Tom Lyche

• Manni, Carla; Speleers, Hendrik & Lyche, Tom (2024). A local simplex spline basis for C3 quartic splines on arbitrary triangulations. Applied Mathematics and Computation. ISSN 0096-3003. 462. doi: 10.1016/j.amc.2023.128330. Fulltekst i vitenarkiv
• Lyche, Tom ; Mørken, Knut Martin & Reif, Ulrich (2023). On the condition of cubic B-splines. Journal of Approximation Theory. ISSN 0021-9045. 289. doi: 10.1016/j.jat.2023.105883.
• Lyche, Tom ; Manni, Carla & Speleers, Hendrik (2022). Construction of C^2 Cubic Splines on Arbitrary Triangulations. Foundations of Computational Mathematics. ISSN 1615-3375. doi: 10.1007/s10208-022-09553-z. Fulltekst i vitenarkiv
• Bressan, Andrea & Lyche, Tom (2020). Local Approximation from Spline Spaces on Box Meshes. Foundations of Computational Mathematics. ISSN 1615-3375. 21, s. 807–848. doi: 10.1007/s10208-020-09467-8. Fulltekst i vitenarkiv
• Lyche, Tom & Muntingh, Agnar Georg P (2019). B-spline-like bases for C2 cubics on the Powell–Sabin 12-split. SMAI Journal of Computational Mathematics (SMAI-JCM). ISSN 2426-8399. S5, s. 129–159. Fulltekst i vitenarkiv Vis sammendrag

  For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell–Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single triangle, which are tied together across triangles in a Bézier-like manner. In this paper we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simplex splines, we derive simple differentiation and recurrence formulas to other S-bases. We establish a Marsden identity that gives rise to various quasi-interpolants and domain points forming an intuitive control net, in terms of which conditions for C 0 -, C 1 -, and C 2 -smoothness are derived.

• Lyche, Tom ; Manni, Carla & Speleers, Hendrik (2019). Tchebycheffian B-Splines Revisited: An Introductory Exposition . I Giannelli, Carlotta & Speleers, Hendrik (Red.), Advanced Methods for Geometric Modeling and Numerical Simulation. Springer Nature. ISSN 978-3-030-27330-9. s. 179–216. doi: 10.1007/978-3-030-27331-6_8. Fulltekst i vitenarkiv
• Bracco, Cesare; Lyche, Tom ; Manni, Carla & Speleers, Hendrik (2019). Tchebycheffian spline spaces over planar T-meshes: Dimension bounds and dimension instabilities. Journal of Computational and Applied Mathematics. ISSN 0377-0427. 349, s. 265–278. doi: 10.1016/j.cam.2018.10.026. Fulltekst i vitenarkiv
• Lyche, Tom ; Manni, Carla & Speleers, Hendrik (2018). Foundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement, Splines and PDEs: From Approximation Theory to Numerical Linear Algebra, Cetraro, Italy 2017. Springer Nature. ISSN 978-3-319-94910-9. s. 1–76. doi: 10.1007/978-3-319-94911-6_1. Fulltekst i vitenarkiv
• Lyche, Tom Johan & Merrien, Jean-Louis (2018). Simplex-splines on the Clough-Tocher element. Computer Aided Geometric Design. ISSN 0167-8396. 65, s. 76–92. doi: 10.1016/j.cagd.2018.07.004. Fulltekst i vitenarkiv
• Bruvoll, Solveig; Lyche, Tom Johan & Mørken, Knut Martin (2017). Uniformly stable wavelets on nonuniform triangulations. Mathematics and Computers in Simulation. ISSN 0378-4754. 137, s. 177–200. doi: 10.1016/j.matcom.2016.09.006.
• Bracco, Cesare; Lyche, Tom Johan; Manni, Carla; Roman, Fabio & Speleers, Hendrik (2016). On the dimension of Tchebycheffian spline spaces over planar T-meshes. Computer Aided Geometric Design. ISSN 0167-8396. 45, s. 151–173. doi: 10.1016/j.cagd.2016.01.002.
• Bracco, Cesare; Lyche, Tom Johan; Manni, Carla; Roman, Fabio & Speleers, Hendrik (2016). Generalized spline spaces over T-meshes: Dimension formula and locally refined generalized B-splines. Applied Mathematics and Computation. ISSN 0096-3003. 272, s. 187–198. doi: 10.1016/j.amc.2015.08.019.
• Lyche, Tom Johan & Muntingh, Agnar Georg Peder (2016). Stable Simplex Spline Bases for C3 Quintics on the Powell–Sabin 12-Split. Constructive approximation. ISSN 0176-4276. 45(1), s. 1–32. doi: 10.1007/s00365-016-9332-8. Fulltekst i vitenarkiv Vis sammendrag

