Truncated Quadrature Rules Over $(0,\infty)$ and Nyström-Type Methods

Gaussian quadrature rules with exponential weights on (−1, 1)

“…Notice that the functions belonging to C w can increase exponentially at the endpoints of the interval (−1, 1). It is well known that (16) holds true with w replaced by a Jacobi weight v γ,δ (x) = (1 − x) γ (1 + x) δ , γ, δ > 0, but it is false for exponential weights on unbounded intervals (see [3,9,10,12]). Now, we want to investigate whether estimates of the form…”

“…2). This phenomenon appears also in the case of exponential weights on unbounded intervals and in this regard the reader can consult, for instance, [3,9,10,12,13,15] and the references therein. On the other hand, this fact contrasts with what happens on bounded intervals for Jacobi weights.…”

“…Therefore, also following an idea in [9,10], in Sect. 3, we propose a quadrature rule that is as simple as the Gaussian rule but requires a lower computational cost and converges with the order of the best polynomial approximation in L 1 if f ∈ W 1 1 (w) (see Proposition 1 and Theorem 3).…”

A Lagrange-type projector on the real line

“…To overcome such a problem, in [26,17,30] the authors have suggested to interpolate a "finite section" of the function and then to estimate a finite section of the interpolating polynomial in suitable weighted norms. A similar method has been used for Fourier sums in [26,19].…”

Some new applications of truncated Gauss-Laguerre quadrature formulas

“…In [6,7] we have introduced and examined a special class of quadrature formulas, which are associated with the classical Gauss-Laguerre quadrature rules…”