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I have the below problem and I am not sure how to go about proving it.

By considering $ \langle h|h \rangle $, where $h = f + \lambda g $ with $ \lambda = real $ prove that for two functions $f$ and $ g $,

$ \langle f|f \rangle $$ \langle g|g \rangle $ $>=$ $ \frac{1}{4}$[$ \langle f|g \rangle $ + $ \langle g|f \rangle $]$^2$

The function $y(x) $ is real and positive for $x$. Its Fourier cosine tranform $\tilde{y}_{c}(k)$ is defined by

$\tilde{y}_{c}(k)$ = $\int_{-\infty}^{\infty} y(x)cos(kx)dx $

and is given that $\tilde{y}_{c}(0) = 1$ Prove that

$\tilde{y}_{c}(k) \geq 2[\tilde{y}_{c}(k)]^2 $

Appreciate any thoughts on how to start this.

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  • $\begingroup$ @Ramanujan i am not quite following you. can you tell me more? $\endgroup$
    – Wickylk
    Commented Nov 3, 2020 at 23:06
  • $\begingroup$ Please define $\langle f| g \rangle$. $\endgroup$ Commented Nov 5, 2020 at 20:17

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