Function approximation, as a goal in itself or as an ingredient in scientific computing, typically relies on having a basis. However, in many cases of interest an obvious basis is not known or is not easily found. Even if it is, alternative representations may exist with much fewer degrees of freedom, perhaps by mimicking certain features of the solution into the “basis functions" such as known singularities or phases of oscillation. Unfortunately, such expert knowledge typically doesn’t match well with the mathematical properties of a basis: it leads instead to representations which are either incomplete or overcomplete. In turn, this makes a problem potentially unsolvable or ill-conditioned. We intend to show that overcomplete representations, in spite of inherent ill-conditioning, often work wonderfully well in numerical practice. We explore a theoretical foundation for this phenomenon, use it to devise ground rules for practitioners, and illustrate how the theory and its ramifications manifest themselves in a number of applications.
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