Markov-Bernstein-type inequalities

Bernstein's inequality asserts that

$$ \max _ {x \in [ - \pi, \pi ] } {\left | {Q ^ \prime ( x ) } \right | } \leq n \max _ {x \in [ - \pi, \pi ] } {\left | {Q ( x ) } \right | } $$

for every trigonometric polynomial $ Q $ of degree at most $ n $ with complex coefficients. The inequality

$$ \max _ {x \in [ - 1,1 ] } {\left | {Q ^ \prime ( x ) } \right | } \leq n ^ {2} \max _ {x \in [ - 1,1 ] } {\left | {Q ( x ) } \right | } $$

for every (algebraic) polynomial $ Q $ of degree at most $ n $ with complex coefficients is known as Markov's inequality. These inequalities can be extended to higher derivatives. The sharp extension of Bernstein's inequality is easy by induction, while the sharp extension of the Markov inequality requires some serious extra work. S.N. Bernstein (S.N. Bernshtein) proved the first inequality above in 1912 with $ 2n $ in place of $ n $. The sharp inequality appears first in a paper of M. Fekete (1916) who attributes the proof to L. Fejér. Bernstein attributes the proof to E. Landau. The inequality

$$ \max _ {x \in [ - 1,1 ] } {\left | {Q ^ {( m ) } ( x ) } \right | } \leq T _ {n} ^ {( m ) } ( 1 ) \cdot \max _ {x \in [ - 1,1 ] } {\left | {Q ( x ) } \right | } $$

for every (algebraic) polynomial $ Q $ of degree at most $ n $ with complex coefficients was first proved by V.A. Markov in 1892 (here, $ T _ {n} $ denotes the Chebyshev polynomial of degree $ n $; cf. also Chebyshev polynomials). He was the brother of the more famous A.A. Markov, who proved the above inequality for $ m = 1 $ in 1889, thereby answering a question raised by the prominent Russian chemist D. Mendeleev. Bernstein presented a shorter variational proof of V.A. Markov's inequality in 1938 (see [a2]). The simplest known proof of Markov's inequality for higher derivatives is due to R.S. Duffin and A.C. Shaeffer, who gave various extensions as well.

Various analogues of the above two inequalities are known, in which the underlying intervals, the maximum norms, or the family of functions are replaced by more general sets, norms, and families of functions, respectively. These inequalities are called Markov- and Bernstein-type inequalities. If the norms are the same in both sides, the inequality is called Markov-type, otherwise it is called Bernstein-type (this distinction is not completely standard). Markov- and Bernstein-type inequalities are known on various regions of the complex plane and $ n $- dimensional Euclidean space, for various norms, such as weighted $ L _ {p} $ norms, and for many classes of functions such as polynomials with various constraints, exponential sums of $ n $ terms, just to mention a few. Markov- and Bernstein-type inequalities have their own intrinsic interest. In addition, they play a fundamental role in proving so-called inverse approximation theorems (cf. Approximation of functions, direct and inverse theorems).

Many books dealing with approximation theory discuss Markov- and Bernstein-type inequalities in detail.

References

[a1]	P.B. Borwein, T. Erdélyi, "Polynomials and polynomial inequalities" , GTM , Springer (1995)
[a2]	S.N. Bernstein, "Collected Works: Vol 1. Constructive Theory of Functions (1905–1930)" , Atomic Energy Commission (1958) (In Russian)
[a3]	E.W. Cheney, "Introduction to approximation theory" , McGraw-Hill (1966)
[a4]	R.A. DeVore, G.G. Lorentz, "Constructive approximation" , Springer (1993)
[a5]	G.G. Lorentz, M. von Golitschek, Y. Makovoz, "Constructive approximation: Advanced problems" , Springer (1996)
[a6]	G.V. Milovanović, D.S. Mitrinović, Th.M. Rassias, "Topics in polynomials: Extremal problems, inequalities, zeros" , World Sci. (1994)
[a7]	Q.I. Rahman, G. Schmeisser, "Les inégalités de Markoff et de Bernstein" , Presses Univ. Montréal (1983)