Best Constants for Poincaré–Type Inequalities in W 1 n ( 0, 1 ) $$W^{n}_1(0,1)$$

Allal Guessab () and Gradimir V. Milovanović ()
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Allal Guessab: Université de Pau et des Pays de l’Adour
Gradimir V. Milovanović: The Serbian Academy of Sciences and Arts

A chapter in Differential and Integral Inequalities, 2019, pp 391-402 from Springer

Abstract: Abstract For any positive integer n, let T k k = 1 n $$\left \{ T_{k} \right \}_{k=1}^{n}$$ be a given set of linear functionals on W 1 n ( 0 , 1 ) , $$W^{n}_1(0,1),$$ which are unisolvent for polynomials of degree n − 1. We determine the best possible constant c(T 1, …, T n) in the following general higher-order Poincaré-type inequalities ∫ 0 1 f ( x ) d x ≤ c n ( T 1 , … , T n ) ∫ 0 1 | f ( n ) ( x ) | d x , $$\displaystyle \int _{0}^{1} \left | f(x)\right | {\mathrm{d} }x\leq c_n(T_{1},\ldots ,T_{n})\int _{0}^{1} \big |f^{(n)}(x)\big | {\mathrm{d} }x, $$ where f ∈ W 1 n ( 0 , 1 ) $$f\in W^{n}_1(0,1)$$ satisfying the conditions T k f = 0 , $$T_{k}\left [f\right ] =0,$$ k = 1, …, n. Our main result states that the minimal value c n of the constants c n(T 1, …, T n) is just the L ∞-norm of the (properly normalized) perfect B-spline B n of degree n on [0, 1]. We were also able to exhibit one particular set of extremal functionals for which this constant is achieved. Furthermore, comparison of the best constants in the previous inequality for some most frequently used functionals in practice is also given.

Date: 2019
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DOI: 10.1007/978-3-030-27407-8_11

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