A multivariate extension of an inequality of Brenner–Alzer

In this paper, we present a more general and extended form in a multivariate setting of an inequality of Brenner and Alzer.     

A Definiteness Theory for Cubature Formulae of Order Two

Let Ω ⊂ ℝ^d be a compact convex set of positive measure. A cubature formula will be called positive definite (or a pd-formula, for short) if it approximates the integral ∫_Ω f(x) dx of every convex function f from below. The pd-formulae yield a simple sharp error bound for twice continuously differentiable functions. In the univariate case (d = 1), they are the quadrature formulae with a positive semidefinite Peano kernel of order two. As one of the main results, we show that there is a correspondence between pd-formulae and partitions of unity on Ω. This is a key for an investigation of pd-formulae without employing the complicated multivariate analogue of Peano kernels. After introducing a preorder, we establish criteria for maximal pd-formulae. We also find a lower bound for the error constant of an optimal pd-formula. Finally, we describe a phenomenon which resembles a property of Gaussian formulae.     

Enhancement of the algebraic precision of a linear operator and consequences under positivity

Let Ω be a compact convex domain in and let L be a bounded linear operator that maps a subspace of C(Ω) into C(Ω). Suppose that L reproduces polynomials up to degree m. We show that for appropriately defined coefficients a_mrj the operator

Pairs of B-Splines with Small Support on the Four-Directional Mesh Generating a Partition of Unity

Let τ be the four-directional mesh of the plane and let Σ₁ (respectively Λ₁) be the unit square (respectively the lozenge) formed by four (respectively eight) triangles of τ. We study spaces of piecewise polynomial functions in C ^k (R ²) with supports Σ₁ or Λ₁ having sufficiently high degree n, which are invariant with respect to the group of symmetries of Σ₁ or Λ₁ and whose integer translates form a partition of unity. Such splines are called complete Σ₁ and Λ₁-splines. We first give a general study of spaces of linearly independent complete Σ₁ and Λ₁-splines of class C ^k and degree n. Then, for any fixed k≥0, we prove the existence of complete Σ₁ and Λ₁-splines of class C ^k and minimal degree, but they are not unique. Finally, we describe algorithms allowing to compute the Bernstein–Bézier coefficients of these splines.     

A Family of Spline Quasi-Interpolants on the Sphere

In this paper, we construct a local quasi-interpolant Q for fitting a function f defined on the sphere S. We first map the surface S onto a rectangular domain and next, by using the tensor product of polynomial splines and 2-periodic trigonometric splines, we give the expression of Qf. The use of trigonometric splines is necessary to enforce some boundary conditions which are useful to ensure the C ² continuity of the associated surface. Finally, we prove that Q realizes an accuracy of optimal order.     

Existence and Construction of Simple B-Splines of Class C k on a Four-Directional Mesh of the Plane

In this paper we present a study of spaces of splines in C ^k (R ²) with supports the square ₁ and the lozenge ₁ formed respectively by four and eight triangles of the uniform four directional mesh of the plane. Such splines are called ₁ and ₁-splines. We first compute the dimension of the space of ₁-splines. Then we prove the existence of a unique ₁-spline of minimal degree for any fixed k0. By using this last result, we also prove the existence of a unique ₁-spline of minimal degree. Finally, we describe algorithms allowing to compute the Bernstein–Bézier coefficients of ₁-spline and ₁-spline of minimal degree.