The asymptotic behavior of the lebesque constants for A sequence of triangular partial sums of double Fourier series

Lebesgue constants for polyhedral sets and polynomial interpolation on Lissajous–Chebyshev nodes

“…A special case of our result (see Theorem 2.1) is that for all positive integers m 1 , m 2 we can improve (1.11) to L Γ (m 1 ,m 2 ), * ln(m 1 + 1) ln(m 2 + 1). Under the strongly restrictive condition that m 2 is a multiple of m 1 this result appears already in [19].…”

Fourier Analysis and Approximation of Functions

“…Further problems and theorems Carenini and Soardi [1983J and Brandolini [1990J and [1993J. See Kuznetsova [1977]; the case nl = n2 is due to I. K. Daugavet (1970).…”

Asymptotics of the Lebesgue constants for bivariate approximation processes

“…Proof. Equalities (3.25) and (3.26) can be found in [16]. The proofs of relations (3.27)-(3.30) are given in [17].…”

“…First, we investigate the Lebesgue constants for the ℓ 1 -partial sums of Fourier series and then establish relationships between the corresponding Lebesgue constants. For our purposes, we essentially extend a series of equalities and estimates obtained in [16] and [17] for the Dirichlet kernels with the frequencies in the rhombus {(k 1 , k 2 ) : |k 1 |/n 1 + |k 2 |/n 2 ≤ 1 } and apply new methods developed recently in the papers [13] and [12].…”

“…In particular, we are interested in the so-called triangular partial sums, which is also called the ℓ 1 -partial sums. The estimates for the Lebesgue constants of such partial sums were obtained, e.g., in [1,6,12,13,16,20,25]. Different asymptotic formulas for the Lebesge constant in the case of isotropic dilations of a polyhedra were given in [15,20,21,22]; for the anisotropic case see [16] and [17] as well as the recent work [12], in which an asymptotic formula is given for the Lebesgue constants generated by the anisotropically dilated d-dimensional simplex.…”