Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell-Sabin partitions

In this paper we generate and study new cubature formulas based on spline quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations of a polygonal domain in ?². By using a specific refinement of a generic triangulation, optimal convergence orders are obtained for some of these rules. Numerical tests are presented for illustrating the theoretical results.     

Some performances of local bivariate quadratic C1 quasi-interpolating splines on nonuniform type-2 triangulations

We investigate spline quasi-interpolants defined by C¹ bivariate quadratic B-splines on nonuniform type-2 triangulations and by discrete linear functionals based on a fixed number of triangular mesh-points either in the support or close to the support of such B-splines.We show they can approximate a real function and its partial derivatives up to an optimal order and we derive local and global upper bounds.We also present some numerical and graphical results.     

Construction of spherical spline quasi-interpolants based on blossoming

A general theory of quasi-interpolants based on quadratic spherical Powell-Sabin splines on spherical triangulations of a sphere-like surface S is developed by using polar forms. As application, various families of discrete and differential quasi-interpolants reproducing quadratic spherical Bézier-Bernstein polynomials or the whole space of the spherical Powell-Sabin quadratic splines of class C¹ are presented.     

Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

In this paper we generate and study new cubature formulas based on spline quasi-interpolants defined as linear combinations of C ¹ bivariate quadratic B-splines on a rectangular domain Ω, endowed with a non-uniform criss-cross triangulation, with discrete linear functionals as coefficients. Such B-splines have their supports contained in Ω and there is no data point outside this domain. Numerical results illustrate the methods.     

On the construction of local quadratic spline quasi-interpolants on bounded rectangular domains

In this paper local bivariate C¹$C^1$ spline quasi-interpolants on a criss-cross triangulation of bounded rectangular domains are considered and a computational procedure for their construction is proposed. Numerical and graphical tests are provided.     

On the f-vectors of the pyramidal triangulations of point configurations

A triangulation of a point configuration is called pyramidal if all its simplices have a vertex in common. A series of inequalities are derived for the components of the f-vectors of pyramidal triangulations, and, for every d ≥ 3, a d-dimensional convex polytope and its triangulation are constructed whose f-vector cannot be realized as the f-vector of a pyramidal triangulation.     

Geometric Realization of a Triangulation on the Projective Plane with One Face Removed

Let M be a map on a surface F ². A geometric realization of M is an embedding of F ² into a Euclidean 3-space ?³ such that each face of M is a flat polygon. We shall prove that every triangulation G on the projective plane has a face f such that the triangulation of the Möbius band obtained from G by removing the interior of f has a geometric realization.     

Hierarchical spline spaces: quasi-interpolants and local approximation estimates

A local approximation study is presented for hierarchical spline spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement in any dimensionality. We provide approximation estimates for general hierarchical quasi-interpolants expressed in terms of the truncated hierarchical basis. Under some mild assumptions, we prove that such hierarchical quasi-interpolants and their derivatives possess optimal local approximation power in the general q-norm with \(1\leq q\leq \infty \). In addition, we detail a specific family of hierarchical quasi-interpolants defined on uniform hierarchical meshes in any dimensionality. The construction is based on cardinal B-splines of degree p and central factorial numbers of the first kind. It guarantees polynomial reproduction of degree p and it requires only function evaluations at grid points (odd p) or half-grid points (even p). This results in good approximation properties at a very low cost, and is illustrated with some numerical experiments.     

On near-best discrete quasi-interpolation on a four-directional mesh

Spline quasi-interpolants are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete quasi-interpolants which are based on Ω-splines, i.e. B-splines with octagonal supports on the uniform four-directional mesh of the plane. These quasi-interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters. We show that this problem has always a solution, in general nonunique. Concrete examples of such quasi-interpolants are given in the last section.     

On the convergence of Kergin and Hakopian interpolants at Leja sequences for the disk

We prove that Kergin interpolation polynomials and Hakopian interpolation polynomials at the points of a Leja sequence for the unit disk D of a sufficiently smooth function f in a neighbourhood of D converge uniformly to f on D. Moreover, when f∈C ^∞(D), all the derivatives of the interpolation polynomials converge uniformly to the corresponding derivatives of f.     

Lacunary interpolation by quartic splines on uniform meshes

The interpolation of a discrete set of data on the interval [0, 1], representing the first and the second derivatives (except at 0) of a smooth function f is investigated via quartic C²-splines. Error bounds in the uniform norm for ∥s⁽ⁱ⁾ − f⁽ⁱ⁾∥, i=0(1)2, if f ∈ C^l[0, 1], l=3, 5 and ⁽³⁾ ∈ BV[0, 1], together with computational examples will also be presented.     

