Functions on circles
نویسنده
چکیده
1. A confusing example 2. Natural function spaces 3. Topology on limits C∞(S1) 4. Distributions (generalized functions) 5. Invariant integration, periodicization 6. Hilbert space theory of Fourier series 7. Completeness in L(S) 8. Sobolev lemma, Sobolev imbedding 9. C∞(S1) = limC(S) = limHs(S) 10. Distributions, generalized functions, again 11. The confusing 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/Z, which inherits its 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/Z→ C× and are eigenfunctions for d/dx on R/Z. Finite or infinite linear combinations ∑
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