zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Bernstein-type inequalities. (English) Zbl 1258.41007

Let E[-π,π] be compact and symmetric with respect to the origin and let Γ E ={e it |tE}. Denote by ω Γ E the equilibrium density of Γ E on the unit circle 𝕋. It is proved that if θE is an inner point of E then for any trigonometric polynomial T n of degree at most n we have

|T n ' (θ)|n2πω Γ E (e iθ )T n E ·(1)

This Bernstein-type inequality is sharp. In particular, if E=[-β,-α][α,β] with some 0α<βπ, then

ω Γ E (e iθ )=1 2π|sinθ| |cosθ-cosα||cosθ-cosβ|·

So for E=[-β,β], he gets from (1) the Videnskii inequality. Also, it is shown that the original Bernstein inequality implies its Szegő variant as well as both Videnskii’s inequality and its half-integer variant.

MSC:
41A17Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
References:
[1]Baran, M.: Bernstein type theorems for compact sets in rn, J. approx. Theory 69, 156-166 (1992) · Zbl 0748.41008 · doi:10.1016/0021-9045(92)90139-F
[2]Benko, D.; Dragnev, P.; Totik, V.: Convexity of harmonic densities, Rev. mat. Iberoam. 28, No. 4, 1-14 (2012)
[3]Bernstein, S. N.: On the best approximation of continuos functions by polynomials of given degree, Izd. akad. Nauk SSSR 1 (1952)
[4]Borwein, P.; Erdélyi, T.: Polynomials and polynomial inequalities, Graduate texts in mathematics 161 (1995)
[5]T. Erdélyi, Note on Bernstein-type inequalities on subarcs of the unit circle (personal communication).
[6]Garnett, J. B.; Marshall, D. E.: Harmonic measure, new mathematical monographs 2, (2008)
[7]Lukashov, A. L.: Estimates for derivatives of rational functions and the fourth zolotarev problem, St. Petersburg math. J. 19, 253-259 (2008) · Zbl 1181.26032 · doi:10.1090/S1061-0022-08-00997-7
[8]Milovanovic, G. V.; Mitrinovic, D. S.; Rassias, Th.M.: Topics in polynomials: extremal problems, inequalities, zeros, (1994) · Zbl 0848.26001
[9]Ransford, T.: Potential theory in the complex plane, (1995)
[10]Riesz, M.: Eine trigonometrische interpolationsformel und einige ungleichungen für polynome, Jahresber. deutsch. Math.-verein. 23, 354-368 (1914) · Zbl 45.0405.02
[11]Saffari, B.: Some polynomial extremal problems which emerged in the twentieth century, NATO sci. Ser. II math. Phys. chem. 33, 201-233 (2001) · Zbl 0996.42001
[12]Schaake, G.; Van Der Corput, J. G.: Ungleichungen für polynome und trigonometrische polynome, Compos. math. 2, 321-361 (1935) · Zbl 0013.10802 · doi:numdam:CM_1935__2__321_0
[13]Szego, G.: Über einen satz des herrn serge Bernstein, Schriften königsberger gelehrten ges. Naturwiss. kl. 5, 59-70 (1928–29) · Zbl 54.0311.03
[14]Totik, V.: Polynomial inverse images and polynomial inequalities, Acta math. 187, 139-160 (2001) · Zbl 0997.41005 · doi:10.1007/BF02392833
[15]Videnskii, V. S.: Extremal estimates for the derivative of a trigonometric polynomial on an interval shorter than its period, Sov. math. Dokl. 1, 5-8 (1960) · Zbl 0087.06203
[16]Videnskii, V. S.: On trigonometric polynomials of half-integer order, Izv. akad. Nauk armjan. SSR ser. Fiz.-mat. Nauk 17, 133-140 (1964)