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A dimension series for multivariate splines

1991
*
Discrete & Computational Geometry
*

We consider the formal power

doi:10.1007/bf02574678
fatcat:akzpx43u3nhq7eso55xx7uulca
*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 ...##
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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. ...##
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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". ...##
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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*dimensionality*”*for*high-*dimensional*problems. ...##
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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. ...##
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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) ...##
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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

2013
*
BMC Medical Research Methodology
*

Methods: In this contribution we propose

doi:10.1186/1471-2288-13-1
pmid:23297754
pmcid:PMC3599933
fatcat:fg4w53qiknc53agek3rzx5zjx4
*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. ...##
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Reviews and Descriptions of Tables and Books

1990
*
Mathematics of Computation
*

Chui,

doi:10.1090/s0025-5718-90-99790-x
fatcat:jsqobpy6ifb3dkkfg545rfrrie
*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. ...##
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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

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

doi:10.1186/1471-2288-13-43
pmid:23510439
pmcid:PMC3626842
fatcat:vd26vv3aijejnobppau7gmdl6y
*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*. ...##
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Multivariate B-splines on triangulated rectangles

1983
*
Journal of Mathematical Analysis and Applications
*

In the two-

doi:10.1016/0022-247x(83)90265-2
fatcat:vzymvqs5x5bvpkudpyubbbzrxu
*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] . ...##
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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. ...##
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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. ...##
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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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