  For the space of \(C^3\) quintics on the Powell–Sabin 12-split of a triangle, we determine explicitly the six symmetric simplex spline bases that reduce to a B-spline basis on each edge and have a positive partition of unity, a Marsden identity that splits into real linear factors, and an intuitive domain mesh. The bases are stable in the \(L_\infty \) norm with a condition number independent of the geometry and have a well-conditioned Lagrange interpolant at the domain points and a quasi-interpolant with local approximation order 6. We show an \(h^2\) bound for the distance between the control points and the values of a spline at the corresponding domain points. For one of these bases, we derive \(C^0\), \(C^1\), \(C^2\), and \(C^3\) conditions on the control points of two splines on adjacent macrotriangles.

• Lyche, Tom & Muntingh, Agnar Georg Peder (2014). A Hermite interpolatory subdivision scheme for C2-quintics on the Powell-Sabin 12-split. Computer Aided Geometric Design. ISSN 0167-8396. 31(7-8), s. 464–474. doi: 10.1016/j.cagd.2014.03.004. Fulltekst i vitenarkiv Vis sammendrag

  In order to construct a C1C1-quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell–Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme (Dyn and Lyche, 1998). In this paper we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally C3C3 and globally C2C2. For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements

• Lyche, Tom & Mazure, Marie-Laurence (2013). PIECEWISE CHEBYSHEVIAN MULTIRESOLUTION ANALYSIS. East Journal on Approximations. ISSN 1310-6236. 17(4), s. 410–426.
• Cohen, Elaine; Lyche, Tom & Riesenfeld, Richard F (2013). A B-SPLINE-LIKE BASIS FOR THE POWELL-SABIN 12-SPLIT BASED ON SIMPLEX SPLINES. Mathematics of Computation. ISSN 0025-5718. 82(283), s. 1667–1707. doi: 10.1090/S0025-5718-2013-02664-6. Fulltekst i vitenarkiv
• Dokken, Tor; Lyche, Tom & Pettersen, Kjell Fredrik (2013). Polynomial splines over locally refined box-partitions. Computer Aided Geometric Design. ISSN 0167-8396. 30(3), s. 331–356. doi: 10.1016/j.cagd.2012.12.005. Vis sammendrag

  We address progressive local refinement of splines defined on axes parallel box-partitions and corresponding box-meshes in any space dimension. The refinement is specified by a sequence of mesh-rectangles (axes parallel hyperrectangles) in the mesh defining the spline spaces. In the 2-variate case a mesh-rectangle is a knotline segment. When starting from a tensor-mesh this refinement process builds what we denote an LR-mesh, a special instance of a box-mesh. On the LR-mesh we obtain a collection of hierarchically scaled B-splines, denoted LR B-splines, that forms a nonnegative partition of unity and spans the complete piecewise polynomial space on the mesh when the mesh construction follows certain simple rules. The dimensionality of the spline space can be determined using some recent dimension formulas.

