J. Marshall Ash

Very slowly varying functions

AbstractA real-valued functionf of a real variable is said to beϕ-slowly varying (ϕ-s.v.) if limx→∞ϕ(x) [f(x+α)−f(x)]=0 for each α. It is said to be uniformlyϕ-slowly varying (u.ϕ-s.v.) if limx→∞ supα ∈ Iϕ(x) |f(x+α)−f(x)|=0 for every bounded intervalI. It is supposed throughout that ϕ is positive and increasing. It is proved that ifϕ increases rapidly enough, then everyϕ-s.v. functionf must be u.ϕ-s.v. and must tend to a limit at ∞. Regardless of the rate of increase ofϕ, a measurable functionf must be u.ϕ-s.v. if it isϕ-s.v. Examples of pairs (ϕ,f) are given that illustrate the necessity for the requirements onϕ andf in these results.

Generalizations of the Riemann derivative

Introduction. In ?1 of this paper a derivative generalizing the Riemann derivative is considered. The existence of this derivative on a set is shown to imply the existence of the Peano derivative almost everywhere on the set. In ?2 the LI norm (1 ?p < oo) replaces the LX norm of ?1 and the same result is proved. A special case of this result is that the existence of the Riemann LI derivative implies the existence of the Peano LI derivative almost everywhere. In ?3 a generalization of smoothness is shown to imply smoothness almost everywhere. We consider only measurable sets of real numbers and real valued functions of a real variable.

On the nth Quantum Derivative

The nth quantum derivative Dnf (x) of the real-valued function f is defined for each real non-zero x as

Optimal numerical differentiation using three function evaluations

An optimal choice ofu for approximating thedth derivative,d=1,2, of a real valued function of a real variable by a difference quotient of the formh−d∑i=1nwif(x+uih) is given. Ifd=1 andn is odd, or ifd=2 andn is even, this choiceu turns out to be surprisingly asymmetric.

Some spherical uniqueness theorems for multiple trigonometric series

We prove that if a multiple trigonometric series is spherically Abel summable everywhere to an everywhere flnite functionf(x) which is bounded below by an integrable function, then the series is the Fourier series of f(x )i f the coe‐cients of the multiple trigonometric series satisfy a mild growth condition. As a consequence, we show that if a multiple trigonometric series is spherically convergent everywhere to an everywhere flnite integrable function f(x), then the series is the Fourier series of f(x). We also show that a singleton is a set of uniqueness. These results are generalizations of a recent theorem of J. Bourgain and some results of V. Shapiro.

Lp norm local estimates for exponential sums

Abstract We prove that for every real number p>1 there is an explicitly calculated constant Kp>0 such that, for any arc J of the one-dimensional torus T=R/Z with |J|>0, no matter how small, one can find some 1-periodic exponential sum f(x)= ∑ m k=1 exp (2iπN k x) so that ( ∫ J |f(x)| p d x) 1/p ≥K p ·( ∫ J |f(x)| p d x) 1/p (Lp norm “local concentration” on J). For p≥2 the result remains true (with another Kp) if the arc J is replaced by any set E⊂ T with |E|>0. The special case p=2 of our results had been studied by many authors about twenty years ago, but the general case p>1 had remained open. The limit case p=1 remains unsettled to this date.

Growth of

For any convex polyhedron W in ℝ m , p ∈ (1,∞), and N > 1, there are constants γ 1 (W, p, m) and γ 2 (W, p,m) such that Similar results hold for more general regions. These results are various special cases of the inequalities where φ(N) = N p(m-1)/2 when p ∈ (1,2m/m+1), φ(N) = N p(m―1)/2 log N when p = 2m/m+1, and φ(N) = N m(p-1) when p > 2m/m+1 where B is a bounded subset of ℝ m with non-empty interior.

L^{p}

We give an effective procedure for determining whether or not a series Σ N n=M r (n) telescopes when r (n) is a rational function with complex coefficients. We give new examples of series (*)Σ∞ n=1 r (n), where r (n) is a rational function with integer coefficients, that add up to a rational number. Generalizations of the Euler phi function and the Riemann zeta function are involved. We give an effective procedure for determining which numbers of the form (*) are rational. This procedure is conditional on 3 conjectures, which are shown to be equivalent to conjectures involving the linear independence over the rationals of certain sets of real numbers. For example, one of the conjectures is shown to be equivalent to the well-known conjecture that the set {ζ (s): s = 2,3,4,...} is linearly independent, where ζ (s) = Σn -s is the Riemann zeta function. Some series of the form Σ n s ( r √n, r √n+1,..., r √n+k), where s is a quotient of symmetric polynomials, are shown to be telescoping, as is Σ1/(n! + (n-1)!). Quantum versions of these examples are also given.

Lebesgue constants for convex polyhedra and other regions

The main result of this paper is a generalization of the property that, for smooth u, u xy = 0 implies (*) u(x, y) = a(x) + b(y). Any function having generalized unsymmetric mixed partial derivative identically zero is of the form (*). There is a function with generalized symmetric mixed partial derivative identically zero not of the form (*), but (*) does follow here with the additional assumption of continuity. These results connect to the theory of uniqueness for multiple trigbnometric series. For example, a double trigonometric series is the L 2 generalized symmetric mixed partial derivative of its formal (x, y)-integral

Telescoping, rational-valued series, and zeta functions

Various generalized derivatives are defined and related. Some of these are the Peano derivatives, the symmetric (Peano) derivatives, the symmetric Riemann derivatives, a generalized derivative from numerical analysis, the very large family of A derivatives, symmetric quantum derivatives, and quantum symmetric Riemann derivatives. Additionally, L p , 1 ≤ p < ∞ versions of many of these derivatives are considered. Relations between some of these derivatives are mentioned. Some counterexamples showing that other relations are not true are also given.