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I'm interested in entire functions of exponential type $\sigma$ (Bernstein space $B_\sigma^1$) following

$$\int_{-\infty}^{\infty} |f(x)|dx=1,$$

whose norm is as small as possible outside a range $[-A,A]$, or even attain the minimum of

$$\int_{|x|>A} |f(x)|dx$$

Is there any result in the bibliography about this problem? If not, a numerical method to approximately obtain such $f$ would be helpful.

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This survey article is related to your question: MR0776471 (86g:42005) Vaaler, Jeffrey D. Some extremal functions in Fourier analysis. Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 183–216.

See also later papers of the same author, joint with E. Carneiro, Some extremal functions in Fourier analysis.

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