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I heard that functional analysis can be applied to many problems in signal processing. I'm trying to explain to my engineer friend why it is useful, but I learnt it in a pure math setting. Can anyone give me some insight on how functional analysis can be applied in the domain of engineering?

user119264
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  • Well, on the one hand it gives insight into the treatment of PDEs, which model... hmm... every underlying phenomenon in technology? – GPerez Apr 05 '15 at 17:54
  • @GPerez Thank you for your response! But I was look for something more in-depth and intuitive. Telling my friend that won't convince him more than what I've already said. – user119264 Apr 05 '15 at 17:59
  • I know, which is why I left a comment and not an answer. Unfortunately I don't know enough about functional analysis to give the kind that kind of answer. – GPerez Apr 05 '15 at 18:02
  • Ask your friend if he/she ever used the finite element method. How do they know it's "an approximation" of a/the solution. Heck how does your friend even know that the pde can have a solution and what type of solution is it? Also maybe tell you friend that a better question to ask is: "How can I use X branch of maths to help me better understand the subtleties of the problem I am working on?" – user3371583 Apr 05 '15 at 18:02

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A wide range of physical processes are linear in a regime of small perturbations. Linearity is equivalently described by the principle of superposition: If you double a cause, you double the perturbed effect, and if you add two independent causes, then you can add their effects to obtain the full effect. That describes a linear operator. Differentiable functions are locally linear.

One definition of Functional Analysis is the study of linear spaces that are given a topology defined by neighborhoods in order to consider notions of closeness/stability/convergence. One studies such spaces through continuous linear functionals and their representations, and also studies operators (i.e., system response functions) on such spaces using these tools.

Notions of closeness allow one to apply successive approximations to converge to full descriptions/solutions of the phenomena being studied. Closeness is often defined in the sense of physical measurement, and this is a natural way to define topology, approximation, stability, and convergence.

Nonlinear effects may also be studied through nonlinear Functional Analysis, which brings to bear a whole different set of tools, many of which are global in nature.

Hence, Functional Analysis provides a general framework for studying general Physical systems in some regime, while addressing notions of superposition, measurement, error, stability, and approximation.

Disintegrating By Parts
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There are many applications of functional analysis. One that immediately springs to mind is Quantum Mechanics, here measurements can be interpreted as being operators on some Hilbert space. Hence operator theory (bounded and unbounded) and spectral theory are often used in advanced Quantum mechanics.

Even simple things as derivatives can be studied using functional analysis, notice that the map $\frac{\mathrm{d}}{\mathrm{d}x}:C^{\infty}(\mathbb{C})\rightarrow C^{\infty}(\mathbb{C}): f\mapsto \frac{\mathrm{d}f}{\mathrm{d}x}$ is indeed a linear map.

Another important application is the theory of Wavelets invented by Ingrid Daubechies, this theory is useful in signal analysis and can be used for compression of images (computer stuff).

Furthermore functional analysis also has it's role in multiple domains in mathematics itself. For example, functional analysis has it's use in classifying infinite groups. Notions such as amenability have become fundamental in many topics.

I am no expert in any of these applications, but functional analysis definitely is an extremely useful subject.

Mathematician 42
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