• Cohen, Elaine; Lyche, Tom & Riesenfeld, Richard (2010). MCAD: Key historical developments. Computer Methods in Applied Mechanics and Engineering. ISSN 0045-7825. 199(5-8), s. 224–228. doi: 10.1016/j.cma.2009.08.003.
• Cohen, E.; Martin, T; Kirby, R.M.; Lyche, Tom & Riesenfeld, R. F. (2010). Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis. Computer Methods in Applied Mechanics and Engineering. ISSN 0045-7825. 199(5-8), s. 334–356. doi: 10.1016/j.cma.2009.09.010.
• Lyche, Tom ; Mørken, Knut Martin & Pelosi, Francesca (2009). Stable, linear spline wavelets on nonuniform knots with vanishing moments. Computer Aided Geometric Design. ISSN 0167-8396. 26(2), s. 203–216. doi: 10.1016/j.cagd.2008.04.002.
• Lyche, Tom ; Manni, Carla & Sablonniere, Paul (2008). Quasi-interpolation projectors for box splines. Journal of Computational and Applied Mathematics. ISSN 0377-0427. 221(2), s. 416–429. doi: 10.1016/j.cam.2007.10.029.
• Floater, Michael S. & Lyche, Tom (2008). Divided differences of inverse functions and partitions of a convex polygon. Mathematics of Computation. ISSN 0025-5718. 77, s. 2295–2308.
• Floater, Michael S. & Lyche, Tom (2007). Two chain rules for divided differences and Faa di Bruno's formula. Mathematics of Computation. ISSN 0025-5718. 76, s. 867–877.
• Lyche, Tom Johan & Merrien, JL (2006). C-1 interpolatory subdivision with shape constraints for curves. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 44, s. 1095–1121.
• Lyche, Tom Johan & Mazure, ML (2006). Total positivity and the existence of piecewise exponential B-splines. Advances in Computational Mathematics. ISSN 1019-7168. 25, s. 105–133.
• Lyche, Tom & Scherer, Karl (2006). Addendum to ”On the L1-Condition Number of the Univariate Bernstein Basis". Constructive approximation. ISSN 0176-4276. 24, s. 239–243. doi: 10.1007/s00365-005-0615-8.
• Lyche, Tom & Merrien, Jean-Louis (2006). Hermite subdivision with shape constraints on a rectangular mesh. BIT Numerical Mathematics. ISSN 0006-3835. 46(4), s. 831–859. doi: 10.1007/s10543-006-0105-9.
• Lyche, Tom & Scherer, Karl (2006). Mixed norm condition number for the univariate Bernstein basis. Banach Center Publications. ISSN 0137-6934. 72, s. 177–188.
• Lyche, Tom & Merrien, Jean-Louis (2006). Hermite subdivision with shape constraints on a rectangular mesh. BIT Numerical Mathematics. ISSN 0006-3835. 46, s. 831–859.
• Costantini, Paolo; Lyche, Tom & Manni, Carla (2005). On a class of weak Tchebycheff systems. Numerische Mathematik. ISSN 0029-599X. 101(2), s. 333–354.

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• Lyche, Tom ; Muntingh, Georg & Ryan, Øyvind (2020). Exercises in Numerical Linear Algebra and Matrix Factorizations. Springer Nature. ISBN 978-3-030-59788-7. 255 s.
• Lyche, Tom (2020). Numerical Linear Algebra and Matrix Factorizations. Springer Nature. ISBN 978-3-030-36467-0. 363 s.
• Lyche, Tom ; Manni, Carla & Speleers, Hendrik (2018). Splines and PDEs: From Approximation Theory to Numerical Linear Algebra, Cetraro, Italy 2017. Springer Nature. ISBN 978-3-319-94910-9. 2219(1). 325 s.
• Floater, Michael S.; Lyche, Tom Johan; Mazure, Marie-Laurence; Mørken, Knut Martin & Schumaker, Larry L. (2017). Mathematical Methods for Curves and Surfaces: 9th International Conference, MMCS 2016 Tønsberg, Norway, June 23-28, 2016 Revised Selected Papers. Springer Publishing Company. ISBN 978-3-319-67885-6. 333 s. Vis sammendrag

  The 9th International Conference on Mathematical Methods for Curves and Surfaces took place during June 23–28, 2016, in Tønsberg, Norway. The earlier conferences in the series took place in Oslo (1988), Biri (1991), Ulvik (1994), Lillehammer (1997), Oslo (2000), Tromsø (2004), Tønsberg (2008), and Oslo (2012). This conference series is integrated with the French conferences Curves and Surfaces organized by SMAI-SIGMA (Signal-Image-Géométrie-Modélisation-Approximation), and the last conference in France was in Paris in 2014. The conference gathered 140 participants from 27 countries who presented a total of 115 talks. This includes nine invited talks and seven mini-symposia. This book contains 17 original articles based on talks presented at the conference. The topics range from mathematical theory to industrial applications. The papers have been subject to the usual peer-review process, and we thank both the authors and the reviewers for their hard work and helpful collaboration. We wish to thank all those who supported and helped organize the conference. It is a pleasure to acknowledge the generous financial support from the Research Council of Norway and the Department of Mathematics at the University of Oslo. We also thank Elisabeth Seland, Espen Sande, and Espen Lybekk for their help with administrative and technical matters, and Anne Berg Floater for help with the registration.

