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Approximations in \(L^1\) with convergent Fourier series. (English) Zbl 1478.42001

Summary: For a separable finite diffuse measure space \({\mathcal{M}}\) and an orthonormal basis \(\{\varphi_n\}\) of \(L^2({\mathcal{M}})\) consisting of bounded functions \(\varphi_n\in L^\infty ({\mathcal{M}})\), we find a measurable subset \(E\subset{\mathcal{M}}\) of arbitrarily small complement \(|{\mathcal{M}}{\setminus } E|<\epsilon \), such that every measurable function \(f\in L^1({\mathcal{M}})\) has an approximant \(g\in L^1({\mathcal{M}})\) with \(g=f\) on \(E\) and the Fourier series of \(g\) converges to \(g\), and a few further properties. The subset \(E\) is universal in the sense that it does not depend on the function \(f\) to be approximated. Further in the paper this result is adapted to the case of \({\mathcal{M}}=G/H\) being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of \(n\)-spheres with spherical harmonics is discussed. The construction of the subset \(E\) and approximant \(g\) is sketched briefly at the end of the paper.

MSC:

42A10 Trigonometric approximation
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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