Gavin Brown

Properties of certain trigonometric series arising in numerical analysis

H,(O)= f KDLsinn6, n=l where 0 is real, tx > 0 and /I > - 1. Series of this kind have obvious connec- tions with the Riemann zeta function and with Bernoulli polynomials and are important for fractional integration [6, XII, Sect. 83; they have also arisen recently [2,4] in the study of certain numerical methods for bound- ary integral equations on smooth curves in the plane. For the applications in [2] we are interested firstly in knowing that G, has a unique zero in the interval (0, rc), so that a quadrature rule employing this zero is well defined, and for later applications we need to know that this zero increases monotonically with c(. In the case where o! 3 1 these questions were addressed in [l], as were the corresponding (trickier) questions about the partial sums. For O<b< 1 we write g’(e,p)=~{gol(e,B)~g,(e, 4)) me, D) = fvbut fi) f h,(e, -mi and for 0 < p d f define k: (4 B) = f + p=g: (0, PI. Of crucial importance to the applications in [2] are the facts (proved in Theorem 2 below) that h, + is nonnegative and h; non-positive, and that k’ are both positive except in special cases. In essence, these properties ensure that the numerical method treated in [2] is stable. The rate of convergence of the numerical methods of [2] is determined by the behaviour of g,‘(f?, /I) for small values of p; the salient properties follow from (2) below. In Section 5 of the paper we indicate some extensions of our results to other classical orthogonal polynomials. 2.

Some inequalities that arise in measure theory

The Lebesgue measure, λ ( E + F ), of the algebraic sum of two Borel sets, E, F of the classical “middle-thirds’ Cantor set on the circle can be estimated by evaluating the Cantor meaure, μ of the summands. For example log λ ( E + F ) exceeds a fixed scalar multiple of log μ ( E )+ log μ ( F ). Several numerical inequalities which are required to prove this and related results look tantalizingly simple and basic. Here we isolate them from the measure theory and present a common format and proof.

Ergodic measures are of weak product type

There are many results which discuss ergodicity in terms of approximate product properties. Here we work throughout with stochastically independent σ-algebras (ℰ, ℱ are independent of μ if μ (E ∩ F) = μ (E) μ(F) for all E ∊ ℰ, F ∊ ℱ ) , to obtain an exact product property characteristic of (a large class of) ergodic measures. The ideas are based on work of A. V. Skorokhod on admissible translates of probabilities on Hilbert space.

Coin tossing and sum sets

We consider the distribution μ of numbers whose binary digits are generated from infinitely many tosses of a biased coin. It is shown that, if E has positive μ measure, then some n -fold sum of E with itself must contain an interval. This contrasts with the known result that all convolution powers of μ are singular.

Positivity of Cotes numbers at more Jacobi abscissas

The positivity of certain finite sums of even ultraspherical polynomials has been identified by Askey as a specially interesting case of a more general problem concerning positivity of Cotes numbers at Jacobi abscissas. The authors establish several new inequalities of this type.

A decomposition theorem for numbers in which the summands have prescribed normality properties

If A is a set of integers, each exceeding unity, then every real number can be expressed as a sum of four numbers, each of which is non-normal with respect to every base belonging to A and is normal to every base which is not multiplicatively dependent on any element of A. This result is proved and generalized to allow noninteger bases.

On

Techniques from the authors previous paper [ BD1 ] aren applied to uniquen

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-measures and product measures

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An extension of the Fejér-Jackson inequality

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