# A Sharp Bernstein-type inequality for exponential sums

Borwein, Peter and Erdelyi, Tamas (1995) A Sharp Bernstein-type inequality for exponential sums. [Preprint]

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A difficult Bernstein-type extremal problem is solved by establishing the equality $$\sup_{0 \neq f \in {\widetilde E}_{2n}} \frac{|f^\prime(0)|}{\|f\|_{[-1,1]}} = 2n - 1$$ where $${\widetilde E}_{2n} := \Biggl\{f : f(t) = a_0 + \sum^n_{j=1} \bigl(a_j e^{\lambda_j t} + b_j e^{-\lambda_j t}\bigr), \quad a_j, b_j, \lambda_j \in {\Bbb R} \Biggr\}.$$ This settles a conjecture of Lorentz and others and it is surprising to be able to provide a sharp solution. It follows fairly simply from the above that $$\frac {1}{e-1}\frac {n-1}{\min\{y-a,b-y\}} \leq \sup_{0 \neq f \in E_n} \frac{|f^\prime(y)|}{\|f\|_{[a,b]}} \leq \frac {2n - 1}{\min\{y-a,b-y\}}$$ for every $y \in (a,b)$, where $$E_n := \Biggl\{f : f(t) = a_0 + \sum^n_{j=1} a_j e^{\lambda_j t}, \quad a_j, \lambda_j \in {\Bbb R} \Biggr\}.$$ The proof relies on some subtle properties of the particular Descartes system $$(\sinh \lambda_0t,\sinh \lambda_1t, \ldots , \sinh \lambda_nt), \qquad 0 < \lambda_0 < \lambda_1 < \cdots < \lambda_n$$ for which certain comparison theorems can be proved. Also critical to the proof is the interesting observation that the zeros of the Chebyshev polynomials associated with such systems have various lexicographic properties. Essentially sharp Nikolskii-type inequalities are also proved for $E_n$.