Error bounds for Gauss-type quadratures with Bernstein–Szegő weights

Error estimates of Gaussian quadrature formulae with the third class of Bernstein-Szegő weights

“…1−t dt from ( [10], p. 4), in the analogous way to ( [5], p. 5) we obtain by direct calculation that for n ≥ 2 the kernel can be expressed as (7)…”

“…I Thus in the cases when 1 < β/α < 2 and β/α > 2, δ > β/2, we analyze the expression (13) in the purpose of determining as precisely as possible the minimal value of ρ * such that for each ρ > ρ * , x ∈ [0, 1] holds I(x) > 0. This will be done using the same method as in ( [6], p. 11/12) or in ( [5], p. 16/17). There we had the polynomial of the 4-th or the 3-th degree and here we have the polynomial of the 6-th degree, but the procedure gives equally satisfactory results, which are presented in the Tables 1 and 2.…”

“…In order to ensure that J 1 is non-negative for each ρ greater than ρ (for each θ ∈ [0, π]), we will use the same method as in the (2.b) in ( [5], p. 19). This method was also used in the purpose of proving the last Gautschi's conjecture ( [7]).…”

“…which were considered in [6] and [5] respectively. The results obtained by Schira [9] cannot be applied in general for all Bernstein-Szegő weight functions, since those are not symmetric.…”

Error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

“…Our first published paper on this topic was Milovanović and Spalević [39]. In the meanwhile, we and our collaborators published a number of related papers; see [9], [32]- [34], [39]- [46], [37], [49]- [55], [69], [73]- [81], [86]- [92]. We considered in our papers mainly error bounds and estimates of the type L ∞ , L 1 , and the ones based on expanding the remainder term into a series for the quadrature formulas with multiple and simple nodes of Gaussian type, including Kronrod extensions, Radau, and Lobatto modifications, mainly with the specific weight functions such as the generalized and ordinary Chebyshev weights (see [5,70]), the Gori-Micchelli weights (see [24]), the Bernstein-Szegő weight functions (see [19]), and some of their modifications; see [17].…”