On the spectrum and eigenfunctions of the operator $(Vf)(x) = \int_0^{x^\alpha} f(t) dt$

TL;DR

This paper determines the spectrum and eigenfunctions of a specific integral operator involving a power transformation in the interval [0,1], providing insights into its spectral properties.

Contribution

It explicitly finds the spectrum and eigenfunctions of the operator $(Vf)(x) = extstyle loatint_0^{x^ ext{alpha}} f(t) dt$ in $L_2[0,1]$, a problem not previously solved.

Findings

Spectrum characterized explicitly

Eigenfunctions derived in closed form

Spectral properties depend on alpha

Abstract

We find spectrum and eigenfunctions of the operator $(Vf)(x) = \int_0^{x^\alpha} f(t) dt$ in $L_2[0,1]$