Friedrich Littmann

Gaussian subordination for the Beurling-Selberg extremal problem

We determine extremal entire functions for the problem of ma- jorizing, minorizing, and approximating the Gaussian function e x 2 by en- tire 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 asjxj for 1 < ; log (x 2 + 2 )=(x 2 + 2 ) , for 0 < ; log x2 + 2 ; andx2n logx2 , forn2 N. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomi- als and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one.

Quadrature and Extremal Bandlimited Functions

Let

Polynomials with coefficients from a finite set

\mathcal{A}(\delta)

Entire majorants via Euler–Maclaurin summation

be the class of functions of exponential type

Extremal functions in de Branges and Euclidean spaces

\delta>0

Extremal Functions with Vanishing Condition

. We prove that for integrable

One-sided approximation by entire functions

F\in\mathcal{A}(2\pi \delta)

L1-approximation to Laplace transforms of signed measures

\int_{-\infty}^\infty F(x) dx = \delta^{-1}\sum_{\xi\in\mathcal{T}_{\gamma,r}} (1-\frac{\gamma}{\pi(\xi^2+\gamma^2)+\gamma}) F(\delta^{-1}\xi)

Interpolation formulas with derivatives in de Branges spaces II

, where