• Boissonnat, Jean-Daniel; Cohen, Albert; Gibaru, Olivier; Gout, Christian; Lyche, Tom Johan & Mazure, Marie-Laurence [Vis alle 7 forfattere av denne artikkelen] (2015). Curves and Surfaces: 8th International Conference, Paris, France, June 12-18, 2014. Springer. ISBN 978-3-319-22803-7. 492 s.
• Lyche, Tom & Merrien, Jean-Louis (2014). Exercises in Computational Mathematics with MATLAB. Springer. ISBN 978-3-662-43510-6. 372 s.
• Floater, Michael S.; Lyche, Tom ; Mazure, Marie-Laurence; Mørken, Knut Martin & Schumaker, Larry (2014). Mathematical Methods for Curves and Surfaces. 8th International Conference, MMCS 2012, Oslo, Norway, June 28 – July 3, 2012, Revised Selected Papers. Springer. ISBN 978-3-642-54381-4. 511 s.
• Dæhlen, Morten; Floater, Michael S.; Lyche, Tom ; Merrien, Jean-Louis; Mørken, Knut Martin & Schumaker, Larry (2010). Mathematical Methods for Curves and Surfaces. Springer. ISBN 978-3-642-11619-3. 465 s.
• Chenin, Patrick; Lyche, Tom & Schumaker, Larry (2007). Curve and Surface Design: Avignon 2006. Nashboro Press, Ltd.. ISBN 978-0-9728482-7-5. 285 s.

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• Lyche, Tom ; Merrien, Jean-Louis & Sauer, Tomas (2019). Polynomial Splines with Arbitrary Smoothness on Plane Triangulations. Vis sammendrag

  Given a triangulation on a polygonal domain. We consider polynomial splines of global smoothness $r$ on a refined triangulation, where each triangle is split into sub patches. Examples are the WangShi split and the Powell-Sabin 6 and 12 splits. We propose a partition of unity simplex spline basis of degree 3r and smoothness r on the Clough-Tocher split. This is joint work with Jean-Louis Merrein and Tomas Sauer.

• Lyche, Tom ; Merrien, Jean-Louis & Sauer, Tomas (2019). Simplex--Splines on the Clough--Tocher Split with Arbitrary Smoothness,
• Lyche, Tom ; Manni, Carla & Speleers, Hendrik (2019). Interesting Splits II.
• Lyche, Tom ; Manni, Carla & Speleers, Hendrik (2019). Interesting splits.
• Muntingh, Agnar Georg Peder & Lyche, Tom (2018). B-spline-like simplex spline bases on the Powell-Sabin 12-split.
• Lyche, Tom (2018). Tchebycheffian B-splines. Vis sammendrag

  We give a short survey of Tchebycheffian Splines and Tchebycheffian B-splines. These piecewise functions are natural generalizations of polynomial splines and polynomial B-splines. Each segment of the spline belongs to a piecewise Ex- tended Complete Tchebycheff (ECT)-space. We discuss various ways to define them. And chose a recursive definition, allowing different spaces on the seg- ments. Several examples will be given to illustrate the concepts.

• Lyche, Tom (2018). Tchebycheffian Splines and Tchebycheffian B-splines . Vis sammendrag

  We give a survey of Tchebycheffian Splines and Tchebycheffian B-splines. These piecewise functions are natural generalizations of polynomial splines and polynomial B-splines. Each segment of the spline belongs to a piecewise Extended Complete Tchebycheff (ECT)-space. An ECT-space can also be considered as the null space of a linear differential operator with variable coefficients defined on a suitable interval. ECT-spaces can either be spanned by a set of basis functions that are a natural generalization of the polynomial power basis or an alternative basis with similar properties to the Bernstein basis for polynomials. We define Tchebycheffian B-splines recursively, allowing different spaces on the segments. If time allows we also consider Tchebycheffian divided differences. Several examples will be given to illustrate the concepts.

