• Corpus ID: 252531405

Almost everywhere convergence questions of series of translates of non-negative functions

@inproceedings{Buczolich2022AlmostEC,
  title={Almost everywhere convergence questions of series of translates of non-negative functions},
  author={Zolt{\'a}n Buczolich},
  year={2022}
}
This survey paper is based on a talk given at the 44th Summer Symposium in Real Analysis in Paris. This line of research was initiated by a question of Haight and Weizs¨aker concerning almost everywhere convergence properties of series of the form P ∞ n =1 f ( nx ). A more general, additive version of this problem is the following: Suppose Λ is a discrete infinite set of nonnegative real numbers. We say that Λ is of type 1 if the series s ( x ) = P λ ∈ Λ f ( x + λ ) satisfies a zero-one law. This… 

Figures from this paper

References

SHOWING 1-10 OF 49 REFERENCES

On Series of Translates of Positive Functions. III

Suppose Λ is a discrete infinite set of nonnegative real numbers. We say that Λ is of type 1 if the series $$s(x) = \sum\nolimits_{\lambda \in \wedge } {f(x + \lambda )} $$s(x)=∑λ∈∧f(x+λ) satisfies a

On Series of Translates of Positive Functions

AbstractFor Λ, a discrete infinite set of nonnegative real numbers, and a nonnegative measurable function f: R → R+, consider $$C = C\left( {f,\Lambda } \right) = \left\{ {x:\sum {_{\lambda \in

On strong uniform distribution

Let a= (ai)i=1 be a strictly increasing sequence of natural numbers and let be a space of Lebesgue measurable functions defined on [0,1). Let {y} denote the fractional part of the real number y. We

On the convergence of Σ ∞ n = 1 f(nx) for measurable functions

Questions of Haight and of Weizsacker are answered in the following result. There exists a measurable function f: (0, + ∞) → {0,1} and two non-empty intervals I F I ∞ ⊂[½,1) such that Σ ∞ n = 1 f(nx)

Fractional parts of powers of large rational numbers

A linear set of infinite measure with no two points having integral ratio

It is not difficult to construct an unbounded set E on the positive real line such that, if x 1 , x 2 belong to E , then x 1 / x 2 is never equal to an integer. Our object is to show that it is

Convergence of ∑ c k f ( kx ) and the Lip α class

By Carleson’s theorem a trigonometric series ∑ ∞ k=1 ck cos 2πkx or ∑ ∞ k=1 ck sin 2πkx is a.e. convergent if ∞ ∑ k=1 c2k < ∞. (1) Gaposhkin generalized this result to series of the form ∞ ∑ k=1

ON SERIES OF DILATED FUNCTIONS

Given a periodic function $f$, we study the almost everywhere and norm convergence of series $\sum_{k=1}^\infty c_k f(kx)$. As the classical theory shows, the behavior of such series is determined by

On series Σckf(kx) and Khinchin’s conjecture

We prove the optimality of a criterion of Koksma (1953) in Khinchin’s conjecture on strong uniform distribution. This verifies a claim of Bourgain (1988) and leads also to a near optimal a.e.