Convergence of Series of Dilated Functions and Spectral Norms of Gcd Matrices
نویسنده
چکیده
We establish a connection between the L norm of sums of dilated functions whose jth Fourier coefficients are O(j−α) for some α ∈ (1/2, 1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L and for the almost everywhere convergence of series of dilated functions.
منابع مشابه
Gcd Sums from Poisson Integrals and Systems of Dilated Functions
(nknl) are established, where (nk)1≤k≤N is any sequence of distinct positive integers and 0 < α ≤ 1; the estimate for α = 1/2 solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for α = 1/2. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues ...
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