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# Box Splines by Carl de Boor

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## Summary

Compactly supported smooth piecewise polynomial functions provide an efficient tool for the approximation of curves and surfaces and other smooth functions of one and several arguments.

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## Box Splines Summary

### Box Splines by Carl de Boor

Compactly supported smooth piecewise polynomial functions provide an efficient tool for the approximation of curves and surfaces and other smooth functions of one and several arguments. Since they are locally polynomial, they are easy to evaluate. Since they are smooth, they can be used when smoothness is required, as in the numerical solution of partial differential equations (in the Finite Element method) or the modeling of smooth sur faces (in Computer Aided Geometric Design). Since they are compactly supported, their linear span has the needed flexibility to approximate at all, and the systems to be solved in the construction of approximations are 'banded'. The construction of compactly supported smooth piecewise polynomials becomes ever more difficult as the dimension, s, of their domain G ~ IRs, i. e. , the number of arguments, increases. In the univariate case, there is only one kind of cell in any useful partition, namely, an interval, and its boundary consists of two separated points, across which polynomial pieces would have to be matched as one constructs a smooth piecewise polynomial function. This can be done easily, with the only limitation that the num ber of smoothness conditions across such a breakpoint should not exceed the polynomial degree (since that would force the two joining polynomial pieces to coincide). In particular, on any partition, there are (nontrivial) compactly supported piecewise polynomials of degree ~ k and in C(k-l), of which the univariate B-spline is the most useful example.

I * Box splines defined.- II * The linear algebra of box spline spaces.- III * Quasi-interpolants & approximation power.- IV * Cardinal interpolation & difference equations.- V * Approximation by cardinal splines & wavelets.- VI * Discrete box splines & linear diophantine equations.- VII Subdivision algorithms.- References.

NLS9781441928344
9781441928344
1441928340
Box Splines by Carl de Boor
New
Paperback
Springer-Verlag New York Inc.
2010-12-03
201
N/A
Book picture is for illustrative purposes only, actual binding, cover or edition may vary.
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