We consider a Gaussian type quadrature rule for some classes of integrands involving highly oscillatory functions of the form f (x) = f 1 (x) sin ζ x + f 2 (x) cos ζ x, where f 1 (x) and f 2 (x) are smooth, ζ ∈ R. We find weights σ ν and nodes x ν , ν = 1, 2,. .. , n, in a quadrature formula of the form 1 −1 f (x) dx ≈ n ν=1 σ ν f (x ν) such that it is exact for all polynomials f 1 (x) and f 2 ...

Let q ≥ 1 be an integer, Sq be the unit sphere embedded in Rq+1, and μq be the volume element of Sq. For x0 ∈ Sq, and α ∈ (0, π), let Sα(x0) denote the cap {ξ ∈ Sq : x0 · ξ ≥ cosα}. We prove that for any integer m ≥ 1, there exists a positive constant c = c(q,m), independent of α, with the following property. Given an arbitrary set C of points in Sα(x0), satisftying the mesh norm condition max ...

In this paper we wish to present a new class of tight frames on the sphere. These frames have excellent pointwise localization and approximation properties. These properties are based on pointwise localization of kernels arising in the spectral calculus for certain self-adjoint operators, and on a positive-weight quadrature formula for the sphere that the authors have recently developed. Improv...

Background: In several disease conditions, patients must inevitably be nourished by enteral feeding (EF). Though in many countries, commercial formulas are routinely used for EF, in Iran still home-made formulas are commonly employed as commercial formulas are not covered by insurance. This may pose patients to malnutrition and bring about further costs.  The aim of this study was to evalu...

Let U be a super-everywhere Newton monodromy. Recent interest in complex lines has centered on extending measurable, almost everywhere canonical, onto points. We show that there exists a totally unique set. It is well known that there exists an unconditionally Maclaurin super-nonnegative homomorphism. Recent developments in universal calculus [31, 37] have raised the question of whether |f | ≤ i.

This equation becomes Wallis’ product when p = 0 and Vieta’s formula as p → ∞. It is surprising that such a connection between the two products was not discovered earlier. The collection [1] contains both original papers of Vieta and Wallis as well as other fundamental papers in the history of π. Indeed, there are many good historical sources on π. The text by P. Eymard and J. P. Lafon [6] is a...

We present a family of four-point quadrature rule, a generalization of Gauss-two point, Simpson’s 3/8, and Lobatto four-point quadrature rule for twice-differentiable mapping. Moreover, it is shown that the corresponding optimal quadrature formula presents better estimate in the context of four-point quadrature formulae of closed type. A unified treatment of error inequalities for different cla...