We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function e −πλx 2 by entire functions of exponential type. The combination of the Gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously known examples (for instance [3], [4], [10] and [17]), plus a variety of new interesting functions such as |x| α for −1 < α; log (x 2 + α 2 )/(x 2 + β 2 ) , for 0 ≤ α < β; log x 2 +α 2 ; and x 2n log x 2 , for n ∈ N. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomials and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one.
Abstract.We determine a class of real valued, integrable functions fix) and corresponding functions MA[x) such that fix) < MA[x) for all x, the Fourier transform MA[t) is zero when |/| > 1, and the value of MA[0) is minimized. Several applications of these functions to number theory and analysis are given.
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