On the Lebesgue constant of subperiodic trigonometric interpolation

Polynomial approximation and quadrature on geographic rectangles

“…In this paper, we study cubature rules of product Gaussian type on regions of S d defined by longitudes and (co)latitudes ("geographic rectangles"), with caps and collars (also called zones) as special cases. In particular we will determine cubature rules that are exact on all algebraic polynomials of total degree not greater than n, by using "subperiodic" trigonometric Gaussian rules, that are rules with n + 1 angular nodes, exact on trigonometric polynomials of degree not greater than n on subintervals of the period, [α, β] ⊆ [0, 2π] (see [8,9,10,11]). We show the quality of the cubature rules by numerical tests on some examples with integrands on S 2 and S 4 .…”

Polynomial fitting and interpolation on circular sections

“…10). As already observed, it is theoretically known that the Lebesgue constant of univariate trigonometric interpolation at the Chebyshev-like angles (11) is independent of x [17]. This implies that also the bounds 18,19 are independent of x.…”

“…Moreover, in [17] it was proved that their Lebesgue constant, say C n , is independent of x and increases logarithmically with respect to the degree, C n ¼ Oðlog nÞ. Clearly, all these properties are shift-invariant and hence are inherited by any angular interval ½a; b with b À a 6 2p.…”

“…Let L the lune defined as the difference of the unit disk with of a disk of radius r centered at ðÀd; 0Þ; d > 0, where j1 À rj < d < 1 þ r. Let H n ða; bÞ be the set of the 2n þ 1 angular nodes (4), x 1 ; x 2 the angles defined in (11), r i ; i ¼ 1; 2; 3, the transformations (13)-(17).The sequence of finite subsets A ð1Þ n ¼ r 1 ðH n ðÀx 1 ; x 1 Þ Â H n ð0; x 2 ÞÞ is a WAM of L, with constant CðA ð1Þ n Þ ¼ Oðlog 2 nÞ and cardinality cardðA ð1Þ n Þ ¼ 4n 2 þ 4n þ 1. Moreover, for i ¼ 1; 2 the sequence of finite subsets A ðiÞ n ¼ r i ðH n ðÀx 1 ; x 1 Þ H n ðÀx 2 ; x 2 ÞÞ is a WAM of L, with constant CðA ðiÞ n Þ ¼ Oðlog 2 nÞ and cardinality cardðA ðiÞ n Þ ¼ 2n 2 þ n þ 1.…”

Product Gaussian Quadrature on Circular Lunes

“…Concerning efficiency of the method, using the computational tricks for trigonomet ric Gaussian quadrature discussed in [6], in particular a fast version of the Golub-Welsh algorithm, recently proposed in [12] for Gaussian rules with symmetric weight func tion, we are able to compute the product Gaussian formulas for lunes in just 10-2 seconds up to exactness degree n = 200, that is up to tens of thousands nodes and weights.…”