Asymptotic behavior of Lebesgue functions of two variables

Asymptotics of the Lebesgue constants for bivariate approximation processes

“…Equalities (3.25) and (3.26) can be found in [16]. The proofs of relations (3.27)-(3.30) are given in [17]. Now we prove (3.32).…”

“…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].…”

“…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.…”