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Moreover, he showed that there is equality in (1.4) if and only if F(z) = B(z). As an application Beurling found an interesting inequahty for almost periodic functions (we include it here in Theorem 15), but his results were never published. In 1974 A. Selberg used the function B(z) to obtain a sharp form of the large sieve inequahty. Selberg noted that if… (More)

Q × ́ and induces a metric topology in this group. We show that the completion of this metric space is a Banach space over the field R of real numbers. We further show that this Banach space is isometrically isomorphic to a co-dimension one subspace of L(Y,B, λ), where Y is a certain totally disconnected, locally compact space, B is the σ-algebra of Borel… (More)

We count algebraic numbers of fixed degree over a fixed algebraic number field. When the heights of the algebraic numbers are bounded above by a large parameter H, we obtain asymptotic estimates for their cardinality as H → ∞.

We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function e−πλx 2 by entire 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… (More)

- Michael J. Mossinghoff, Christopher G. Pinner, Jeffrey D. Vaaler
- Math. Comput.
- 1998

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 x2n + cxn + 1 with c… (More)

- Jeffrey D. Vaaler
- The American Mathematical Monthly
- 2008

Let F (z) = N n=0 a n z n be a polynomial with complex coefficients and roots α 1 ,. .. , α N , let F p denote its L p norm over the unit circle, and let F p denote Mahler's measure of F. Gonçalves' inequality asserts that

We obtain extremal majorants and minorants of exponential type for a class of even functions on R which includes log |x| and |x|α, where −1 < α < 1. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms,… (More)

We obtain extremal majorants and minorants of exponential type for a class of even functions on R which includes log |x| and |x|α, where −1 < α < 1. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms,… (More)