It is the purpose of this talk to demonstrate that there is a new
(alternative) approach to the theory of Fourier multipliers for
$L^p$-spaces, i.e. a characterization of the space of all
multipliers, i.e. bounded operators on $L^p(R^d)$ which commute
with translations.

The key tool is the Herz-space (which is in fact even an algebra
with respect to pointwise multiplication) $A_p(R^d)$, which can be
characterized as follows: Starting from $$ A_p(R^d) = \{ h =
\sum_{n=1}^ïnfty f_n \ast g_n \}, $$ where sequence $(f_n)_{n \geq
1}$ in $\L^p(R^d)$ and the sequence $(g_n)_{n \geq 1}$ in the dual
space $L^q(R^d)$ satisfies $$\sum_{n=1}^\infty \|f_n\|_p \|g_n\|_q
< \infty.$$ There is a natural infimum (i.e. quotient) norm for
this space, turning $A_p(R^d)$ into a Banach space (in fact, a
Banach algebra with pointwise multiplication, which is called the
Herz-Algebra). Using it we have:

A bounded operator $T: L^p(R^d) \to \L^p(R^d)$ commutes with
translations if and only if it is a convolution operator with some
(mild) distribution which defines a functional on $A_p(R^d)$, and
the norm of the functional and the operator norm of $T$ on
$L^p(R^d)$ are equivalent.

The new proof that will be discussed in the presentation makes use
the Banach Gelfand Triple $(S_0,L^2,S_0*)$ and is based on the
observation that the multipliers of $L^p(R^d)$ can be shown to be
also multipliers of $A_p(R^d)$ and vice versa.




DETAILED TEXT:

For mathematicians convolution is one of the central subjects of
Fourier analysis and many of its properties are derived from the
fact that it is turned into pointwise multiplication via the
Fourier transform (first for integrable function, then in a more
general context, of either tempered, or easier, for so-called
``mild distribution'', i.e. element of $S_0*$, the dual of the
Feichtinger algebra $S_0(R^d)$.

For engineers convolution operators appear naturally at the
beginning of system-theory courses, where it is shown (mostly by
some vague, but mathematically not strict arguments) that every
translation-invariant system, i.e. any linear operator $T$ which
commutes with translations, can be represented as a convolution
operator. The convolution kernel generating this operator (via a
moving average construction) is known as the impulse response of
the system $T$. Given the convolution theorem (say for $L^1$) one
can of course show that $T$ can also be understood as a pointwise
multiplication operator on the Fourier transform side, and that
this multiplication by the so-called ``transfer function'' suggests
that it is just the Fourier transform of the impulse response.

Trying a more strict mathematical characterization of the situation
we find that - depending on the setting - that we have quite a few
different cases which can be treated in a similar way, but only for
few of the classical cases one can give a complete characterization
of either the corresponding class of impulse response function
(convolution kernels) and their corresponding transfer functions.

We consider from now on only operators (systems) which commute with
translation. The known cases are: 1) $T$ is bounded on $L^2(R^d)$,
if and only if $ FT(T(f)) = h . FT(f)$, for some (essentially)
bounded function $h$ (on the frequency side). 2) Secondly Wendel's
Theorem provides a characterization for systems which are bounded
on $L^1(R^d)$. Similar to the case $C_0(R^d)$