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Approximations in $$L^1$$ with convergent Fourier series
- Source :
- MATHEMATISCHE ZEITSCHRIFT
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
Abstract
- For a separable finite diffuse measure space $${\mathcal {M}}$$ M and an orthonormal basis $$\{\varphi _n\}$$ { φ n } of $$L^2({\mathcal {M}})$$ L 2 ( M ) consisting of bounded functions $$\varphi _n\in L^\infty ({\mathcal {M}})$$ φ n ∈ L ∞ ( M ) , we find a measurable subset $$E\subset {\mathcal {M}}$$ E ⊂ M of arbitrarily small complement $$|{\mathcal {M}}{\setminus } E| | M \ E | < ϵ , such that every measurable function $$f\in L^1({\mathcal {M}})$$ f ∈ L 1 ( M ) has an approximant $$g\in L^1({\mathcal {M}})$$ g ∈ L 1 ( M ) with $$g=f$$ g = f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of $${\mathcal {M}}=G/H$$ 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.
- Subjects :
- Measurable function
General Mathematics
010102 general mathematics
Hausdorff space
Second-countable space
Space (mathematics)
01 natural sciences
Functional Analysis (math.FA)
Separable space
Mathematics - Functional Analysis
010101 applied mathematics
Combinatorics
Mathematics and Statistics
Bounded function
41A99, 43A15, 43A50, 43A85, 46E30
Homogeneous space
FOS: Mathematics
Orthonormal basis
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 14321823 and 00255874
- Volume :
- 299
- Database :
- OpenAIRE
- Journal :
- Mathematische Zeitschrift
- Accession number :
- edsair.doi.dedup.....36b5b48ea912b88cf9d88b870ff738b6