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Let $\xi_1,\dotsc,\xi_n$ and $\eta_1,\dotsc,\eta_n$ be real numbers.

Is there a complex valued function $f\in L^1(\mathbb{R})$ such that:

  1. $\int f(x)e^{2\pi i\xi_k x}dx=1$ for every $1\leq k\leq n$.
  2. $\int f(x)e^{2\pi i\eta_k x}dx=0$ for every $1\leq k\leq n$.
  3. $|\int f(x)dx|\leq C$ for some absolute constant $C$ (not depending on $n$)

If the answer to the question above is negative: Is there a sequence of functions $f_n\in L^1(\mathbb{R})$ such that:

  1. $\int f_n(x)e^{2\pi i\xi_k x}dx\rightarrow 1$ as $n\rightarrow\infty$, for every $1\leq k\leq n$.
  2. $\int f_n(x)e^{2\pi i\eta_k x}dx\rightarrow 0$ as $n\rightarrow\infty$, for every $1\leq k\leq n$.
  3. $|\int f_n(x)dx|\leq C$ for every $n$, for some absolute constant $C$ (not depending on $n$)
Terry
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  • is it a joke, for every $i$ ?! come on ! here $i^2 = -1$ – reuns May 04 '16 at 02:56
  • @user1952009: It's not a joke. Here, $i$ appears in two meanings. In "$2\pi $i" it's a solution for $X^2+1=0$. In $\xi_i$ and $\eta_i$ it's a index ranging from $1$ to $n$. I will change the index to $k$ instead of $i$ for clarity. – Terry May 04 '16 at 06:12
  • so, do you know the Fourier transform, in particular the Fourier inversion theorem, for example when both $f$ and $\hat{f} \in L^1$ ? and do you know a way for constructing $f$ such that $\hat{f}(\xi_0) = 0$ ? – reuns May 04 '16 at 07:30
  • @user1952009: Maybe I can take $g(x)=e^{2\pi i x}$ and an $L^1$ function $h(x)$ such $\int h(x)dx=\hat{f}(\xi_0)$, and then $\int g(x)h(x) e^{-2\pi I x}=\hat{f}(\xi_0)$. Now I can "correct" $f$ using $g(x)h(x)$. I'm not sure how to do it for all $\xi_i$ and $\eta_i$, especially while keeping $\int fdx$ bounded. – Terry May 04 '16 at 09:47
  • I understood nothing of what you said. Note that with $h(x) = e^{-x^2}$ we have $\hat{h}(\xi) = \int_{-\infty}^\infty e^{-2 i \pi \xi x} h(x) dx = \sqrt{\pi} e^{-\pi^2\xi^2}$, which is never $0$. hence, if $f \in L^1$ and $\hat{f} \in L^1$ then $\hat{f} - \frac{\hat{f}(\xi_0)}{\hat{h}(\xi_0)} \hat{h}$ vanishes at $\xi_0$. – reuns May 04 '16 at 10:44
  • I see. Can we extend this method to control the values at all $\xi_i$ and $eta_i$ and not just at one place $\xi_0$? Can we do so while keeping $\int fdx$ bounded? – Terry May 04 '16 at 12:05

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