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Of all publications in the section: 2
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Article
Beckermann B., Matos A., Wielonsky F. et al. Mathematics of Computation. 2011. Vol. 80. No. 274. P. 931-958.

In order to reduce the Gibbs phenomenon exhibited by  the partial Fourier sums of a periodic function, discontinuous in 0, Driscoll and Fornberg considered so-called singular Fourier-Padé approximants constructed from the Hermite-Padé approximants. Convincing numerical experiments have been obtained by these authors, but no error estimates have been proven so far. In the present paper we study the special case of Nikishin systems and their Hermite-Padé approximants, both theoretically and numerically. We obtain rates of convergence by using orthogonality properties of the functions involved along with results from logarithmic potential theory. In particular, we address the question of how to choose the degrees of the approximants, by considering diagonal and row sequences, as well as linear Hermite-Padé approximants. Our theoretical findings and numerical experiments confirm that these Hermite-Padé approximants are more efficient than the more elementary Padé approximants, particularly around the discontinuity of the goal function. 

Added: Nov 23, 2012
Article
Protasov V. Mathematics of Computation. 2017. Vol. 86. No. 305. P. 1499-1524.

For an arbitrary set  of nonnegative integers, we consider the Euler binary partition function  which equals the total number of binary expansions of an integer  with ``digits'' from . By applying the theory of subdivision schemes and refinement equations, the asymptotic behaviour of  as  is characterized. For all finite , we compute the lower and upper exponents of growth of , find when they coincide, and present a sharp asymptotic formula for  in that case, which is done in terms of the corresponding refinable function. It is shown that  always has a constant exponent of growth on a set of integers of density one. The sets  for which  has a regular power growth are classified in terms of cyclotomic polynomials.

Added: May 11, 2017