• Floater, Michael S.; Lyche, Tom Johan Wiborg; Mazure, Marie-Laurence; Mørken, Knut Martin & Schumaker, Larry L. (2017). Preface. Lecture Notes in Computer Science (LNCS). ISSN 0302-9743. 1052:16191 LNCS, s. 1–1. doi: 10.1016/j.cam.2009.02.019.
• Muntingh, Agnar Georg Peder & Lyche, Tom Johan (2017). B-spline-like bases for C2 cubics on the Powell-Sabin 12-split. Vis sammendrag

  For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, Cohen, Lyche and Riesenfeld recently discovered so-called S-bases. These are simplex spline bases with B-spline-like properties on a single macrotriangle, which are tied together across macrotriangles in a Bézier-like manner. In this talk we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simplex splines, we derive simple differentiation and recurrence formulas to other S-bases. We establish a Marsden identity that gives rise to various quasi-interpolants and domain points forming an intuitive control net, in terms of which conditions for C0, C1, and C2-smoothness are derived. Although the cubic bases can only be used to define smooth surfaces over specific triangulations, we envision applications for local constructions, such as hybrid meshes and extra-ordinary points, with the potential to be used in isogeometric analysis.

• Muntingh, Agnar Georg Peder & Lyche, Tom Johan (2015). Stable Simplex Spline Bases for C3 Quintics on the Powell-Sabin 12-Split.
• Muntingh, Agnar Georg Peder & Lyche, Tom Johan (2015). A Hermite Subdivision Scheme for C2 Quintics on the Powell-Sabin 12-Split.
• Muntingh, Agnar Georg Peder & Lyche, Tom Johan (2015). Simplex Spline Bases on the Powell-Sabin 12-Split: Part II.
• Bruvoll, Solveig; Lyche, Tom Johan & Mørken, Knut Martin (2015). Uniformly stable wavelets on nonuniform triangulations.
• Muntingh, Agnar Georg Peder & Lyche, Tom (2014). A Hermite subdivision scheme for C2 quintic macro-elements on the Powell-Sabin 12-split.
• Lyche, Tom ; Manni, Carla & Speleers, Hendrik (2014). Tchebycheffian Splines and Local Refinenement.
• Lyche, Tom (2014). Simplex Splines.
• Floater, Michael S.; Lyche, Tom; Mazure, Marie-Laurence; Mørken, Knut Martin & Schumaker, Larry L. (2014). Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics): Preface. Lecture Notes in Computer Science (LNCS). ISSN 0302-9743. 8177 LNCS.
• Muntingh, Agnar Georg Peder & Lyche, Tom (2013). A Hermite subdivision scheme for smooth macro-elements on the Powell-Sabin 12-split.
• Lyche, Tom (2013). Smooth Simplex Splines for the Powell-Sabin 12-split.
• Lyche, Tom (2013). Polynomial splines over locally refined box-partitions.
• Lyche, Tom (2013). Polynomial Splines over Locally Refined Box-Partitions.
• Lyche, Tom ; Dokken, Tor & Pettersen, Kjell Fredrik (2013). Polynomial Splines over locally refined box partitions.
• Lyche, Tom (2012). B-splines: A Fundamental Tool for Analysis and Computation.
• Dokken, Tor; Lyche, Tom & Pettersen, Kjell Fredrik (2012). A local refinement algorithm on box partitions.
• Bruvoll, Solveig; Lyche, Tom & Mørken, Knut Martin (2012). Uniformly stable linear wavelets on nonuniform triangulations.
• Lyche, Tom (2012). Polynomial Splines on Locally Refined Box-Partitions.
• Lyche, Tom (2012). Locally Refinable Splines on Box Partitions.
• Mørken, Knut Martin; Bruvoll, Solveig & Lyche, Tom (2011). Uniformly stable ’wavelets’ on general triangulations.
• Mørken, Knut Martin; Bruvoll, Solveig & Lyche, Tom (2011). Uniformly stable ’wavelets’ on general triangulations.
• Lyche, Tom ; Dokken, Tor & Pettersen, Kjell Fredrik (2011). LR-Splines.
• Lyche, Tom ; Cohen, Elaine & Riesenfeld, Richard (2011). Simplex splines for the Powell Sabin 12 split.
• Lyche, Tom ; Dokken, Tor & Pettersen, Kjell Fredrik (2011). Locally Refined B-Splines.
• Lyche, Tom (2011). A Subdivision Scheme based on the PS12 split.
• Lyche, Tom (2011). B-splines: A Fundamental Tool for Analysis and Computation.
• Lyche, Tom ; Cohen, Elaine & Riesenfeld, Richard (2011). Simplex splines on triangulations.
• Lyche, Tom ; Dokken, Tor & Pettersen, Kjell Fredrik (2011). Locally refined B-splines.
• Lyche, Tom (2011). Quadratic C1 Splines on Triangulations.
• Cohen, Elaine; Lyche, Tom & Riesenfeld, Richard (2011). Simplex splines for the Powell-Sabin 12 split.
• Lyche, Tom ; Dokken, Tor & Pettersen, Kjell Fredrik (2011). Locally refined B-splines; recent progress.
• Floater, Michael S. & Lyche, Tom (2010). Special Issue: International Conference on Multivariate Approximation and Interpolation with Applications (MAIA 2007). Journal of Computational and Applied Mathematics. ISSN 0377-0427. 233(7), s. 1619–1619.
• Lyche, Tom (2009). The sensitivity of spline approximations to perturbations of data and knots.
• Lyche, Tom (2009). Analysis aware modeling; Model Quality.
• Lyche, Tom (2009). The sensitivity of spline approximations to perturbations of the knots.
• Lyche, Tom (2009). Divided Differences; Recent Progress.
• Lyche, Tom (2008). New formulas for divided differences and partitions of a convex polygon.
• Lyche, Tom (2008). New formulas for divided differences with combinatorial applications.
• Floater, Michael S. & Lyche, Tom (2008). Divided Differences of the Inverse Function and Partitions of a Convex Polygon.
• Floater, Michael S. & Lyche, Tom (2007). New Formulas for Divided Di®erences and Partitions of a Convex Polygon. Vis sammendrag

