An everywhere divergent Fourier-Walsh series of the class 𝐿(𝑙𝑜𝑔⁺𝑙𝑜𝑔⁺𝐿)^{1-𝜖}

Moon, K. H.

doi:10.1090/S0002-9939-1975-0377406-8

An everywhere divergent Fourier-Walsh series of the class $L(\textrm {log}^{+}\textrm {log}^{+}L)^{1-\varepsilon }$
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by K. H. Moon PDF

Proc. Amer. Math. Soc. 50 (1975), 309-314 Request permission

Abstract:

Let $\Phi$ be a function satisfying (a) $\Phi (t) \geq 0$, convex and increasing; (b) $\Phi ({t^{1/2}})$ is a concave function of $t,0 \leq t < \infty$; and (c) $\Phi (t) = 0(t\log \log t)$ as $t \to \infty$. We construct a function in the class \[ \Phi (L) = \{ f \in L(0,1):\int _0^1 {\Phi (|f(x)|)dx < \infty } \} \]

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