The results of this paper concern the expected norm of random polynomials on the boundary of the unit disc (equivalently of random trigonometric polynomials on the interval ). Specifically, for a random polynomial
let
Assume the random variables , are independent and identically distributed, have mean 0, variance equal to 1 and, if 2$">, a finite moment . Then
and
as .
In particular if the polynomials in question have coefficients in the set (a much studied class of polynomials), then we can compute the expected norms of the polynomials and their derivatives
and
This complements results of Fielding in the case, Newman and Byrnes in the case, and Littlewood et al. in the case.
The principal result of this paper is the following Markov-type inequality for Müntz polynomials. Theorem. Let be an integer. Let be distinct real numbers. Let . Then
where the supremum is taken for all (the span is the linear span over ).
with on the half-space as a consequence of a new Liouville theorem of elliptic type for solutions of () on . This new result is in turn a consequence of other new phenomena established for nonlinear evolution problems. In particular, we prove that the inequality
has no nontrivial solutions on when We also show that the inequality
has no nontrivial nonnegative solutions for , and it has no solutions on bounded below by a positive constant for 1.$">
where is a bounded domain in with smooth boundary , is continuous and p-sublinear at and is a large parameter.
and
where and are complex variables in suitably defined cut planes, were first evaluated by Watson in 1939 for the special cases and , respectively. In the present paper simple direct methods are used to prove that can be expressed in terms of squares of complete elliptic integrals of the first kind for general values of and . It is also shown that and are related by the transformation formula
where
Thus both of Watson's results for are contained within a single formula for .
as goes to infinity. Here is a smooth bounded domain of . Among other results we give a positive answer to a question raised by Chen, Ni, and Zhou (2000) by showing that .
(where are independent complex Steinhaus random variables), then there exists a set , with , such that
for all (). The dependence on of the threshold proportion is sharp.
then for each nontrivial closed invariant subspace for the shift on .
in remains a frame when the dilation matrices and the translation parameters are perturbed. As a special case of our result, we obtain that if is a frame for an expansive matrix and an invertible matrix , then is a frame if and for sufficiently small 0$">.
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.
The theorem is a multi-dimensional analogue for some well-known operator moment problems due to Sebestyén in case or, recently, to Gavruta and Paunescu in case .
where is a certain cube associated with the dyadic cube and . We characterize the pair of weights for which the maximal operator applies into weak- for .
where is a sequence of independent random variables taking on values and with equal probability. Moreover, it is shown that
The paper concludes by providing an example indicating that, if , then the estimate
is the best possible.
where . Using the concept of -Gateaux derivative, we apply this result to characterize orthogonality in the sense of James in , and to give an easy proof of the characterization of smooth points in .