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. In this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on R N . These extremal functions minimize the L 1 (R N , |x| 2ν+2−N dx)-distance to the original function, where ν > −1 is a free parameter. To achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function F λ (x) = e −λ|x| , where λ > 0, in the general framework of de Branges spaces of entire functions. We then specialize the construction to a particular family of homogeneous de Branges spaces to approach the multidimensional Euclidean case. Finally, we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter λ > 0 against suitable measures. Applications of the results presented here include multidimensional versions of Hilbert-type inequalities, extremal one-sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one-sided approximations by polynomials for a class of functions on the sphere S N−1 with an axis of symmetry.
Abstract. Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function N (T, β) defined to be the number of pairs γ and γ of ordinates of nontrivial zeros of the Riemann zetafunction satisfying 0 < γ, γ ≤ T and 0 < γ − γ ≤ 2πβ/ log T as T → ∞. In this paper, assuming the Riemann hypothesis, we prove upper and lower bounds for N (T, β), for all β > 0, using Montgomery's formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval [−β, β] in a way to minimize the L 1 R, 1 − sin πx πx 2 dx -error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work by P. X. Gallagher [19] in 1985, where the case β ∈ 1 2 N was considered using non-extremal majorants and minorants.
Let A(δ) be the class of functions of exponential type δ > 0. We prove that for integrable F ∈ A(2πδ) ∞ −∞ F (x)dx = δ −1 ξ∈Tγ,r 1 − γ π(ξ 2 + γ 2) F (δ −1 ξ) where Tγ,r is the set of zeros of Bγ,r(z) = z sin π(z + r) − γ cos π(z + r). Let a > (2δ) −1. It is shown that for any Polya-Laguerre entire function E with E(±a) = 0 there exist two integrable functions G−, G+ ∈ A(2πδ) such that for all real x E(x){G−(x) − χ [−a,a] (x)} ≤ 0, E(x){G+(x) − χ [−a,a] (x)} ≥ 0. Combining these results we find the minimal value of ||S − T ||1 where S, T ∈ A(2πδ) satisfy S(x) ≤ χ [−a,a] (x) ≤ T (x) for all real x. We determine extremal functions for which the minimal value is assumed. As an application we give an explicit expression for C(δ, α) = inf g∈A 2 (δ) sup x∈[−α,α] ||g|| 2 2 |g(x)| 2 where A2(δ) is the set of square integrable functions in A(δ). This constant occurs in work of Donoho and Logan regarding reconstruction of bandlimited functions. 2010 Mathematics Subject Classification. primary 42A05, secondary 30D15, 26A51. Key words and phrases. entire functions of exponential type, totally positive functions, extremal majorants, quadrature, de Branges spaces.
Abstract. For a variation diminishing function g which is analytic on a set containing the real line and any real polynomial P , we prove that g + P has at most deg(P ) + 2 real zeros.Based on this estimate, we present a way to construct entire approximations Gn to the truncated powers x
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