Inequalities with exponential weights

Abstract

Let R = (−∞, ∞) and let Q ∈ C 2 : R → R + = [0, ∞) be an even function. Then in this paper we consider the infinite-finite range inequality, an estimate for the Christoffel function, and the Markov-Bernstein inequality with the exponential weights w (x)= |x| e −Q(x) , x ∈ R.

The Markov–Bernstein inequality and Hermite–Fejér interpolation for exponential-type weights

“…Let S n,ρ (x) ∈ P n (n ≥ 2) be a polynomial satisfying that S n,ρ (x) ∼ |x| + a n n ρ , x ∈ (−a 2n , a 2n ) and |S n,ρ (x)| |x| + a n n ρ−1 , |x| ≤ a 2n (see [3,Lemma 3.1]). Then we have for |x| ≤ a βn , (Pw) (x) |x| + a n n ρ ϕ αn (x) (P S n,ρ w) (x)ϕ αn (x) + (PwS n,ρ )(x)ϕ αn (x) .…”

“…We prove the results of Section 2 in Section 4. Finally the Appendix contains various estimates and known theorems from [3,4].…”

Higher order Hermite-Fejér interpolation polynomials with Laguerre-type weights

“…There are many properties of P n, r* (t) = P n (W r* ; t) with respect to W r* (t), [2,3,7,[11][12][13]. They were obtained by transformations from the results in [5,6].…”

On general Bernstein and Nikol’skiî type inequalities

“…Up till now, inequalities of the same type have been established for various systems of functions. We refer the reader to Borwein [3], Borwein and Erdélyi [4][5][6], Baranov[1], Pesenson [18], Jung [12], and Erdélyi [10,11].…”