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3

You are not required to. Functions can be decomposed into a wide array of orthogonal basis functions, including the Bessel functions (in the Hankel transform) and the Legendre functions. The sine function just happens to be the overall simplest to deal with in the general case.

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You refer to Fourier Series. The brilliance of Fourier was to use sin to express a function.You know that you can create any vector from the sum of some unit vectors.Exactly the same think happens here. The number you multiply the unit vectors is the coefficients in F.S. To answer to your question of why we use sin and cos is that they have (mathematicly) ...

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If at every time $t$, $\phi(\mathbf{x},t)$ is a nice enough function that it has a Fourier transform, then $$\phi(\mathbf{x},t)=\int_{-\infty}^{\infty}\frac{d^{3}k}{(2\pi)^{3}}\widetilde{\phi}(\mathbf{k},t)e^{i\mathbf{k}\cdot\mathbf{x}},$$ where $\widetilde{\phi}(\mathbf{k},t)$ is just the Fourier coefficient at that time $t.$ But now you ask that the whole ...

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Right now, I'm using a book entitled " An Introduction to Wavelets through Linear Algebra" by Micheal Frazier. Published in 1999, it's still a pretty good book, and contains a nifty refresher to linear algebra as well.

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Are these transformations different to Lorentz and coordinate transformations when discussing symmetry? Yes and no. They have in common that we use them to look at the problem from another point of view. When one study wave phenomena it is very common (if not automatic) to use the Fourier transform (and sometimes the Laplace transform). For example the ...

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