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Atiyah–Singer index theorem - An example on the circle | | Atiyah–Singer index theorem - An example on the circle: Encyclopedia II - Atiyah–Singer index theorem - An example on the circle | | We start by considering complex-valued functions on the circle that are "square integrable"(i.e., elements of L2) and have no Fourier coefficients with negative phase (equivalently, that extend to be holomorphic in the disk). We want an operator that takes one of these functions and gives us back another one. Given a continuous function f, let Tf be the operator that multiplies by f and then kills off the negative Fourier coefficients of the resulting function. For ...
See also:Atiyah–Singer index theorem, Atiyah–Singer index theorem - An example on the circle, Atiyah–Singer index theorem - More formal statement, Atiyah–Singer index theorem - History, Atiyah–Singer index theorem - Proof techniques, Atiyah–Singer index theorem - Further developments | | | Atiyah–Singer index theorem, Atiyah–Singer index theorem - An example on the circle, Atiyah–Singer index theorem - Further developments, Atiyah–Singer index theorem - History, Atiyah–Singer index theorem - More formal statement, Atiyah–Singer index theorem - Proof techniques | | |
| | | Atiyah–Singer index theorem: Encyclopedia II - Atiyah–Singer index theorem - An example on the circle
Atiyah–Singer index theorem - An example on the circle
We start by considering complex-valued functions on the circle that are "square integrable"(i.e., elements of L2) and have no Fourier coefficients with negative phase (equivalently, that extend to be holomorphic in the disk). We want an operator that takes one of these functions and gives us back another one. Given a continuous function f, let Tf be the operator that multiplies by f and then kills off the negative Fourier coefficients of the resulting function. For these operators, f is known as the symbol of Tf.
If f never takes the value zero on the circle, then Tf has a well-defined index, which can be shown to equal the number of times that f goes around zero (in the complex plane) when its argument goes around the circle once (i.e., the winding number of f).
Other related archives1950s, 1960, 1962, 1965, 2004, Abel Prize, Alain Connes, Annals of Mathematics, Atiyah, Atiyah–Bott fixed-point theorem, Cauchy-Riemann operators in several variables, Dirac equation, Fourier coefficients, Fredholm operator, Grothendieck-Riemann-Roch theorem, Hirzebruch-Riemann-Roch theorem, Hodge theory, Isadore Singer, Israel Gelfand, K-theory, L2, Laplacian, Lefschetz fixed-point theorem, Lie groups, M. C. Escher, Michael Atiyah, Norwegian Academy of Science and Letters, Raoul Bott, Riemann-Roch theorem, Singer, analysis, cobordism theory, cokernel, compact, cotangent bundle, differential operator, differential operators, elliptic differential operators, functional analysis, functoriality, harmonic functions, heat equation, holomorphic, homomorphism, inclusion map, kernel, manifold, manifolds, mathematics, microeconomics, pseudo-differential operators, quadratic form, spheres, submanifold, theoretical physics, topological invariants, topology, vector bundle, winding number, zero section
 Adapted from the Wikipedia article "An example on the circle", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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