Mellin Transform - As A Unitary Operator On L2

As A Unitary Operator On L2

In the study of Hilbert spaces, the Mellin transform is often posed in a slightly different way. For functions in (see Lp space) the fundamental strip always includes, so we may define a linear operator as

In other words we have set

This operator is usually denoted by just plain and called the "Mellin transform", but is used here to distinguish from the definition used elsewhere in this article. The Mellin inversion theorem then shows that is invertible with inverse

Furthermore this operator is an isometry, that is to say for all (this explains why the factor of was used). Thus is a unitary operator.