The Lebesgue constant for the periodic Franklin system

• Autorzy: Markus Passenbrunner
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 41A15: Spline approximation
• 41A44: Best constants

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 205
• Numer: 3
• Strony: 251-279

Daty

wydano

2011

Twórcy

autor

Markus Passenbrunner

• Department of Analysis, J. Kepler University, Altenberger Strasse 69, A-4040 Linz, Austria

Identyfikatory

DOI

10.4064/sm205-3-3