Let \(f\) be a continuous function defined on the compact set \(K\) consisting of the union of several disjoint intervals in the complex plane. Using the overconvergence technique the author estimates lower bounds of the error \(E_n(f,K)=\inf_{P\in {\mathcal{P}}_n}\| f-P\|\) where inf is taken over all polynomials of degree at most \(n\). He obtains also the convergence result for this class of functions proving the formulae for \(\limsup_{n\rightarrow\infty} (E_n(f,K))^{\frac{1}{n}}\). In addition, his proof provides an algorithm to build the polynomial of near best approximation. This work was motivated by problems occuring in digital signal processing where approximation by polynomials on disjoint intervals is used for the dessign of digital filters.