of them launching whole areas of Harmonic and Complex Analysis. For instance,
it is natural to consider the support of a measure and pronounce it small if it is
bounded, or has finite volume (length) or if it is thin (porous) in some sense. The
most basic statement of that type, the one that Wiener relays to non-mathematical
audience using musical analogies in his memoir [149], is that the supports of f and
f cannot both be compact. Further statements in the same direction are given in
the next subsection.
Another natural way to understand the smallness of f is in terms of its decay
near infinity. In many situations the Fourier transform
f an analytic function that
can be considered to be small if it has a large zero set. Of course, the smallness of
f and
f does not have to be understood in the same way and many deep problems
appear when the criteria for smallness applied to these two objects is different.
As was mentioned above, the Fourier transform does not have to be a part of
the statement and similar questions can be formulated about other transforms or,
more generally, a variety of pairs of ‘dual’ properties of f. Despite rather extensive
literature, many of such questions remain open.
In the rest of this section we give further examples of particular problems of UP,
with smallness in the general statement understood in terms of support, decay or
the size of the zero set. We then discuss an immediate connection of such questions
with completeness problems.
2.2. Support/Support. As was mentioned above, perhaps the simplest
statement of that type is the observation that the supports of f and
f on the
real line cannot both be bounded. A more advanced statement is a corollary of a
theorem by F. and M. Riesz that says that if μ M has semi-bounded support
then the support of ˆ μ contains the whole line R.
The next significant step in the same direction begins with results of Amrein-
Berthier [5] and Benedicks [12] that imply that the Lebesgue measures of the
supports of f and
f cannot both be finite. These results were later improved by
Nazarov [114] who showed that for any two sets T, S
||f||L2(Rn) C exp (C|S||T |)(||f||L2(Sc) + ||
where T c,Sc denote the complements of T, S correspondingly. If one of the sets is
convex, the product |S||T| in the above inequality may be raised into the power
1/n. It is unknown if the same is true in general.
As we can see, even in this case of UP there are subcases corresponding to
different meanings of smallness of the supports of f and
f Another set of problems
appears if one understands such smallness in terms of porosity. One of the classical
results in this group is the following theorem by Beurling from 1961 [14].
A collection of disjoint intervals {In} on R (or the union of these intervals,
when it is more convenient) is called long if
1 +
or short if the sum is finite. Here |In| denotes the length of In. Such collections of
intervals appear in many results in the area of UP including those discussed further
in this text.
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