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 theMöbius 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 1 ns f Tn(x) if and only if f (x) = ∞ n=1 μ(n) ns g Tn(x) . Some other similar results are given. © 2005 Elsevier Inc. All rights reserved