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∈Λcλ(f)e 2πiλ·x. There is a natural sequence of finite subsets Λn 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.
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Research supported in part by the National Science Foundation, Grant DMS-0140194.
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Strichartz, R.S. Convergence of mock Fourier series. J. Anal. Math. 99, 333–353 (2006). https://doi.org/10.1007/BF02789451
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DOI: https://doi.org/10.1007/BF02789451