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\ce{\bf Computer Sciences Department} \smallskip

\def\title#1{\vskip .5cm\noindent{\bf #1}\smallskip}
\ce{\bf CS514, Spring 02}
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\ce{\bf Prof. Ron}
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\ce {\bf Midterm Exam: sample questions}
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\ce{\bf Date: March 11, 7:00-9:00 p.m.}
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Closed books. Calculators are allowed.

The sample is merely aiming at illustrating the kind of questions 
you might expect on this exam. Of course, the sample is not 
comprehensive, only illustrative.


\next{}
Define the notion of a CQF mask. Check whether the function
$$H(\ome)=\cos^2(\ome/2)$$
is a CQF.

\next{} Check whether  the following assignment:
$$\lam: f\mapsto \int_0^1 t^3 f'(t)\, dt$$
is a linear functional.

\next{} Let $\calD_a$ be the normalized dilation operator:
$$\calD_a f(t):=\sqrt{a} f(at).$$
Now, assume that $f$ is $2\pi$-periodic. Is $\calD_2 f$ necessarily
a $2\pi$-periodic function, too? Explain, and discuss at list one example.

\next{} Find the Fourier transform of the function
$$f(t):=\cases{1,&$0\le t\le 10$,\cr
               0,&otherwise.\cr}$$

\next{} You are given the function:
$$f(t)={1\over 1+t^2}.$$
Without computing $\hatf$, what can you say about (i) the smoothness,
(ii) decay, of $\hatf$? Explain.

\next{} Is the function $f$ whose Fourier transform is given by
$$\hatf(\ome)={\sin(\ome/2)\over \ome/2}$$
refinable? If your answer is `yes' find its refinement mask. If your answer
is `no', explain why.

\next{} You are given the following {\tt matlab} code:....
Explain what the code does, and write down its output.

\next{} You are given the following {\tt Matlab} code. Note that
the code contains three loops. Rewrite the code, using as few loops as 
possible.

\next{} Let $H_0(\ome)$ be a trigonometric polynomial that satisfies
$$|H_0(\ome)|^2=\cos^6(\ome/2)+3\cos^4(\ome/2)\sin^2(\ome/2).$$
Your friend Jeff found a wavelet mask $H_1$ such that
the pair $(H_0,H_1)$ satisfies the unitary extension principle.
What can  you say about $H_1$ (without finding explicitly $H_0$)?
what can you say about the wavelet $\psi$ whose   mask is $H_1$?

\bye
