Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Are best approximations really better than Chebyshev?   Best and Chebyshev approximations play an important role in approximation
theory. From the viewpoint of measuring approximation error in the maximum
norm, it is evident that best approximations are better than their Chebyshev
counterparts. However, the situation may be reversed if we compare the
approximation quality from the viewpoint of either the rate of pointwise
convergence or the accuracy of spectral differentiation. We show that when the
underlying function has an algebraic singularity, the Chebyshev projection of
degree n converges one power of n faster than its best counterpart at each
point away from the singularity and both converge at the same rate at the
singularity. This gives a complete explanation for the phenomenon that the
accuracy of Chebyshev projections is much better than that of best
approximations except in a small neighborhood of the singularity. Extensions to
superconvergence points and spectral differentiation, Chebyshev interpolants
and other orthogonal projections are also discussed.

17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
26
The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an