Markov-Duffin-Schaeffer inequality for polynomials with a circular majorant
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- by Q. I. Rahman and G. Schmeisser PDF
- Trans. Amer. Math. Soc. 310 (1988), 693-702 Request permission
Abstract:
If $p$ is a polynomial of degree at most $n$ such that $|p(x)| \leqslant \sqrt {1 - {x^2}}$ for $- 1 \leqslant x \leqslant 1$, then for each $k$, $\max |{p^{(k)}}(x)|$ on $[ - 1, 1]$ is maximized by the polynomial $({x^2} - 1){U_{n - 2}}(x)$ where ${U_m}$ is the $m$th Chebyshev polynomial of the second kind. The purpose of this paper is to investigate if it is enough to assume $|p(x)| \leqslant \sqrt {1 - {x^2}}$ at some appropriately chosen set of $n + 1$ points in $[ - 1, 1]$. The problem is inspired by a well-known extension of Markov’s inequality due to Duffin and Schaeffer.References
- D. L. Berman, Solution of an extremal problem of the theory of interpolation, Doklady Akad. Nauk SSSR (N.S.) 87 (1952), 167–170 (Russian). MR 0051889
- R. J. Duffin and A. C. Schaeffer, A refinement of an inequality of the brothers Markoff, Trans. Amer. Math. Soc. 50 (1941), 517–528. MR 5942, DOI 10.1090/S0002-9947-1941-0005942-4 R. J. Duffin and L. A. Karlovitz, The Markoff-Duffin-Schaeffer inequalities abstracted, Proc. Nat. Acad. Sci. U.S.A. 82 (1985), 955-957.
- H. Ehlich and K. Zeller, Auswertung der Normen von Interpolationsoperatoren, Math. Ann. 164 (1966), 105–112 (German). MR 194799, DOI 10.1007/BF01429047
- P. Erdös, On the uniform distribution of the roots of certain polynomials, Ann. of Math. (2) 43 (1942), 59–64. MR 5947, DOI 10.2307/1968879
- R. Haverkamp, Zur Konvergenz der Ableitungen von Interpolationspolynomen, Computing 32 (1984), no. 4, 343–355 (German, with English summary). MR 748935, DOI 10.1007/BF02243777 W. A. Markoff, Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen, Math. Ann. 77 (1916), 218-258.
- R. Pierre and Q. I. Rahman, On a problem of Turan about polynomials, Proc. Amer. Math. Soc. 56 (1976), 231–238. MR 412362, DOI 10.1090/S0002-9939-1976-0412362-6
- R. Pierre and Q. I. Rahman, On a problem of Turán about polynomials. II, Canadian J. Math. 33 (1981), no. 3, 701–733. MR 627652, DOI 10.4153/CJM-1981-055-8
- Q. I. Rahman, On a problem of Turán about polynomials with curved majorants, Trans. Amer. Math. Soc. 163 (1972), 447–455. MR 294586, DOI 10.1090/S0002-9947-1972-0294586-5
- Theodore J. Rivlin, The Chebyshev polynomials, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0450850
- T. J. Rivlin, Optimally stable Lagrangian numerical differentiation, SIAM J. Numer. Anal. 12 (1975), no. 5, 712–725. MR 408203, DOI 10.1137/0712053
- A. C. Schaeffer and R. J. Duffin, On some inequalities of S. Bernstein and W. Markoff for derivatives of polynomials, Bull. Amer. Math. Soc. 44 (1938), no. 4, 289–297. MR 1563728, DOI 10.1090/S0002-9904-1938-06747-X
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 310 (1988), 693-702
- MSC: Primary 26D05
- DOI: https://doi.org/10.1090/S0002-9947-1988-0946426-8
- MathSciNet review: 946426