Statistical limit of Lebesgue measurable functions at ∞ with applications in Fourier Analysis and Summability

This is basically a survey paper on recent results indicated in the title. A function s: [a, ∞) → ℂ, measurable in Lebesgue’s sense, where a ≥ 0, is said to have statistical limit l at ∞ if for every ɛ > 0, $$\mathop {\lim }\limits_{b \to \infty } (b - a)^{ - 1} |\{ \nu \in (a,b):|s(\nu ) - \ell | > \varepsilon \} | = 0$$ . We briefly summarize the main properties of this new concept of statistical limit at ∞. Then we demonstrate its applicability in Fourier Analysis. For example, the classical inversion formula involving the Fourier transform \(\hat s\) of a function s ∈ L1(ℝ) remains valid even in the general case when \(\hat s\not \in L^1 (\mathbb{R})\). We also present Tauberian conditions, under which the ordinary limit of a function s ∈ Lloc1[1,∞) follows from the existence of the statistical limit of its logarithmic mean at ∞.