The Lebesgue constant for the periodic Franklin system

Markus Passenbrunner

Studia Mathematica, Tome 204 (2011), p. 251-279 / Harvested from The Polish Digital Mathematics Library

Résumé

We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_{j}$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n,ν}$ be the space of piecewise linear continuous functions on the torus with knots ${t_{j}: 0 ≤ j ≤ N-1}$. Finally, let $P_{n,ν}$ be the orthogonal projection operator from L²([0,1)) onto $V_{n,ν}$. The main result is $lim_{n→∞,ν=1} ||P_{n,ν}: L^{∞} → L^{∞}|| = sup_{n∈ℕ,0≤ν≤n} ||P_{n,ν}: L^{∞} → L^{∞}|| = 2 + (33-18√3)/13$. This shows in particular that the Lebesgue constant of the classical Franklin orthonormal system on the torus is 2 + (33-18√3)/13.

Détail

Publié le : 2011-01-01

Zbl 1232.41032

EUDML-ID : urn:eudml:doc:285864