Generalized spline spaces over T-meshes: Dimension formula and locally refined generalized B-splines

Univariate generalized splines are smooth piecewise functions with sections in certain extended Tchebycheff spaces. They are a natural extension of univariate (algebraic) polynomial splines, and enjoy the same structural properties as their polynomial counterparts. In this paper, we consider general...

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Published in:	Applied mathematics and computation Vol. 272; pp. 187 - 198
Main Authors:	Bracco, Cesare, Lyche, Tom, Manni, Carla, Roman, Fabio, Speleers, Hendrik
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01.01.2016
Subjects:	Dimension formula Generalized splines LR-meshes T-meshes Generalized splines T-meshes LR-meshes Dimension formula
ISSN:	0096-3003, 1873-5649
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Abstract	Univariate generalized splines are smooth piecewise functions with sections in certain extended Tchebycheff spaces. They are a natural extension of univariate (algebraic) polynomial splines, and enjoy the same structural properties as their polynomial counterparts. In this paper, we consider generalized spline spaces over planar T-meshes, and we deepen their parallelism with polynomial spline spaces over the same partitions. First, we extend the homological approach from polynomial to generalized splines. This provides some new insights into the dimension problem of a generalized spline space defined on a prescribed T-mesh for a given degree and smoothness. Second, we extend the construction of LR-splines to the generalized spline context.
AbstractList	Univariate generalized splines are smooth piecewise functions with sections in certain extended Tchebycheff spaces. They are a natural extension of univariate (algebraic) polynomial splines, and enjoy the same structural properties as their polynomial counterparts. In this paper, we consider generalized spline spaces over planar T-meshes, and we deepen their parallelism with polynomial spline spaces over the same partitions. First, we extend the homological approach from polynomial to generalized splines. This provides some new insights into the dimension problem of a generalized spline space defined on a prescribed T-mesh for a given degree and smoothness. Second, we extend the construction of LR-splines to the generalized spline context.
Author	Bracco, Cesare Manni, Carla Speleers, Hendrik Lyche, Tom Roman, Fabio
Author_xml	– sequence: 1 givenname: Cesare surname: Bracco fullname: Bracco, Cesare email: cesare.bracco@unifi.it organization: Department of Mathematics and Computer Science, University of Florence, Italy – sequence: 2 givenname: Tom surname: Lyche fullname: Lyche, Tom email: tom@math.uio.no organization: Department of Mathematics, University of Oslo, Norway – sequence: 3 givenname: Carla orcidid: 0000-0002-1519-4106 surname: Manni fullname: Manni, Carla email: manni@mat.uniroma2.it organization: Department of Mathematics, University of Rome ‘Tor Vergata’, Italy – sequence: 4 givenname: Fabio surname: Roman fullname: Roman, Fabio email: fabio.roman@unito.it organization: Department of Mathematics, University of Turin, Italy – sequence: 5 givenname: Hendrik surname: Speleers fullname: Speleers, Hendrik email: speleers@mat.uniroma2.it organization: Department of Mathematics, University of Rome ‘Tor Vergata’, Italy
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Keywords	Generalized splines T-meshes LR-meshes Dimension formula
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Snippet	Univariate generalized splines are smooth piecewise functions with sections in certain extended Tchebycheff spaces. They are a natural extension of univariate...
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Title	Generalized spline spaces over T-meshes: Dimension formula and locally refined generalized B-splines
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