Functional analysis and signal processing have a strong bond and I am trying to understand how and when it all started. Technically, signal processing is heavily based on Fourier analysis, which helps model natural phenomena using signals. Eventually, functional analysis became somehow interconnected to signal processing and communications but it is not clear to me how this type of bond originated.

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    $\begingroup$ Somehow? Signals are modelled as functions, systems as operators. The bond is very natural. $\endgroup$ Oct 10 at 19:46
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    $\begingroup$ Fourier analysis was one of the early motivations for functional analysis, in Hilbert's 1904-1906 work on spectral theory of bounded operators especially. Development of wavelets and Gabor frames since 1940s was directly influenced by work on bases in Banach and Hilbert spaces, see Heil's review. Gradual entanglement of functional analysis with sampling theory since Weiss and Brown in 1960s is described in Butzer-Stens review. $\endgroup$
    – Conifold
    Oct 10 at 21:16

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