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$">, a finite moment . Then
and
as .
In particular if the polynomials in question have coefficients in the set (a much studied class of polynomials), then we can compute the expected norms of the polynomials and their derivatives
and
This complements results of Fielding in the case, Newman and Byrnes in the case, and Littlewood et al. in the case.
is considered subject to the boundary conditions
We assume that is positive and that is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to for , or equivalently up to for , the eigenvalues of the above boundary value problem.
Theorem 1. The following are equiconsistent:
(i) a Jónsson cardinal;
(ii) a sufficiently elementary submodel of the universe of sets with not homeomorphic to
The reverse direction is a corollary to:
Theorem 2. is Jónsson hereditarily separable, hereditarily Lindelöf, with .
We further consider the large cardinal consequences of the existence of a topological space with a proper substructure homeomorphic to Baire space.
In this paper, we are concerned with the boundedness of all the solutions and the existence of quasi-periodic solutions for second order differential equations
where the 1-periodic function is a smooth function and satisfies sublinearity:
Assume the dimension of is greater than one. About 20 years ago the author asked the following questions:
Can this aposyndetic decomposition raise dimension? Can it lower dimension? We answer these questions by proving the following theorem.
Theorem. The dimension of the quotient space is one.
Conjecture 0.1(Erdos and Turán). Suppose that is an increasing sequence of integers and
Suppose that
If 0$"> for all , then is unbounded.
Our main purpose is to show that the sequence cannot be bounded by . There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.
It is also shown, via a connection between the operator and Laguerre functions, that
The bilinear Hilbert transform is given by
It satisfies estimates of the type
In this paper we prove such estimates for a discrete model of the bilinear Hilbert transform involving the Walsh Fourier transform. We also reprove the well-known fact that the Walsh Fourier series of a function in , with converges pointwise almost everywhere. The purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time-frequency analysis.
In this paper we give exact formulae for for various values of . We also give a variety of related results for different classes of polynomials including polynomials of fixed height H, polynomials with coefficients and reciprocal polynomials. The results are surprisingly precise. Typical of the results we get is the following.
Theorem 0.1. For , we have
and
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.
Closely related and equally interesting notion is that of , which is the collection of numbers which can be represented as a sum of different elements of :
The goal of this paper is to investigate the structure of and , where is a subset of . As application, we solve two conjectures by Erdös and Folkman, posed in 1960s.
where , then
In the proof, the zeros of the function are redistributed to minimize the large values of .
We prove in this paper a Bochner integral representation theorem for bounded linear operators
which satisfy the following condition:
where is the conjugate space of . In the particular case where , this condition is obviously satisfied by every bounded linear operator
and the result reduces to the classical Riesz representation theorem.
If the dimension of is greater than , we show by a simple example that not every bounded linear admits an integral representation of the type above, proving that the situation is different from the one dimensional case.
Finally we compare our result to another representation theorem where the integration process is performed with respect to an operator valued measure.
A class of functions and the corresponding solutions of
are obtained as a special case of the solutions of
where is defined as .