Uniquely and faithfully embeddable projective-planar triangulations

Let G be a 5-connected graph not isomorphic to the complete graph K₆ with 6 vertices and triangularly embedded in a projective plane P². Then it will be shown that for any embedding f: G → P², there is a homeomorphism h: P² → P² such that h| G = f. In our terminology, the result states that every 5-connected projective-planar triangulation is uniquely and faithfully embeddable in a projective plane unless it is isomorphic to K₆.     

Non-real zeros of higher derivatives of real entire functions of infinite order

Letf be a real meromorphic function of infinite order in the plane such thatf has finitely many poles. Then for eachk≥3, at least one off andf ^(k) has infinitely many non-real zeros. Together with a result of Edwards and Hellerstein, this establishes the analogue for higher derivatives of a conjecture going back to Wiman around 1911.     

Weighted L p-Approximation of Derivatives by the Method of Gammaoperators

We consider Gammaoperators G _n on suitable Sobolev type subspaces of L _p(0, ∞) and characterize the global rate of approximation of derivatives f ^(r) through corresponding derivatives (G _n f)^(r) in an appropriate weighted L _p — metric by the rate of Ditzian and Totik’s second order weighted modulus of smoothness.     

Dyadic Hermite interpolants on a triangulation

Givenf and f on the vertices of a triangulation, we build an interpolating functionf by means of a subdivision algorithm. Infinite products of matrices are used to prove the convergence to aC ¹ function for some classes of triangulations.     

Algorithms for finding connected separators between antipodal points

A set (or a collection of sets) contained in the Euclidean space Rm is symmetric if it is invariant under the antipodal map. Given a symmetric unicoherent polyhedron X (like an n-dimensional cube or a sphere) and an odd real function f defined on vertices of a certain symmetric triangulation of X, we algorithmically construct a connected symmetric separator of X by choosing a subcollection of the triangulation. Each element of the subcollection contains the vertices v and u such that f(v)f(u)?0.     

Blow-up results for a class of first-order nonlinear evolution inequalities

We investigate the non-existence of solutions to a class of evolution inequalities; in this case, as it happens in a relatively small number of blow-up studies, nonlinearities depend also on time-variable t and spatial derivatives of the unknown. The present results, which in great part do not require any assumption on the regularity of data, are completely new and shown with various applications. Some of these results referring to the problem u_t=Δu+a(x)|u|^p+λf(x) in R^N, t>0 include the non-existence results of positive global solutions obtained by Fujita and others when a≡1 and f≡0, Bandle-Levine and Levine-Meier when a≡|x|^m and f≡0, Pinsky when either f≡0 or f?0 and λ>0, Zhang and Bandle-Levine-Zhang when a≡1 and λ=1.     

On the univalence of derivatives of functions which are univalent in angular domains

We consider functions f that are univalent in a plane angular domain of angle απ, 0 < α ≤ 2. It is proved that there exists a natural number k depending only on α such that the kth derivatives f ^(k) of these functions cannot be univalent in this angle. We find the least of the possible values of for k. As a consequence, we obtain an answer to the question posed by Kir’yatskii: if f is univalent in the half-plane, then its fourth derivative cannot be univalent in this half-plane.     

The ultrahyperbolic Bessel operator: Some basic properties

In this note, we establish several relations between the Ultrahyperbolic Bessel operator B^αf, the Klein-Gordon operator, and the generalized Bessel derivatives D^αf.     

Direct sums of finitary permutation groups

The interpolation problem under consideration is connected with the finite element method in ?³. In most cases, when finite elements are constructed by means of the partition of a given domain in ?² into triangles and interpolation of the Hermite or Birkhoff type, the sine of the smallest angle of the triangle appears in the denominators of the error estimates for the derivatives. In the case of ? ^m (m ≥ 3), the ratio of the radius of the inscribed sphere to the diameter of the simplex is used as an analog of this characteristic. This makes it necessary to impose constraints on the triangulation of the domain. The recent investigations by a number of authors reveal that, in the case of triangles, the smallest angle in the error estimates for some interpolation processes can be replaced by the middle or the greatest one, which makes it possible to weaken the triangulation requirements. There are fewer works of this kind for m ≥ 3, and the error estimates are given there in terms of other characteristics of the simplex. In this paper, methods are suggested for constructing an interpolation third-degree polynomial on a simplex in ?³. These methods allow one to obtain estimates in terms of a new characteristic of a rather simple form and weaken the triangulation requirements.