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by jschveibinz
342 days ago
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An engineering, signal processing extension/perspective: An infinite sequence approximates a general function, as described in the article (see the slider bar example). In signal processing applications, functions can be considered (or forced) to be bandlimited so a much lower-order representation (i.e. vector) suffices: - The subspace of bandlimited functions is much smaller than the full L^2 space
- It has a countable orthonormal basis (e.g., shifted sinc functions)
- The function can be written as (with sinc functions): x(t) = \sum_{n=-\infty}^{\infty} f(nT) \cdot \text{sinc}\left( \frac{t - nT}{T} \right) - This is analogous to expressing a vector in a finite-dimensional subspace using a basis (e.g. sinc) Discrete-time signal processing is useful for comp-sci applications like audio, SDR, trading data, etc. |
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Basically, define a finite dimensional function space V_N of dimension N, in such a way that you could grow N to be arbitrarily large. Solve not the PDE (originally defined over an infinite dimensional function space V, such as H^1), but its discretization as if it dealt only with functions in V_N rather than all functions. The PDE is then simply a linear system, easy to solve. And you can prove, for instance in the case of elliptic PDEs, that the solution to the discrete problem is the orthogonal projection of the true solution of the PDE (in V) onto V_N (Céa's Lemma). Finally, you can produce estimations of the error this projection incurs as a function of N, and thus give theoretical guarantees that the algorithm converges to the true solution as N goes to infinity.
(N in this case is the number of vertices in a mesh that is used to define the basis functions of V_N)