A transform involving Chebyshev polynomials and its inversion formula

We define a functional analytic transform involving the Chebyshev polynomials Tn(x), with an inversion formula in which the Möbius function μ(n) appears. If s∈C with Re(s)>1, then given a bounded function from [−1,1] into C, or from C into itself, the following inversion formula holds:g(x)=∑n=1∞1nsf(Tn(x)) if and only iff(x)=∑n=1∞μ(n)nsg(Tn(x)). Some other similar results are given.