generate . Finally, we discuss the problem of finding the minimum number of monomials , , which have the property that the translates of the functions , , generate , for a given .
where is normalized Lebesgue measure on the unit circle, is a nonnegative integrable function, and ranges over the trigonometric polynomials with frequencies in
or
These distances are related to other extremal problems, and are shown to be positive if and only if is integrable. In some cases they are expressed in terms of the series coefficients of the outer functions associated with .
for all . We study its properties, their relation to the ``Lebesgue measure" defined on by R. Baker in 1991, and the associated Hausdorff dimension. Finally, we give some examples.
Then, for each for which the set is not convex and for each convex set dense in , there exist and 0$"> such that the equation
has at least three solutions.
for the highest exponent of the system, where
The previous best known value and the substantially smaller values of are reduced to the still smaller value.
where , converges to a period two solution.
where denotes the Fourier transform and if , and if . We determine all pairs such that on of negative order is bounded from to . To be more precise, we prove that for the estimate holds if and only if , where
We also obtain some weak-type results for .
where are any nonnegative real numbers with 0$">. We prove that there exists a positive integer such that every positive solution of this equation is eventually periodic of period .
(i) Suppose is an ordered Banach space with weakly normal closed cone and assume there exists such that for all . If the local resolvent admits a bounded analytic extension to the right half-plane 0\}$">, then for all and we have
(ii) Suppose is a rearrangement invariant Banach function space over with order continuous norm. If is an element such that defines an element of , then for all and we have