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The Lebesgue constant for the periodic Franklin system

Markus Passenbrunner (2011)

Studia Mathematica

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 l i m n , ν = 1 | | P n , ν : L L | | = s u p 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...

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