of Müntz theorem, J. Approx. Theory 39 (1983), 394-395. MR 723231 (85b:41005)
  • [11] D. Leviatan, Improved estimates in Müntz-Jackson theorems, Progress in Approximation Theory (P. Nevai and A. Pinkus, eds.), Academic Press, 1991, pp. 567-573. MR 1114798 (93c:41014)
  • [12] G. G. Lorentz, Approximation by incomplete polynomials (problems and results), Padé and Rational Approximation, Theory and Applications (E. B. Saff and R. S. Varga, eds.), Academic Press, New York, 1977. MR 0467089 (57:6956)
  • [13] P. C. McCarthy, J. E. Sayre, and B. L. R. Shawyer, Generalized Legendre polynomials, manuscript, 1989.
  • [14] P. Nevai, Géza Freud, orthogonal polynomials and Christoffel functions. A case study, J. Approx. Theory 48 (1986), 3-167. MR 862231 (88b:42032)
  • [15] P. Nevai, V. Totik, and J. Zhang, Orthogonal polynomials: their growth relative to their sums, J. Approx. Theory 67 (1991), 215-234. MR 1133061 (92k:42034)
  • [16] P. Nevai and J. Zhang, Rate of relative growth of orthogonal polynomials, J. Math. Anal. Appl. 175 (1993), 10-24. MR 1216740 (94d:42028)
  • [17] D. J. Newman, Derivative bounds for Müntz polynomials, J. Approx. Theory 18 (1976), 360-362. MR 0430604 (55:3609)
  • [18] -, Approximation with rational functions, CBMS Regional Conf. Ser. in Math., vol. 41, Amer. Math. Soc., Providence, R.I., 1978.
  • [19] A. Pinkus and Z. Ziegler, Interlacing properties of zeros of the error functions in best $ {L^p}$ approximations, J. Approx. Theory 27 (1979), 1-18. MR 554112 (81c:41063)
  • [20] E. B. Saff and R. S. Varga, On incomplete polynomials, Numerische Methoden der Approximationstheorie (L. Collatz, G. Meinardus, and H. Weiner, eds.), Birkhäuser-Verlag, Basel, 1978. MR 527107 (80d:41008)
  • [21] L. Schwartz, Etude des sommes d'exponentielles, Hermann, Paris, 1959. MR 0106383 (21:5116)
  • [22] P. W. Smith, An improvement theorem for Descartes systems, Proc. Amer. Math. Soc. 70 (1978), 26-30. MR 0467118 (57:6985)
  • [23] G. Somorjai, A Müntz-type problem for rational approximation, Acta Math. Hungar. 27 (1976), 197-199. MR 0430617 (55:3622)
  • [24] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
  • [25] A. K. Taslakyan, Some properties of Legendre quasi-polynomials with respect to a Müntz system, Mathematics 2, Erevan University, Erevan, 1984, pp. 179-189. (Russian, Armenian Summary) MR 875260 (88e:33008)
  • [26] T. T. Trent, A Müntz-Szász theorem for $ C(D)$, Proc. Amer. Math. Soc. 83 (1981), 296-298. MR 624917 (82h:46071)
  • [27] J. Zhang, Relative growth of linear iterations and orthogonal polynomials on several intervals, Linear Algebra Appl. 186 (1993), 97-115. MR 1217201 (94h:42041)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42C05, 39A10, 41A17

Retrieve articles in all journals with MSC: 42C05, 39A10, 41A17


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1227091-4
Keywords: Exponential polynomials, Müntz systems, Müntz-Szász Theorem, Müntz-Legendre polynomials, recurrence formulae, orthogonal polynomials, Christoffel functions, Markov-type inequalities, Bernstein-type inequalities, Nikolskii-type inequalities, interlacing properties of zeros, lexicographic properties of zeros
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society