On the decay rate of Chebyshev coefficients

Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces

“…(ii) When the fractional regularity exponential s → 1, our results improve the existing bounds in usual Sobolev spaces (see, e.g., [34,38,35,24]). In this sense, the fractional Sobolev-type space with regularity exponential m + s (s ∈ (0, 1) and integer m ≥ 0) can be regarded as an intermediate space inbetween the spaces with regularity exponentials m + 1 and m in [35].…”

“…we obtain (4.28) from (4.30) immediately. Thanks to (4.28), we derive from Theorem 2.1 that [24]). Here, we first estimate the Chebyshev expansion errors in the L ∞ -norm and L 2 ω -norm for functions with interior singularities.…”

“…(Ω) W m (Ω), m ≥ 1. which has been used in [35,24] for Chebyshev approximation of functions with limited regularity.…”

“…[34,35]). In fact, as shown in [34, and [35, Thms 7.1-7.2] (also see Lemma 5.1 below), one should choose B r ⊂ BV(Ω) (the space of functions of bounded variation) to achieve optimality (see Section 5, and refer to [34,38,35,24] for more details). Unfortunately, the Sobolev spaces therein were defined through integer-order derivatives, so they could not best characterise the regularity of e.g., u(x) = |x| α with non-integer α > 0.…”

“…Here, we put the emphasises on estimating the decay rate of expansion coefficients for the reason that the errors of spectral expansions in various norms, and the related interpolation and quadratures can be estimated directly from the sums of the coefficients (cf. [34,24]). The essential ideas and main contributions of this study are summarised as follows.…”