After more consultation with some colleagues, we can at last tell you
our decision about topics which will NOT be part of the official
syllabus and of course will NOT appear on examination questions
during this particular semester. (Winter 2001/2).
You can see what the syllabus will be by "subtracting" the topics
in the list below from the usual syllabus in the catalogue and on the
Technion general website for courses.

We shall soon also prepare a more detailed version of the reduced
syllabus.

Thanks for your patience.

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The material on inner product spaces will be the same as is usually
taught in other semesters.

List of topics which will NOT be taught and NOT be examined:

(Some of these topics are also sometimes not taught in a regular
semester. There is not always enough time to do them properly.)

(1) The proof of Dirichlet's theorem.
But, please note that you must know the statement of this theorem and be
able to use it to solve problems. (see further remarks after the list.)
(2) The proof that the trigonometric system is complete for the space of
continuous (or piecewise continuous) functions on the interval
[ - pi,pi ].
(3) Fourier series on other intervals. This is not a difficult topic.
You will need it as a tool for solving PDEs in other courses.
WE WILL MENTION IT IN THE LECTURES, but we will not have time to deal
with it in tirgulim. We will also make available some notes and
explanations to you, but again this material will not be required for
the examination.
(4) Gibbs' phenomenon
(5) Applications of Fourier series to solution of PDE.
(6) Shannon Sampling theorem.
(7) Application of Fourier transforms to solving PDE (e.g. the heat equation
on the line, or Laplace's equation in a half plane.)
(8) The use of complex variable methods for calculating Fourier and
Laplace transforms.
(9) The special "improper" line integral which gives the inverse Laplace
transform.
(10) The Dirac delta "function" and its Laplace transform.


FURTHER REMARKS:
As "compensation" for not teaching the proof of Dirichlet's theorem I
will make available on my website a relatively short and simple proof (3
pages) of the special case of this theorem for the case of a 2 pi
periodic function which is differentiable at every point. It does not
use the Dirichlet kernel, only the Riemann-Lebesgue lemma, and other
basic facts. This proof will NOT be required material for the
examination.

Using this simpler proof means that you will not encounter the Dirichlet
kernel ("garin") and the idea of representing and studying functions
with the aid of a kernel. (This is useful in other situations also.) Nor
will you see the analogue of the Dirichlet kernel in the context of
Fourier transforms. On the other hand this proof is a good way of
illustrating and emphasizing the important interaction between
translation (hazazah) of functions and what it does to their Fourier
series (or Fourier transforms)
Of course full details about the usual proof using the Dirichlet kernel
are available in the course text book by A.Pinkus and S.Zafrani, also
available on the web.

Concerning the proof of the theorem for finding the inverse of the
Fourier transform (the "Fourier inversion theorem"): Here we will
probably only give a simplified proof, in the style of the simplified
proof of Dirichlet's theorem mentioned above. Again we will only
consider the case of relatively nice functions with extra properties.
This material will also be made available on the internet.

In the material about Fourier and Laplace transforms, we use various
basic results about single or double integrals on infinite intervals or
infinite rectangles: interchange of order of integration (Fubini's
theorem), differentiation under the integral sign, Lebesgue dominated
convergence, etc. In a normal semester we do not have time or tools to
completely justify these results. This semester we may have to present
them even more hastily than usual.