  Divided di®erences are a basic tool in approximation theory and numerical analysis: they play an important role in interpolation and approximation by polynomials and in spline theory. So it is worthwhile to look for identities that are analogous to identities for derivatives. An exam- ple is the Leibniz rule for di®erentiating products of functions. This rule was generalized to divided di®erences by Popoviciu and Ste®ensen 70 years ago. To our surprise it was discovered that there were no analog of a 150 year old formula for di®erentiating composite functions (Faa di Bruno's formula) and for di®erentiating the inverse of a function. In this talk I will discuss chain rules and inverse rules for divided di®erences. The inverse rule turns out to have a surprising and beautiful structure: it is a sum over partitions of a convex polygon into smaller polygons using only nonintersecting diagonals. This provides a new way of enumerating all partitions of a convex polygon with a speci¯ed number of triangles, quadrilaterals, and so on.

• Floater, Michael S. & Lyche, Tom (2006). Chain rules for divided differences.
• Lyche, Tom (2005). The chain rule for divided differences.
• Lyche, Tom (2005). Differences divisees des fontions composees.
• Lyche, Tom (2005). Subdivision with shape constraints.
• Lyche, Tom (2005). Constrained curve and surface design.
• Lyche, Tom (2005). Chain rules for divided differences and Faa di Bruno's formula.
• Lyche, Tom (2005). Hermite subdivison and total positivity.
• Lyche, Tom (2005). On a Class of Weak Tchebycheff Systems.
• Lyche, Tom (2005). Shape preserving Hermite subdivision.
• Lyche, Tom (2005). Hermite Subdivision with Shape Constraints.
• Dokken, Tor; Lyche, Tom Johan & Pettersen, Kjell Fredrik (2012). Locally Refinable Splines over Box-Partitions . SINTEF. ISSN 9788214052824. Fulltekst i vitenarkiv Vis sammendrag

  We address progressive local refinement of splines defined on axes parallel box-partitions and corresponding box-meshes in any space dimension. The refinement is specified by a sequence of mesh-rectangles (axes parallel hyperrectangles) in the mesh defining the spline spaces. In the 2-variate case a mesh-rectangle is a knotline segment. When starting from a tensor-mesh this refinement process builds what we denote an LR-mesh, a special instance of a box-mesh. On the LR-mesh we obtain a collection of hierarchically scaled B-splines, denoted LR B-splines, that forms a nonnegative partition of unity and spans the complete piecewise polynomial space on the mesh when the mesh construction follows certain simple rules. The dimensionality of the spline space can be determined using recent dimension formulas. Oppdragsgiver: SINTEF ; Research Council of Norway

• Mørken, Knut Martin; Bruvoll, Solveig & Lyche, Tom (2011). Uniformly stable wavelets on nonuniform triangulations. CMA.
• Dæhlen, Morten; Lyche, Tom ; Mørken, Knut Martin & Seidel, Hans-Peter (2006). Multiresolution analysis based on quadratic Hermite interpolation Part 1: Piecewise polynomial Curves. Department of Informatics, University of Oslo. ISSN 82-7368-273-0.

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