Corpus ID: 119316223

On Some Generalizations of B-Splines

@article{Massopust2019OnSG,
  title={On Some Generalizations of B-Splines},
  author={Peter R. Massopust},
  journal={arXiv: Metric Geometry},
  year={2019}
}
  • 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

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References

SHOWING 1-10 OF 33 REFERENCES
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
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