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I fairly understand the heuristic approach to get to equation 5 in picture 2 from the fourier series in picture 1. Can the explanation be made a little more rigorous? Thank you.

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I've never seen a rigorous development of Fourier integral as a limit of Fourier series. (This does not necessarily mean one does not exist). The books I've read develop Fourier analysis on the circle and on the line separately: An introduction to harmonic analysis by Katznelson is one such book. –  user31373 Jun 16 '12 at 17:10

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Well, maybe I will not be mathematically absolutely strict (or even in some aspects correct), but still in some textbooks I read the following line of reasoning. First the authors start with the derivation of Fourier series for periodic signals (I am speaking about book, dealing with physical processes). Second they come to nonperiodic signals, defined (lets say for easiness) on some interval and periodically (with the period, equal to that interval) complement it to form a periodic one. Then they expand the obtained "periodic" signal in Fourier series and at last they take the limit with the period approaching infinity (to get in the limit an initial nonperiodic signal). And in the limiting form the series representation turns into integral one.

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