Jeffrey D. Vaaler

Some Extremal Functions in Fourier Analysis, III

We obtain the best approximation in L1(ℝ), by entire functions of exponential type, for a class of even functions that includes e−λ|x|, where λ>0, log |x| and |x|α, where −1<α<1. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.

A class of extremal functions for the Fourier transform

We determine a class of real valued, integrable functions f(x) and corresponding functions M

Counting algebraic numbers with large height II

x) such that f(x) 1, and the value of MfO) is miimi. Several applications of these functions to number theory and analysis are given.

Note on a Diophantine inequality in several variables

Let ℚ denote the field of rational numbers, Open image in new window an algebraic closure of ℚ, and H : Open image in new window the absolute, multiplicative, Weil height. For each positive integer d and real number \( \mathcal{H} \geqslant 1 \), it is well known that the number Open image in new window of points α in Open image in new window having degree d over ℚ and satisfying \( H\left( \alpha \right) \leqslant \mathcal{H} \) is finite. This is the one-dimensional case of Northcott’s Theorem [8] (see also [5, page 59]). The systematic study of the counting function Open image in new window , and that of related functions in higher dimensions, was begun by Schmidt [10]. It is relatively easy to prove the existence of a positive constant C = C(d) such that Open image in new window (1) and also the existence of positive constants c = c(d) and \( \mathcal{H}_0 = \mathcal{H}_0 \left( d \right) \) such that Open image in new window (2)

Polynomials with Low Height and Prescribed Vanishing

We establish estimates for the number of points that belong to an aligned box in (R/Z)N in terms of certain exponential sums. These generalize previous results that were known only in case N = 1.

Small zeros of quadratic forms over number fields. II

In a recent paper [2] we obtained an improved formulation of Siegel’s classical result([9],Bd. I,p. 213, Hilfssatz) on small solutions of systems of linear equations. Our purpose here is to illustrate the use of this new version of Siegel’s lemma in the problem of constructing a simple type of auxiliary polynomial. More precisely, let k be an algebraic number field, O k its ring of integers, α1,α2,…,αJ distinct, nonzero algebraic numbers (which are not necesarily in k), and m1,m2,…,mJ positive integers. We will be interested in determining nontrivial polynomials P(X) in 0 K [X] which have degree less than N, vanish at each αj with multiplicity at least mj and have low height. In particular, the height of such plynomials will be bounded from above by a simple function of the degrees and heights of the algebraic numbers αj and the remaining data in the problem: m1,m2,…mJ, N and the field constants associated with k.

Gaussian subordination for the Beurling-Selberg extremal problem

Let F be a nontrivial quadratic form in N variables with coefficients in a number field k and let A be a K x N matrix over k. We show that if the simultaneous equations F(x) = 0 and Ax = 0 hold on a subspace X of dimension L and L is maximal, then such a subspace X can be found with the height of X relatively small. In particular, the height of X can be explicitly bounded by an expression depending on the height of F and the height of A. We use methods from geometry of numbers over adele spaces and local to global techniques which generalize recent work of H. P. Schlickewei.

Inequalities for Heights Of Algebraic Subspaces And the Thue-Siegel Principle

We determine extremal entire functions for the problem of ma- jorizing, minorizing, and approximating the Gaussian function e x 2 by en- tire functions of exponential type. The combination of the Gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously known examples (for instance (3), (4), (10) and (17)), plus a variety of new interesting functions such asjxj for 1 < ; log (x 2 + 2 )=(x 2 + 2 ) , for 0 < ; log x2 + 2 ; andx2n logx2 , forn2 N. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomi- als and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one.

Perturbing polynomials with all their roots on the unit circle

Let k be an algebraic number field and let k N denote the vector space of N x 1 column vectors over k. In his fundamental paper [16] W.M. Schmidt introduced a concept of height on linear subspaces A of k N . The idea is to apply the so-called Weil-height (or the absolute Weil-height) to the vector of Grassmann coordinates of any basis for A. In this way Schmidt was able to formulate and prove many of the basic theorems of Diophantine approximation in a very general setting. In the present paper we prove a new inequality for heights on subspaces and apply it to the problem of constructing certain auxiliary polynomials in two variables. Let H(A) denote the height of the subspace A \( \subseteq \) k N , which is precisely defined in section 2. We adopt the convention that Our inequality has both a local and global version and the global version can be stated as follows.

Some extremal functions in Fourier analysis. II

Given a monic real polynomial with all its roots on the unit circle, we ask to what extent one can perturb its middle coefficient and still have a polynomial with all its roots on the unit circle. We show that the set of possible perturbations forms a closed interval of length at most 4, with 4 achieved only for polynomials of the form x 2n + cx n + 1 with c in [-2,2]. The problem can also be formulated in terms of perturbing the constant coefficient of a polynomial having all its roots in [-1,1]. If we restrict to integer coefficients, then the polynomials in question are products of cyclotomics. We show that in this case there are no perturbations of length 3 that do not arise from a perturbation of length 4. We also investigate the connection between slightly perturbed products of cyclotomic polynomials and polynomials with small Mahler measure. We describe an algorithm for searching for polynomials with small Mahler measure by perturbing the middle coefficients of products of cyclotomic polynomials. We show that the complexity of this algorithm is O(C√ d ), where d is the degree, and we report on the polynomials found by this algorithm through degree 64.