The Lebesgue constants for the Franklin orthogonal system

Abstract

Abstract. To each set of knots t i = i/2n for i = 0, . . . , 2ν and t i = (i − ν)/n for i = 2ν + 1, . . . , n + ν, with 1 ≤ ν ≤ n, there corresponds the space S ν,n of all piecewise linear and continuous functions on I = [0, 1] with knots t i and the orthogonal projection P ν,n of L 2 (I) onto S ν,n . The main result isThis shows that the Lebesgue constant for the Franklin orthogonal system is 2+(2− √ 3) 2 .

Sample path properties of fractional Riesz–Bessel field of variable order

“…Integrodifferential operators of variable order from purely mathematical aspect were first introduced by Samko and Ross. In contrast, the more widely known fractional Levy Brownian field 27,28 is not intermittent. [9][10][11][12][13] We remark that for fractional Brownian motion of variable order, or multifractional Brownian motion, 19,20 and multifractional Ornstein-Uhlenbeck process, the fractional integrodifferential operators used followed the definitions introduced by Samko and Ross.…”

The Lebesgue constant for the periodic Franklin system

“…By (6), exact formulae for the entries of the inverse (a jk ) of the Gram matrix are absolutely necessary in determining the exact value of the Lebesgue constant. We will provide this information in Proposition 6 for the periodic case.…”

“…We will provide this information in Proposition 6 for the periodic case. In the non-periodic dyadic case, such exact formulae for the inverse of the Gram matrix were given in [3] and they were used in the calculation of the corresponding Lebesgue constant in [6]. For the general Franklin system, there are important estimates both for the non-periodic case and for the periodic case (see [11] and [12] respectively).…”

“…Since every row in (a jk ) is equal up to shifts, formula (6) does not depend on j in this case. So while we consider equally spaced knots, we write κ to denote the value of κ(j) for arbitrary 0 ≤ j ≤ N − 1.…”

“…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]. For splines of higher degree (d ≥ 2), a problem was the mere existence of a bound C d for the L ∞ −norms of orthogonal projections onto splines of degree d with arbitrary knots, where C d depends only on d and not on the partition.…”