The size of exponential sums on intervals of the real line

Details The size of exponential sums on intervals of the real line The size of exponential sums on intervals of the real line

2010-06-22

arxiv.org/abs/1006.4323v2

Mathematics

Classical Analysis and ODEs

Fixed minor problems and added a new remark

Scientific paper

We prove that there is a constant $c > 0$ depending only on $M \geq 1$ and $\mu \geq 0$ such that $$\int_y^{y+a}{|g(t)| \, dt} \geq \exp (-c/(a\delta))\,, a \in (0,\pi]\,,$$ for every $g$ of the form $$g(t) = \sum_{j=0}^n{a_j e^{i\lambda_jt}}, a_j \in {\Bbb C}, \enskip |a_j| \leq Mj^\mu\,, \enskip |a_0|=1\,, \enskip n \in {\Bbb N} \,,$$ where the exponents $\lambda_j \in {\Bbb C}$ satisfy $$\text{\rm Re}(\lambda_0) = 0\,, \qquad \text{\rm Re}(\lambda_j) \geq j\delta > 0\,, j=1,2,\ldots\,,$$ and for every subinterval $[y,y+a]$ of the real line. Establishing inequalities of this variety is motivated by problems in physics.

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