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Harmonic analysis is concern with generalization of Fourier series(of any function) while harmonic functions are solutions to the Laplacian.

Are these matters related somehow? I cant really see how, they seams to be rather unrelated subjects.

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Fourier series are decompositions of functions $f$ into eigenfunctions of the Laplacian $\nabla^2 g_i = \lambda_i g_i$. We write $f(\mathbf{x}) = \sum_i f_i g_i(\mathbf{x})$. For example, on a rectangular domain with periodic boundary conditions, $\nabla^2 \exp(i \mathbf{k} \cdot \mathbf{x}) = -|\mathbf{k}|^2$ and writing $f(\mathbf{x}) = \sum_\mathbf{k} f_\mathbf{k} \exp(i \mathbf{k} \cdot \mathbf{x})$ is the Fourier decomposition.

Harmonic functions are functions annihilated by the Laplacian (its kernel), i.e. solutions to $\nabla^2 h = 0$. This means they are the zero eigenfunctions, $g_i$ with $\lambda_i = 0$.

So harmonic functions are a special subset of the Fourier basis.


For a simple example of where this subset might arise, suppose that you study the heat equation $$\partial u/\partial t = \nabla^2 u \qquad \text{with initial conditions} \qquad u(t=0,\mathbf{x}) = \sum u_i(t=0) g_i(\mathbf{x})$$ where we have Fourier-decomposed the initial conditions. Then in Fourier space, the equation is $\dot{u}_i(t) = \lambda_i u_i(t)$ with solution $$u(t,\mathbf{x}) = \sum_i u_i(t=0) \exp(\lambda_i t) g_i(\mathbf{x})$$ Now assuming that the Laplacian has no positive eigenvalues (exercise: find out why for simple boundary conditions), almost all of the exponentials decay away, so at late times we have $$u(t,\mathbf{x}) \to \sum_{i'} u_{i'}(t=0) h_{i'}(\mathbf{x})$$ where we sum only over harmonic functions with $\lambda_i = 0$.

(In fact, in general there's only one harmonic function given suitable boundary conditions. Another thing to check!)

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  • $\begingroup$ when you say zero eigenfunction you mean eigenfunction related to the eigenvalue zero? $\endgroup$
    – user390930
    Oct 25, 2017 at 10:24
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    $\begingroup$ Yes, as I say, $\lambda_i = 0$. $\endgroup$ Oct 25, 2017 at 11:38
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    $\begingroup$ That's a really nice example, with the heat equation. :) But you need to update your profile, @Sharkos! $\endgroup$
    – Kenny Wong
    Oct 25, 2017 at 17:04

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