Order of growth of the degrees of a~polynomial basis of a~space of continuous functions

V. N. Temlyakov

Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 711-728.

Résumé

The problem under consideration is the one posed independently by C. Foias and I. Singer and by P. L. Ul'yanov concerning the minimal growth of the degrees $\nu_n$ of a polynomial basis $\{t_n(x)\}_0^\infty$ of a space of continuous functions. It is shown that there exist an absolute constant $C$ and a polynomial basis $\{t_n(x)\}_0^\infty$ such that $$ \nu_n\le C(n\ln^+\ln(n+1)+1),\quad n=0,1,2,\dots $$ The feasibility of the method employed is also considered.