Convergence of mock Fourier series

Abstract

For certain Cantor measures μ on ℝⁿ, it was shown by Jorgensen and Pedersen that there exists an orthonormal basis of exponentialse ^2πiγ·x for λεΛ. a discrete subset of ℝⁿ called aspectrum for μ. For anyL ¹ functionf, we define coefficientsc _γ(f)=∝f(y)e ^−2πiγiy dμ(y) and form the Mock Fourier series ∑_λ∈Λ^cλ(f)e ^2πiλ·x. There is a natural sequence of finite subsets Λ_n increasing to Λ asn→∞, and we define the partial sums of the Mock Fourier series by$s_n (f)(x) = \sum\limits_{\lambda \in \Lambda _n } {c_n (f)e^{2\pi i\lambda \cdot x} } .$

We prove, under mild technical assumptions on μ and Λ, thats _n(f) converges uniformly tof for any continuous functionf and obtain the rate of convergence in terms of the modulus of continuity off. We also show, under somewhat stronger hypotheses, almost everywhere convergence forfεL ¹.