Uniform reconstruction of continuous functions with the RAFU method

Eduardo Corbacho Cortés

doi:10.4995/agt.2017.7263

Authors

Eduardo Corbacho Cortés IES Saenz de Buruaga (Badajoz)

DOI:

https://doi.org/10.4995/agt.2017.7263

Keywords:

RAFU method, RAFU approximation, uniform approximation

Abstract

The RAFU (radical functions) method can be used to obtain the uniform reconstruction of a continuous function from its values at some of the points of partitions of a closed interval. In this work we will prove that we can reconstruct a continuous function from average samples of these points, from linear combinations of them and from local average samples given by convolution. Uniform error bounds will be established. If these data are unknown but approximate values of them are known, uniform reconstruction will be also possible. Error estimates in these cases will be given. The case of a non-uniform net will be treated. Examples and algorithms will be also shown.

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Author Biography

Eduardo Corbacho Cortés, IES Saenz de Buruaga (Badajoz)

Department of Mathematics

References

H. Behforooz, Approximation by integro cubic splines, Appl. Math. Comput. 175 (2006), 8--15.

H. Behforooz, Interpolation by integro quintic splines, Appl. Math. Comput. 216 (2010), 364--367.

J.L. Blasco and A. Moltó, On the uniform closure of

a linear space of bounded real-valued functions, Annali di Matematica Pura ed Applicata IV, vol. CXXXIV (1983) 233--239.

J. Bustamante, R.C. Castillo, A.F. Collar, A regularization

method for polynomial approximation of functions from their approximate values at nodes, J. Numer. Math. Volume 17 Issue 2 (2009), 97--118.

Y. Chen, Y. Huang, W. Han, Function reconstruction

from noisy local averages, Inverse Problems 24 (2008), 025003. http://dx.doi.org./10.1088/0266-5611/24/2/025003. 14

E.W. Cheney, Approximation Theory, AMS Chelsea Publishing.

E. Corbacho, Uniform approximation with radical functions,

SEMA Journal 58 (2012), 97--122.

E. Corbacho, A RAFU linear space uniformly dense in

C[a,b], Appl. Gen. Topology 14 (1) (2013), 53--60.

E. Corbacho, Approximation in different smoothness

spaces with the RAFU method, Appl. Gen. Topology 15 (2) (2014) 221--228.

E. Corbacho, Simultaneous approximation with the RAFU

method (To appear in J. Math. Inequal).

E. Delhez, A spline interpolation technique that

preserves mass budgets, Appl. Math. Lett. 16 (2003), 16--26.

E. Epstein, On obtaining daily climatological values

from monthly means, J. Clim. 4 (1991), 365--8.

M.I. Garrido and F. Montalvo, Uniform approximation theorems for real-valued continuous functions, Topology and Appl. 45 (1992), 145--155.

T. Gassó, S. Hernández and E. Rojas, Representation and approximation by series of continuous functions, Acta Math. Hungar., 123, no. 1-2 (2009), 91-102.

R.C. González, R.E. Woods, Digital image Processing

nd edn, New Jersey, Prentice Hall (2002)

J. Huang, Y. Chen, A regularization method for the

function reconstruction from approximate average fluxes, Inverse Problems 21 (2005), 1667--84.

P. Killworth, Time interpolation of forcing fields

in ocean models, TJ Phys. Oceanogr. 26 (1996), 136--43.

F.G. Lang, X.P. Xu, On integro quartic spline interpolation,

J. Comput. Appl. Math. 236 (2012), 4214--4226.

T. Zhanlav, R. Mijiddorj, The local integro cubic

splines and their approximation properties, Appl. Math. Comput. 216 (2010), 2215--2219.