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Inequalities for exponential sums
T. Erdélyi Department of Mathematics, Texas A&M University, College Station, TX, USA
Abstract:
We study the classes
\begin{gather*}
{\mathscr E}_n:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{\lambda_jt}},
\ a_j, \lambda_j\in {\mathbb C} \biggr\},
\\
{\mathscr E}_n^+:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{\lambda_jt}}, \ a_j, \lambda_j\in {\mathbb C},
\ \operatorname{Re}(\lambda_j) \geqslant 0 \biggr\},
\\
{\mathscr E}_n^-:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{\lambda_jt}},
\ a_j, \lambda_j\in {\mathbb C},
\ \operatorname{Re}(\lambda_j)\leqslant 0 \biggr\},
\end{gather*}
and
$$
{\mathscr T}_n:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{i\lambda_jt}},
\ a_j\in {\mathbb C},
\ \lambda_1<\lambda_2<\dots<\lambda_n \biggr\}.
$$
A highlight of this paper is the asymptotically sharp inequality
$$
|f(0)|\leqslant (1+\varepsilon_n)3n\|f(t)e^{-9nt/2}\|_{L_2[0,1]},
\qquad f\in {\mathscr T}_n ,
$$
where
$\varepsilon_n$ converges to
$0$ rapidly as
$n$ tends to
$\infty$. The inequality
$$
\sup_{0 \not \equiv f\in {\mathscr T}_n}{ \frac{|f(0)|}{\|f\|_{L_2{[0,1]}}}} \geqslant n
$$
is also established. Our results improve an old result due to Halász and a recent result due to Kós. We prove several other related order-sharp results in this paper.
Bibliography: 33 titles.
Keywords:
exponential sums, Nikol'skii-, Bernstein- and Markov-type inequalities, infinite-finite range inequalities.
UDC:
517.518.862
MSC: 11C08,
41A17 Received: 09.02.2016 and 11.11.2016
DOI:
10.4213/sm8670