The degree of approximation by polynomials on some disjoint intervals in the complex plane

Abstract

Let f (z) be a continuous function defined on the compact set K ⊂ C and let E n (f ) = E n (f, K) be the degree of approximation to f, for the supremum norm on K, by polynomials of degree (at most) n. ThusHere P n denotes the space of polynomials of degree at most n and . is the supremum norm on K.For a positive integer s and for 0 < a < b, letWe show that for a large class of piecewise analytic functions f defined on K lim sup n→∞ (E n (f, K)) 1 n = s b s 2 −a s 2 b s 2 +a s 2 , thus recovering several classical results.The proof of this error estimate is then translated into an algorithm that finds the polynomial of near best approximation.

Decay Properties of Spectral Projectors with Applications to Electronic Structure

“…So far we have only considered bounds based on best approximation of analytic functions defined on a single interval. In [61]…”

“…So far we have only considered bounds based on best approximation of analytic functions defined on a single interval. In [61] Proposition 8.9. There exists a positive constant K such that…”

Refined decay bounds on the entries of spectral projectors associated with sparse Hermitian matrices

“…Although the bound given in Theorem (4.3) behaves well in practice, it is not optimal from an asymptotic point of view. Hasson showed in [20] that there exists C > 0 such that…”

“…In [5] one can find rigorous proofs of exponential decay for gapped systems, like insulators. One approach is based on the approximation of the step function with the Fermi-Dirac function, the other is inspired by [10,20] and makes use of the polynomial approximation of piecewise constant function over the union of disjoint intervals.…”