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Let $\mu$ be a singular Borel probability measure on $[0, 1)$, and $f\in L^2(\mu)$. Estimate $$\sup_n\left|\int_0^1 \left(1-e^{i\alpha_n x}\right) f(x)d\mu(x)\right|$$ where $\alpha_n\in\mathbb R$ and $|\alpha_n|\leq L<1$.

This is my approach: $$\sup_n\left|\int_0^1 \left(1-e^{i\alpha_n x}\right) f(x)d\mu(x)\right|\leq \left(\sup_n \left|1-e^{i\alpha_n}\right|\right)\left|\int_0^1 f(x)d\mu(x)\right|=\left(2-2\cos L\right)\left|\int_0^1 f(x)d\mu(x)\right|$$ where the equality is thanks to $|\alpha_n|\leq L<1$.

Probably it is wrong, because it should be

$$\sup_n\left|\int_0^1 \left(1-e^{i\alpha_n x}\right) f(x)d\mu(x)\right|\leq \left(\sup_{n,\mu(x)} \left|1-e^{i\alpha_n x}\right|\right)\left|\int_0^1 f(x)d\mu(x)\right|$$ But here I do not know how to continue. Any suggestions please?

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This is more like comment, but too long.

  1. You probably want to estimate, not to calculate.

  2. $\mu$ is singular --- is this relevant?

  3. You're doing almost right, although there should be absolute value under the integral: \begin{gathered} \left|\int_0^1 \left(1-e^{i\alpha_n x}\right) f(x)d\mu(x)\right|\leq \left(\sup_{x\in[0,1]} \left|1-e^{i\alpha_nx}\right|\right)\int_0^1 |f(x)|d\mu(x)\\ \le \left(2-2\cos L\right)\int_0^1 |f(x)|d\mu(x). \end{gathered}

  4. Since $f\in L^2(\mu)$, you can also use Cauchy--Schwarz: \begin{gathered} \left|\int (1-e^{i\alpha_n x})f(x) d\mu(x)\right|^2 \le \int |1-e^{i\alpha_n x}|^2 d\mu(x) \int |f(x)|^2 d\mu(x)\\ = 2(1-\operatorname{Re}\varphi(\alpha_n)) \int |f(x)|^2 d\mu(x), \end{gathered} where $\varphi$ is the characteristic function of $\mu$.

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  • $\begingroup$ Thank you, I will answer your remarks in the following: 1) You're right, then I've edited my post; 2) maybe no, I written all known informations on the problem; 3) But $L<1$; 4) In what way? $\endgroup$
    – Mark
    Aug 29, 2015 at 7:04
  • $\begingroup$ @Mark, sorry, didn't pay attention at $L\le 1$. I've edited 3 and also 4 to explain about the characteristic function. $\endgroup$
    – zhoraster
    Aug 29, 2015 at 7:16

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