Rodney Nillsen

Difference spaces and invariant linear forms

General and preparatory results.- Multiplication and difference spaces on R n .- Applications to differential and singular integral operators.- Results for L p spaces on general groups.

Banach spaces of functions and distributions characterized by singular integrals involving the Fourier transform

The paper considers a class of Banach spaces ΔS(Fp(Rn)), for s > 0 and 1 ⩽ p < ∞, where Fp(Rn) is the space of tempered distributions whose Fourier transforms are in Lp(Rn). These spaces are defined as subspaces of Fp(Rn) whose elements can be expressed as finite sums of certain “generalized differences” of the form f − μ ∗ f, where fϵFp(Rn) and μ ϵ S, where S is a set of measures. It is shown that they are characterized by the behaviour of the Fourier transform near the origin. In particular, Δ1(L2(Rn)) is spanned by {f−δx ∗ f:fϵL2(Rn) and x ϵRn} and is a Hilbert space in the norm given by ‖g‖ =(∫Rn | g(x)|2 (1+|x|−2) dx)12 and Δ2(L2(Rn)) is spanned by {f − ((δx + δ−x)/2 ∗ f:fϵL2(Rn and xϵRn and is a Hilbert space in the norm given by ‖g‖ =(∫Rn) | g(x)|2 (1+|x|−2)2 dx)12. If Δ denotes the Laplacian, Δs(L2(Rn))= {f:fϵL2(Rn and (−Δ)−s/2 fϵL2(Rn)}, so Δs(L2(Rn)) can be thought of as a Sobolev-type space determined by integrals, rather than derivatives, belonging to L2(Rn). The results can also be regarded as establishing a connection between problems involving invariant forms on Fp(Rn) and a class of Sobolev-type spaces. In fact, a rather more general class of spaces than the above is considered, and some analogous results are presented for Zn and for compact connected abelian groups.

Nets of extreme Banach limits

Let N be the set of natural numbers and let o: N -* N be an injection having no periodic points. Let Mo be the set of a-invariant means on l.. When f E 4r, let do(f) = sup X(f), where the supremum is taken over all X E M0 . It is shown that when f E I., there is a sequence (X., 2 of extreme points of Mo which has no extreme weak* limit points and such that X,(f ) = d0(f ) for s = 2, 3, . As a consequence, the extreme points of M0 are not weak* compact.

Normal numbers without measure theory

Any number can be expanded to the base 10, leading to a sequence of digits between 0 and 9 corresponding to the number. Also, any number can be expanded to the base 2, leading to a sequence of digits, each one being either 0 or 1, corresponding to the number. It is result due to Emile Borel in 1904 that “almost all” numbers have the property that, when expanded to the base 2, each of the digits 0 and 1 appears with an asymptotic frequency of 1/2. That is, if we regard the sequence of digits in the expansion to the base 2 as a sequence of ‘heads’ and ‘tails’ resulting from a coin-tossing experiment, then, in the language of probability theory, the probability of getting heads (that is a 0) is 1/2, and the probability of getting tails (that is a 1) is also 1/2. Numbers with this property are called “simply normal numbers” to the base 2. Traditionally, the proof of Borel’s Theorem relies on a knowledge of measure theory, which generally lies outside the undergraduate curriculum. Here, a proof of Borel’s Theorem is presented which requires only an introductory knowledge of sequences and series, and a knowledge of how to integrate step functions on an interval. This makes it possible to discuss Borel’s theorem at the level of a first or second year course in mathematical analysis.

Adapted sets of measures and invariant functionals on

Let G be a locally compact group. If G is compact, let eo(G) denote the functions in L (G) having zero Haar integral. Let M1(G) denote the probability measures on G and let p 1(G) = M1(G) n L1(G) . If S C M1 (G), let A(Lp(G), S) denote the subspace of Lp(G) generated by functions of the form f u * f, f E Lp(G), u E 5. If G is compact, A(LP(G), S) C Lo(G). When G is compact, conditions are given on S which ensure that for some finite subset F of S, A(LP(G), F) = Lo(G) for all 1 < p < oo. The finite subset F will then have the property that every F-invariant linear functional on LP (G) is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if 1 < p < 00, conditions are given upon G, and upon subsets S of M1 (G) whose elements satisfy certain growth conditions, which ensure that Lp (G) has discontinuous, S-invariant linear functionals. The results are applied to show that for 1 < p < 00, LP(R) has an infinite, independent family of discontinuous translation invariant functionals which are not 3 1 (R)-invariant.

