Sharp constants in weighted 𝐿²-Markov inequalities

Totik, Vilmos

doi:10.1090/proc/13974

Sharp constants in weighted $L^2$-Markov inequalities
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Proc. Amer. Math. Soc. 147 (2019), 153-166 Request permission

Abstract:

The sharp asymptotic constants in $L^2$-Markov inequalities are given for generalized Jacobi weights. They are expressed in terms of the smallest positive zeros of Bessel functions of the first kind and via the equilibrium measure of the ground set.

References

A. I. Aptekarev, A. Dro, and V. A. Kalyagin, On the asymptotics of exact constants in Markov-Bernstein inequalities in integral metrics with classical weight, Uspekhi Mat. Nauk 55 (2000), no. 1(331), 173–174 (Russian); English transl., Russian Math. Surveys 55 (2000), no. 1, 163–165. MR 1751821, DOI 10.1070/rm2000v055n01ABEH000251
A. I. Aptekarev, A. Draux, V. A. Kalyagin, and D. N. Tulyakov, Asymptotics of sharp constants of Markov-Bernstein inequalities in integral norm with Jacobi weight, Proc. Amer. Math. Soc. 143 (2015), no. 9, 3847–3862. MR 3359576, DOI 10.1090/proc/12535
Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
Albrecht Böttcher and Peter Dörfler, Weighted Markov-type inequalities, norms of Volterra operators, and zeros of Bessel functions, Math. Nachr. 283 (2010), no. 1, 40–57. MR 2598592, DOI 10.1002/mana.200810274
Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635
A. A. Markov, Sur une question posée par D.I. Mendeleieff, Izv. Akad. Nauk St. Petersburg, 62(1889), 1–24.
Giuseppe Mastroianni and Vilmos Totik, Weighted polynomial inequalities with doubling and $A_\infty$ weights, Constr. Approx. 16 (2000), no. 1, 37–71. MR 1848841, DOI 10.1007/s003659910002
G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR 1298187, DOI 10.1142/1284
Franz Peherstorfer, Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping, J. Approx. Theory 111 (2001), no. 2, 180–195. MR 1849545, DOI 10.1006/jath.2001.3571
Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
Vilmos Totik, The polynomial inverse image method, Approximation theory XIII: San Antonio 2010, Springer Proc. Math., vol. 13, Springer, New York, 2012, pp. 345–365. MR 3052033, DOI 10.1007/978-1-4614-0772-0_{2}2
Vilmos Totik, Polynomial inverse images and polynomial inequalities, Acta Math. 187 (2001), no. 1, 139–160. MR 1864632, DOI 10.1007/BF02392833
G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110