The convergence in the \(L^ p\) metric of the Fourier-Laplace series
\textit{A. Bonami} and \textit{J.-L. Clerc} [Trans. Am. Math. Soc. 183, 223-263 (1973; Zbl 0278.43015)] proved that for all \(p \neq 2\) there exists a function \(f_ 0 \in L^ p(S^ 3)\) such that its Fourier-Laplace series does not converge in the metric of \(L^ p (S^ 3)\), where \(S^ 3\) is the unit sphere in \(R^ 3\). The present author proves the possibility of a correction of the function in order to eliminate this defect. Given \(\varepsilon>0\), there exists a measurable set \(G \subset S^ 3\) with \(\text{mes} G>4 \pi-\varepsilon\) such that for every function \(f(x) \in L^ p (G)\) for some \(1 \leq p<2\) one can find a function \(g(x) \in L(S^ 3)\) such that (i) \(g(x)=f(x)\) for \(x \in G\), (ii) the Fourier-Laplace series of \(g\) converges to \(f\) in the metric of \(L^ p (G)\) and to \(g\) in the metric of \(L(S^ 3 \backslash G)\).
scientific article; zbMATH DE number 484316