Let f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ , where \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty for some α>2. Let us assume that for some β>1/α, we have f(x)\equal{}O(e^{\minus{}x^{\beta}}) as x→∞. Prove that an is identically 0.
G. Halasz inductionfunctionlogarithmsintegrationlimitreal analysisreal analysis unsolved