Irrational rotations motivate measurable sets

In 1914 Constantin Caratheodory gave his definition of a measurable set, a definition that is crucial in the general theory of integration. This is because a “primitive” notion of area or measure, on a smaller family of sets, can be extended to the larger family of measurable sets, and it is this larger family that has the necessary properties for a natural and complete theory of integration. This more general theory of integration is of enormous practical importance, for it leads to quite broad conditions under which basic operations on integrals are valid. However, Caratheodory’s definition itself remains mysterious, and has long been the subject of comment to this effect, because of the gap between the definition and the consequences which it has. For example, the existence of this gap was the basis upon which Imre Lakatos, in his work “Proofs and Refutations”, was critical of the type of didactic approach often taken in mathematical exposition. This paper examines critically the argument of Lakatos and responds to it by discussing a specific “problem situation” which leads to the Caratheodory definition. The problem is: calculate the outer measure of a subset of the unit circle that is invariant under an irrational rotation of the circle.

Standard deviation of recurrence times for piecewise linear transformations

This paper is concerned with dynamical systems of the form (X,f), where X is a bounded interval and f comes from a class of measure-preserving, piecewise linear transformations on X. If A ⊆ X is a Borel set and x ∈ A, the Poincare recurrence time of x relative to A is defined to be the minimum of {n : n ∈ ℕandfn(x) ∈ A}, if the minimum exists, and ∞ otherwise. The mean of the recurrence time is finite and is given by Kac’s recurrence formula. In general, the standard deviation of the recurrence times need not be finite but, for the systems considered here, a bound for the standard deviation is derived.

Sharp Results Concerning the Expression of Functions as Sums of Finite Differences

A square integrable function on the real line can be written as a finite sum of differences of order s if and only if the Fourier transform of the function vanishes at the origin at a rate comparable with |x|^{-s}. This enables us to see that the space of all such functions becomes a Hilbert space D^s(R) and that this space is the range of the derivative of order s on the Sobolev space of order s. It was proved by the author ( Journal of Functional Analysis 110 (1992), 73-95) that each function in D^s(R) is a sum of 2s+1 differences of order s. It was also known (Springer Lecture Notes in Mathematics vol. 1586, 1994) that in the case s=1, although every function in D^1 (R) is a sum of 3 first order differences, there are functions in D^1(R) that cannot be written as the sum of 2 first order differences – that is, 3 is sharp as an estimate of the minimum number of differences required. It was an open problem as to whether in the case of D^s(R), the number 2s+1 is sharp in this same sense. The main result in this paper shows that 2s+1 is sharp. That is, although every function in D^s (R) is a sum of 2s+1 differences of order s, there are functions in D^s(R) that cannot be written as the sum of 2s differences of order s. In fact, substantially stronger results are proved, and results in related spaces of distributions are discussed. The main techniques involve the Fourier transform and combinatorial methods in harmonic analysis, in particular an estimate of the minimum potential of n points scattered in the unit cube in R^s. This aspect generalizes work of Wolfgang Schmidt. Further ideas related to this work, the behaviour of the Fourier transform near the origin, and differential operators are on the author’s website at http://www.uow.edu.au/~nillsen

Topologically Invariant Linear Forms on Spaces of Abstract Distributions on a Locally Compact Abelian Group

A special case of the main result proved in this paper is the following. IfG is a locally compact, σ-compact, non-compact connected abelian group, thenL2(G)={f−ψ*f:f∈L2(G), ψ∈L1(G), ψ≥0 and ∫Gψ=1}. In this case, any topologically invariant linear form onL2(G) is 0.

Invariant subspaces of some function spaces on a locally compact group

Abstract Let G be a σ-compact and locally compact group. If f ϵ L ∞ ( G ) let U f be the closed subspace of L ∞ ( G ) generated by the left translations of f . Conditions are given which ensure that each function in U f may be expanded in an essentially unique way as an absolutely convergent series of translations of f . In this case U f contains subspaces which are isometrically isomorphic to l 1 . If G is metrizable and nondiscrete there is a continuum Γ in L ∞ ( G ) such that, for each f ϵ Γ , U f contains no non-zero continuous function, and for f , g ϵ Γ with f ≠ g , U f ∩ U g = {0}. If G is non-compact, metrizable, and non-discrete there is a continuum Γ of bounded continuous functions on G such that, for each f ϵ Γ , U f contains no non-zero left uniformly continuous function, and for f , g ϵ Γ with f ≠ g , U f ∩ U g = {0}. The subspaces U f above are translation invariant but are not convolution invariant.