Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes

The focus on locally refined spline spaces has grown rapidly in recent years due to the need in Isogeoemtric analysis (IgA) of spline spaces with local adaptivity: a property not offered by the strict regular structure of tensor product B-spline spaces. However, this flexibility sometimes results in collections of B-splines spanning the space that are not linearly independent. In this paper we address the minimal number of B-splines that can form a linear dependence relation for Minimal Support B-splines (MS B-splines) and for Locally Refinable B-splines (LR B-splines) on LR-meshes. We show that the minimal number is six for MS B-splines, and eight for LR B-splines. The risk of linear dependency is consequently significantly higher for MS B-splines than for LR B-splines. Further results are established to help detecting collections of B-splines that are linearly independent

A fractional spline collocation method for the fractional order logistic equation

We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivative, i.e. the fractional order logistic equation. The use of the fractional B-splines allows us to express the fractional derivative of the approximating function in an analytic form. Thus, the fractional collocation method is easy to implement, accurate and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method.We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivative, i.e. the fractional order logistic equation. The use of the fractional B-splines allows us to express the fractional derivative of the approximating function in an analytic form. Thus, the fractional collocation method is easy to implement, accurate and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method

Approximation and geometric modeling with simplex B-splines associated with irregular triangles

Bivariate quadratic simplical B-splines defined by their corresponding set of knots derived from a (suboptimal) constrained Delaunay triangulation of the domain are employed to obtain a C1-smooth surface. The generation of triangle vertices is adjusted to the areal distribution of the data in the domain. We emphasize here that the vertices of the triangles initially define the knots of the B-splines and do generally not coincide with the abscissae of the data. Thus, this approach is well suited to process scattered data.\ud \ud With each vertex of a given triangle we associate two additional points which give rise to six configurations of five knots defining six linearly independent bivariate quadratic B-splines supported on the convex hull of the corresponding five knots.\ud \ud If we consider the vertices of the triangulation as threefold knots, the bivariate quadratic B-splines turn into the well known bivariate quadratic Bernstein-Bézier-form polynomials on triangles. Thus we might be led to think of B-splines as of smoothed versions of Bernstein-Bézier polynomials with respect to the entire domain. From the degenerate Bernstein-Bézier situation we deduce rules how to locate the additional points associated with each vertex to establish knot configurations that allow the modeling of discontinuities of the function itself or any of its directional derivatives. We find that four collinear knots out of the set of five defining an individual quadratic B-spline generate a discontinuity in the surface along the line they constitute, and that analogously three collinear knots generate a discontinuity in a first derivative.\ud Finally, the coefficients of the linear combinations of normalized simplicial B-splines are visualized as geometric control points satisfying the convex hull property.\ud Thus, bivariate quadratic B-splines associated with irregular triangles provide a great flexibility to approximate and model fast changing or even functions with any given discontinuities from scattered data.\ud An example for least squares approximation with simplex splines is presented

Counterexamples to the B-spline conjecture for Gabor frames

The frame set conjecture for B-splines $B_n$, $n \ge 2$, states that the frame set is the maximal set that avoids the known obstructions. We show that any hyperbola of the form $ab=r$, where $r$ is a rational number smaller than one and $a$ and $b$ denote the sampling and modulation rates, respectively, has infinitely many pieces, located around $b=2,3,\dots$, \emph{not} belonging to the frame set of the $n$th order B-spline. This, in turn, disproves the frame set conjecture for B-splines. On the other hand, we uncover a new region belonging to the frame set for B-splines $B_n$, $n \ge 2$.Comment: Version 2: Lem. 5, Prop. 6, and Thm. 7 added, Version 3: Thm. 8 change