The Fekete polynomials are defined as
where is the Legendre symbol. These polynomials arise in a number of contexts in analysis and number theory. For example, after cyclic permutation they provide sequences with smallest known norm out of the polynomials with coefficients.
The main purpose of this paper is to prove the following extremal property that characterizes the Fekete polynomials by their size at roots of unity.
Theorem 0.1. Let with odd and . If
then must be an odd prime and is . Here
This result also gives a partial answer to a problem of Harvey Cohn on character sums.
We construct an example of a purely 1-unrectifiable AD-regular set in the plane such that the limit
exists and is finite for almost every for some class of antisymmetric Calderón-Zygmund kernels. Moreover, the singular integral operators associated with these kernels are bounded in , where has a positive measure.
where is the best Sobolev constant and is the space obtained by taking the completion of with the norm . We prove here a refined version of this inequality,
where is a positive constant, the distance is taken in the Sobolev space , and is the set of solutions which attain the Sobolev equality. This generalizes a result of Bianchi and Egnell (A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18-24), which was posed by Brezis and Lieb (Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86). regarding the classical Sobolev inequality
A key ingredient in our proof is the analysis of eigenvalues of the fourth order equation
where and is the unique radial function in with . We will show that the eigenvalues of the above equation are discrete:
and the corresponding eigenfunction spaces are
under the hypothesis that is an ample vector bundle on .
where is a ball, and if , if , is unique (up to rotation) if is small enough.
The following dichotomy is established for any pair , of hereditary families of finite subsets of : Given , an infinite subset of , there exists an infinite subset of so that either , or , where denotes the set of all finite subsets of .
then in .
In this paper we deal with the interpolation from Lebesgue spaces and , into an Orlicz space , where and for some concave function , with special attention to the interpolation constant . For a bounded linear operator in and , we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,
where the interpolation constant depends only on and . We give estimates for , which imply . Moreover, if either or , then . If , then , and, in particular, for the case this gives the classical Orlicz interpolation theorem with the constant .
In an earlier paper, we showed that
where denotes linear Lebesgue measure. Here we obtain, for each , the sharp version of this inequality in terms of condenser capacity. In particular, we show that as ,
where
In this paper we determine the groups
for all 0$">. Its rank ranges from to depending on the value of .
for allows every function to be written as an infinite linear combination of translated and modulated versions of the fixed function . In the present paper we find sufficient conditions for to be a frame for , which, in general, might just be a subspace of . Even our condition for to be a frame for is significantly weaker than the previous known conditions. The results also shed new light on the classical results concerning frames for , showing for instance that the condition A>0$">is not necessary for to be a frame for . Our work is inspired by a recent paper by Benedetto and Li, where the relationship between the zero-set of the function and frame properties of the set of functions is analyzed.
Let be a covariant system and let be a covariant representation of on a Hilbert space . In this note, we investigate the representation of the covariance algebra and the -weakly closed subalgebra generated by and in the case of or when there exists a pure, full, -invariant subspace of .
Let be a convex and dominated statistical model on the measurable space , with minimal sufficient, and let . Then , the -algebra of all permutation invariant sets belonging to the -fold product -algebra , is shown to be minimal sufficient for the corresponding model for independent observations, .
The main technical tool provided and used is a functional analogue of a theorem of Grzegorek (1982) concerning generators of .
where is the rational homotopy Lie algebra of and its centre.
Several interesting examples are presented to illustrate our results.
with on the half-space as a consequence of a new Liouville theorem of elliptic type for solutions of () on . This new result is in turn a consequence of other new phenomena established for nonlinear evolution problems. In particular, we prove that the inequality
has no nontrivial solutions on when We also show that the inequality
has no nontrivial nonnegative solutions for , and it has no solutions on bounded below by a positive constant for 1.$">