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Approximate and L p Peano derivatives of nonintegral order

J. Marshall Ash;  Hajrudin Fejzić — 2005

Studia Mathematica

Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. L p , 1 ≤ p ≤ ∞) sense at x ∈ ℝ m if there are numbers f α ( x ) , |α| ≤ n, such that f ( x + h ) - ∑ | α | ≤ n f α ( x ) h α / α ! is O ( h u ) in the approximate (resp. L p ) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or L p sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and f = g on Π....

Exponential sums with coefficients 0 or 1 and concentrated L p norms

B. Anderson;  J. M. Ash;  R. L. Jones;  D. G. Rider;  B. Saffari — 2007

Annales de l’institut Fourier

A sum of exponentials of the form f ( x ) = exp 2 π i N 1 x + exp 2 π i N 2 x + ⋯ + exp 2 π i N m x , where the N k are distinct integers is called an (because the convolution of f with itself is f ) or, simply, an . We show that for every p > 1 , and every set E of the torus 𝕋 = ℝ / ℤ with | E | > 0 , there are idempotents concentrated on E in the L p sense. More precisely, for each p > 1 , there is an constant C p > 0 so that for each E with | E | > 0 and ϵ > 0 one can find an idempotent f such that the ratio ∫ E | f | p / ∫ 𝕋 | f | p 1 / p is greater than C p - ϵ . This is in fact a lower bound result and, though optimal, it is close to the...

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