Corpus ID: 119316223

On Some Generalizations of B-Splines

  title={On Some Generalizations of B-Splines},
  author={Peter R. Massopust},
  journal={arXiv: Metric Geometry},
  • P. Massopust
  • Published 15 February 2019
  • Mathematics
  • arXiv: Metric Geometry
In this article, we consider some generalizations of polynomial and exponential B-splines. Firstly, the extension from integral to complex orders is reviewed and presented. The second generalization involves the construction of uncountable families of self-referential or fractal functions from polynomial and exponential B-splines of integral and complex orders. As the support of the latter B-splines is the set $[0,\infty)$, the known fractal interpolation techniques are extended in order to… Expand

Figures from this paper

Compatibility Conditions for Systems of Iterative Functional Equations with Non-trivial Contact Sets
Systems of iterative functional equations with a non-trivial set of contact points are not necessarily solvable, as the resulting intersections may lead to an overdetermination of the system. ToExpand
Optimal indirect estimation for linear inverse problems with discretely sampled functional data
Optimal mean estimation from noisy independent pathes of a stochastic process that are indirectly observed is an issue of great interest in functional inverse problems. In this paper, minimax ratesExpand


Exponential Splines of Complex Order
We extend the concept of exponential B-spline to complex orders. This extension contains as special cases the class of exponential splines and also the class of polynomial B-splines of complex order.Expand
Splines and fractional differential operators
  • P. Massopust
  • Mathematics
  • International Journal of Wavelets, Multiresolution and Information Processing
  • 2020
Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form [Formula: see text] where [Formula: see text] is a linear differential operator ofExpand
Complex B-splines
We propose a complex generalization of Schoenberg’s cardinal splines. To this end, we go back to the Fourier domain definition of the B-splines and extend it to complex-valued degrees. We show thatExpand
Theory of exponential splines
Abstract Pruess [12, 14] has shown that exponential splines can produce co-convex and co-monotone interpolants. These results justify the further study of the mathematical properties of exponentialExpand
Exponential B-splines and the partition of unity property
An explicit formula is provided for a large class of exponential B-splines that describes the cases where the integer-translates of an exponential A-spline form a partition of unity up to a multiplicative constant. Expand
Interpolation and Approximation with Splines and Fractals
This textbook is intended to supplement the classical theory of uni- and multivariate splines and their approximation and interpolation properties with those of fractals, fractal functions, andExpand
Fractional Splines and Wavelets
The symmetric version of the B-splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative, and may be used to build new families of wavelet bases with a continuously varying order parameter. Expand
Quaternionic B-splines
The relationship between quaternionic B-splines and a backwards difference operator is shown, leading to a recurrence formula, and it is shown that the collection of integer shifts of $B_q$ is a Riesz basis for its span, hence generating a multiresolution analysis. Expand
On the Theory and Application of exponential splines
This paper describes material to appear in [1], and was stimulated by ideas and results of A. Ron who demonstrated that cardinal E-splines are worthy of a detailed investigation. Expand
A Practical Guide to Splines
  • C. D. Boor
  • Mathematics, Computer Science
  • Applied Mathematical Sciences
  • 1978
This book presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines as well as specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. Expand