Rational Functions for the Reconstruction of Exponential Sums from their Fourier Coefficients

We investigate the order of exponential sums involving the coefficients of general [Formula: see text]-functions satisfying a suitable functional equation and give some new estimates, including refining certain results in preceding works [X. Ren and Y. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for [Formula: see text], Sci. China Math. 58(10) (2015) 2105–2124; Y. Jiang and G. Lü, Oscillations of Fourier coefficients of Hecke–Maass forms and nonlinear exponential functions at primes, Funct. Approx. Comment. Math. 57 (2017) 185–204].

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Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ)

International Journal of Number Theory ◽

10.1142/s1793042115500025 ◽

2014 ◽

Vol 11 (01) ◽

pp. 39-49 ◽

Cited By ~ 1

Author(s):

Bin Wei

Keyword(s):

Asymptotic Expansion ◽

Cusp Form ◽

Rational Number ◽

Fourier Coefficients ◽

Bessel Functions ◽

Exponential Sums ◽

Summation Formula ◽

Cusp Forms ◽

Real Numbers ◽

Voronoi Summation

Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n>0λf(n)ϕ(n/X)e(αn), where [Formula: see text]. It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 < X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions.

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Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients

Analysis and Applications ◽

10.1142/s0219530521500196 ◽

2021 ◽

pp. 1-35

Author(s):

Nadiia Derevianko ◽

Gerlind Plonka

Keyword(s):

Rational Approximation ◽

Fourier Coefficients ◽

Exponential Sums ◽

Finite Interval ◽

Large Set ◽

Recovery Method ◽

Exact Reconstruction ◽

Finite Set ◽

Number Formula ◽

Numerical Approaches

In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form [Formula: see text], where the frequency parameters [Formula: see text] are pairwise distinct. In order to reconstruct [Formula: see text] we employ a finite set of classical Fourier coefficients of [Formula: see text] with regard to a finite interval [Formula: see text] with [Formula: see text]. For our method, [Formula: see text] Fourier coefficients [Formula: see text] are sufficient to recover all parameters of [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The recovery is based on the observation that for [Formula: see text] the terms of [Formula: see text] possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [Y. Nakatsukasa, O. Sète and L. N. Trefethen, The AAA Algorithm for rational approximation, SIAM J. Sci. Comput. 40(3) (2018) A1494A1522]. If a sufficiently large set of [Formula: see text] Fourier coefficients of [Formula: see text] is available (i.e. [Formula: see text]), then our recovery method automatically detects the number [Formula: see text] of terms of [Formula: see text], the multiplicities [Formula: see text] for [Formula: see text], as well as all parameters [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], [Formula: see text], determining [Formula: see text]. Therefore, our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method.

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