Functions on circles : Fourier series
نویسنده
چکیده
1. Provocative example 2. Natural function spaces on the circle S = R/2πZ 3. Topology on C∞(S1) 4. Pointwise convergence of Fourier series 5. Distributions: generalized functions 6. Invariant integration, periodicization 7. Hilbert space theory of Fourier series 8. Levi-Sobolev inequality, Levi-Sobolev imbedding 9. C∞(S1) = limC(S) = limH(S) 10. Distributions, generalized functions, again 11. The provocative example explained 12. Appendix: products and limits of topological vector spaces 13. Appendix: Fréchet spaces and limits of Banach spaces 14. Appendix: Urysohn and density of C The simplest physical object with an interesting function theory is the circle, S = R/2πZ, which inherits group structure and translation-invariant differential operator d/dx from the real line R. The exponential functions x → e for n ∈ Z are group homomorphisms R/2πZ → C× and are eigenfunctions for d/dx on R/2πZ. Finite or infinite linear combinations ∑
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