Lebesgue constant for the Strömberg wavelet

The Lebesgue constants for the Franklin orthogonal system

“…In one case the problem was settled in Z. Ciesielski [5]: If there are n + 1 equally spaced knots in [0, 1] and P n denotes the (orthogonal) projection onto the corresponding space of continuous piecewise linear functions, then P n 1 < 2 and P n 1 → 2 as n → ∞. It was also shown in P. Bechler [1] that for the Franklin-Strömberg wavelet (Franklin system on R) the Lebesgue constant is 2 + (2 − √ 3) 2 . It turns out that this result can also be obtained from our result on [0, 1].…”

“…Since φ(t) ≤ 3, the formula analogous to (2.15) for knots on R implies that (4.9) and that P (1) is the orthogonal projection onto S 1 i.e.…”

“…For the L 1 (R) norm of P (1) we have P (1) 1 = 2. Consequently, lim n→∞ P n,n 1 = P (1) 1 , where P n,n is as in Section 3.2.…”

The Lebesgue constant for the periodic Franklin system

“…Some numerical experiments suggested that for the (classical, corresponding to dyadic knots) non-periodic Franklin system, the exact upper bound is 2 + (2 − √ 3) 2 ( [7]). Several years later, P. Bechler ( [1]) proved that for the piecewise linear Strömberg wavelet, the Lebesgue constant is indeed 2 + (2 − √ 3) 2 . Then, Z. Ciesielski and A. Kamont ( [6]) showed that for the classical non-periodic Franklin system, the Lebesgue constant is 2 + (2 − √ 3) 2 , verifying the conjecture in [7].…”