Lebesgue Constants For Cantor Sets

TL;DR

This paper investigates the behavior of Lebesgue constants in polynomial interpolation on Cantor sets, showing they are unbounded, which challenges previous assumptions by Mergelyan.

Contribution

It provides the first evaluation of Lebesgue constants on Cantor sets, demonstrating their unboundedness and disproving Mergelyan's claim.

Findings

Lebesgue constants are unbounded on Cantor sets

Disproves Mergelyan's statement about boundedness

Analyzes three types of Cantor sets

Abstract

We evaluate the values of the Lebesgue constants in polynomial interpolation for three types of Cantor sets. In all cases, the sequences of Lebesgue constants are not bounded. This disproves the statement by Mergelyan.