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A dimension series for multivariate splines

Louis J. Billera, Lauren L. Rose
1991 Discrete & Computational Geometry  
We consider the formal power series ~_,k~O dimR C~,(A) 2k, and show, under mild conditions on A, that this always has the form P(2)/(1 -2) a+ 1, where P(2) is a polynomial in 2 with integral coefficients  ...  For a polyhedral subdivision A of a region in Euclidean d-space, we consider the vector space C[(A) consisting of all C" piecewise polynomial functions over A of degree at most k.  ...  Acknowledgments The authors are grateful to Michael Stillman for pointing out an error in a preliminary version of this work, and to Bernd Sturmfels and GiJnter Ziegler for many constructive suggestions  ... 
doi:10.1007/bf02574678 fatcat:akzpx43u3nhq7eso55xx7uulca

Page 1880 of Mathematical Reviews Vol. , Issue 88d [page]

1988 Mathematical Reviews  
, The limits of multivariate cardinal splines (pp. 47-50); M.  ...  Pfluger, On the dimension of the spline space S} (A) in special cases (pp. 180-190); Manfred von Golitschek, De- gree of best approximation by blending functions (pp. 191-197); T. N. T.  ... 

Page 4130 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews  
By using these schemes, a stable locally supported basis for the multivariate spline space S/,(A) is constructed.  ...  The author investigates the construction of bases for spaces S/,(A) of multivariate splines of degree d and smoothness r, defined on an arbitrary triangulation A in R".  ... 

Page 8096 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
When clustering multivariate observations following a mixture model of Gaussian distributions, projections of the observations onto a linear subspace of lower dimension, called discriminant space (DS),  ...  that are useful in overcoming the “curse of dimensionalityfor high-dimensional problems.  ... 

Page 5767 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
Part II is a generalization of Schoenberg’s univariate cardinal spline theory to the multivariate setting. It is parallel to the univariate development of L-splines by Micchelli.  ...  region in dimension n > 2 is partitioned into subregions using grid partitions of sufficiently smooth (nm — 1)-dimensional mani- folds.  ... 

Page 3453 of Mathematical Reviews Vol. , Issue 94f [page]

1994 Mathematical Reviews  
Some new results on dimensions of multivariate spline spaces and representations of basis functions are obtained.  ...  The results on the upper bound of dimensions for n- variate spline spaces S/ ,(A) generalize related conclusions about bivariate spline spaces. Yu Yu Feng (PRC-HEF; Hefei)  ... 

Page 1664 of Mathematical Reviews Vol. , Issue 97C [page]

1997 Mathematical Reviews  
Applications are given to the basis and dimension problem for spline spaces.” David R.  ...  E is a simplex.” 97¢:41037 41A63 Rose, Lauren L. (1-WLL; Wellesley, MA) Module bases for multivariate splines.  ... 

Reducing and meta-analysing estimates from distributed lag non-linear models

Antonio Gasparrini, Ben Armstrong
2013 BMC Medical Research Methodology  
Methods: In this contribution we propose a method to synthesize DLNMs to simpler summaries, expressed by a reduced set of parameters of one-dimensional functions, which are compatible with current multivariate  ...  , and multivariate meta-analysis, a methodology to pool estimates of multi-parameter associations.  ...  The cross-basis is specified by the choice of two basis, one for each dimension, among a set of possible options such as splines, polynomials, or step functions.  ... 
doi:10.1186/1471-2288-13-1 pmid:23297754 pmcid:PMC3599933 fatcat:fg4w53qiknc53agek3rzx5zjx4

Reviews and Descriptions of Tables and Books

1990 Mathematics of Computation  
Chui, Multivariate Splines, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 54, SIAM, Philadelphia, 1988, vi+189 pp., 25 cm.  ...  In fact, trying to extend particular features of univariate splines into a multivariate setting typically leads to totally different multivariate concepts.  ... 
doi:10.1090/s0025-5718-90-99790-x fatcat:jsqobpy6ifb3dkkfg545rfrrie

Page 4347 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews  
This approach provides simple proofs for the shape- preserving properties of B-spline series. This book is a classic reference in spline theory.  ...  For the purposes of this book, a multivariate spline space is a linear space of functions defined on some domain which has been partitioned into subdomains.  ... 

Applications of functional data analysis: A systematic review

Shahid Ullah, Caroline F Finch
2013 BMC Medical Research Methodology  
One-quarter (25.0%) of the published studies used functional linear models to describe relationships between explanatory and outcome variables and only 8.3% used FDA for forecasting time series data.  ...  Conclusions: Despite its clear benefits for analyzing time series data, full appreciation of the key features and value of FDA have been limited to date, though the applications show its relevance to many  ...  Commonly, time series data are treated as multivariate data because they are given as a finite discrete time series.  ... 
doi:10.1186/1471-2288-13-43 pmid:23510439 pmcid:PMC3626842 fatcat:vd26vv3aijejnobppau7gmdl6y

Multivariate B-splines on triangulated rectangles

Charles K Chui, Ren-Hong Wang
1983 Journal of Mathematical Analysis and Applications  
In the two-dimensional setting, for example, if a domain G' in IFi' is divided into a finite or countable number of cells by a grid partition A, then the space S;(A) of multivariate (or, more precisely  ...  There are already some results on the dimensions of spaces of multivariate spline functions in [3, 6, 111 . A more general result can be obtained using the method in [6] .  ... 
doi:10.1016/0022-247x(83)90265-2 fatcat:vzymvqs5x5bvpkudpyubbbzrxu

Page 5124 of Mathematical Reviews Vol. , Issue 85k [page]

1985 Mathematical Reviews  
approximation, the author gives a comprehensive survey of the history and current status of multivariate B-spline construction for arbitrary knot sets.  ...  An analytical technique for 3-dimensional tion. BIT 24 (1984), no. 1, 119-122.  ... 

Page 4926 of Mathematical Reviews Vol. , Issue 97H [page]

1997 Mathematical Reviews  
, at least, as the polynomial degree tends to infinity.” 97h:41020 41A10 26D10 41A17 Waldron, Shayne (NZ-AUCK; Auckland) A multivariate form of Hardy’s inequality and L ,-error bounds for multivariate  ...  Let S(f) = "fo Ex(f) be the minimax series of f.  ... 

Page 2213 of Mathematical Reviews Vol. , Issue 90D [page]

1990 Mathematical Reviews  
Summary: “A relation between univariate trigonometric and poly- nomial B-splines is extended to higher dimensions and used to construct a class of multivariate trigonometric B-splines from the multivariate  ...  They possess a recurrence relation similar to that of the multivari- ate polynomial B-splines.” 90d:42005 42A10 42A24 Koval'chuk, I. R.  ... 
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