2014/09/01 by Jorge L. deLyra, deLyra, Jorge L. · 1 citation
Mathematics · Engineering · #Algebraic and Geometric Analysis #Differential Equations and Boundary Problems #Material Science and Thermodynamics
paper · pdf · doi:10.48550/arxiv.1409.2582
A correspondence between arbitrary Fourier series and certain analytic\nfunctions on the unit disk of the complex plane is established. The expression\nof the Fourier coefficients is derived from the structure of complex analysis.\nThe orthogonality and completeness relations of the Fourier basis are derived\nin the same way. It is shown that the limiting function of any Fourier series\nis also the limit to the unit circle of an analytic function in the open unit\ndisk. An alternative way to recover the original real functions from the\nFourier coefficients, which works even when the Fourier series are divergent,\nis thus presented. The convergence issues are discussed up to a certain point.\nOther possible uses of the correspondence established are pointed out.\n