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Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm

I can achieve $L(f - g) \leq (\frac{1}{2} + \frac{\pi}{4})\epsilon = (1.285\ldots)\epsilon$. Two reductions: (1) we can assume $|f(t)| < c$ for all $t \in (a,b)$ and (2) we can take $\epsilon = 1$. ...
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2 votes
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Construction of the Lipschitz function with a given Lipschitz constant and given two values

$\newcommand\ep\varepsilon$Yes, it is easy to construct a counterexample here. Indeed, if $g_\ep$ is such a function for each given real $\ep>0$ (so that $|g_\ep| \geq c$, $g_\ep(a)=f(a)$, $g_\ep(b)...
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Bounding the discrete $l^p$ norm by the continuous $L^p$ norm for trigonometric polynomials

Yes, this goes back to the work of Plancherel, M.; Pólya, George, Fonctieres entières et intégrales de Fourier multiples, Comment. Math. Helv. 9, 224-248 (1937). ZBL0016.36004. (see for instance ...
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