The Lebesgue constant for the periodic Franklin system

Markus Passenbrunner. "The Lebesgue constant for the periodic Franklin system." Studia Mathematica 205.3 (2011): 251-279. <http://eudml.org/doc/285864>.

@article{MarkusPassenbrunner2011,
abstract = {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.},
author = {Markus Passenbrunner},
journal = {Studia Mathematica},
keywords = {periodic Franklin system; Lebesgue constant},
language = {eng},
number = {3},
pages = {251-279},
title = {The Lebesgue constant for the periodic Franklin system},
url = {http://eudml.org/doc/285864},
volume = {205},
year = {2011},
}

TY - JOUR
AU - Markus Passenbrunner
TI - The Lebesgue constant for the periodic Franklin system
JO - Studia Mathematica
PY - 2011
VL - 205
IS - 3
SP - 251
EP - 279
AB - 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.
LA - eng
KW - periodic Franklin system; Lebesgue constant
UR - http://eudml.org/doc/285864
ER -