This question is strange enough so I'm beginning to suspect a typo/misinterpretation.

Let $\displaystyle w$ be defined by

$\displaystyle \hat{w}=\{ \begin{array}{c}

1 \text{ if } \xi \in (0.5,1] \\

-1 \text{ if } \xi \in [-1,-0.5) \\

0 \text{ otherwise }\end{array} \}$

Show that $\displaystyle w$ is admissible and compute its normalization constant.

First of all it is very strange to think of the fourier transform to be a function on the reals as opposed to a sequence, right? (I suspect this might be a typo)

Anyway, I interpret admissible to mean: satisfies the Orthonormal Conjugate Quadrature Filter Conditions.

So first I actually find $\displaystyle w$ by taking the inverse fourier transform:

$\displaystyle w=\check{\hat{w}}=\sum_{\xi=-\infty}^{\infty}\hat{w(\xi)}e^{2\pi i \xi x}$

$\displaystyle =e^{2\pi i x}-e^{-2\pi i x}=-2isin(2\pi x)$

and this is zero at all integer values, which makes it a very strange trivial case (which is not orthonormal).

Any thoughts?

Thanks.