To each set of knots
$t_i = {i/ 2n}$ for $i=0,\dots ,2\nu$ and
$t_i= {(i-\nu) / n}$ for $i=2\nu +1,\dots, n+\nu $,
with $1\leq \nu\leq n$, there corresponds the space
${\cal S}_{\nu ,n}$ of all
piecewise linear and continuous functions on $I=[0,1]$ with knots $t_i$
and the orthogonal projection $P_{\nu ,n}$ of
$L^2(I)$ onto ${\cal S}_{\nu ,n}$. The main result is
$$
\lim_{(n-\nu)\wedge\nu\to\infty}\|P_{\nu ,n}\|_1 =
\sup_{\nu ,n\,:\,1\leq \nu\leq n}\|P_{\nu ,n}\|_1 = 2+(2-\sqrt{3})^2.
$$
This shows that the Lebesgue constant for
the Franklin orthogonal system is $2+(2-\sqrt{3})^2$.
Mots-clés :
each set knots dots i dots leq leq there corresponds space cal piecewise linear continuous functions knots orthogonal projection cal main result lim n wedge infty sup leq leq sqrt shows lebesgue constant franklin orthogonal system sqrt
Affiliations des auteurs :
Z. Ciesielski
1
;
A. Kamont
1
1
Institute of Mathematics Polish Academy of Sciences Abrahama 18 81-825 Sopot, Poland
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author = {Z. Ciesielski and A. Kamont},
title = {The {Lebesgue} constants for the {Franklin} orthogonal system},
journal = {Studia Mathematica},
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year = {2004},
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doi = {10.4064/sm164-1-4},
language = {de},
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Z. Ciesielski; A. Kamont. The Lebesgue constants for the Franklin orthogonal system. Studia Mathematica, Tome 164 (2004) no. 1, pp. 55-73. doi: 10.4064/sm164-1-4
