An upper bound on the Kolmogorov widths of a certain family of integral operators

Abstract

We consider the family of integral operators (Kαf)(x) from Lp[0,1] to Lq[0,1] given by (Kαf)(x)=∫01(1−xy)α−1f(y)dy,0<α<1.The main objective is to find upper bounds for the Kolmogorov widths of these operators; these are then used to derive upper bounds for their entropy numbers.We find upper bounds for the nth Kolmogorov widths in question that decrease faster than exp(−κn), and for the nth entropy numbers that decrease faster than exp(−cn3), for some positive constants κ and c.