This paper investigates the existence of elements of the best n-term approximation in infinite dimensional Hilbert spaces. The notion of uniform linear independence (ULI) for a dictionary is introduced. It is shown that if the dictionary used for approximation satisfies the Bessel inequality and has the ULI property, then for every element of the Hilbert space there exists an element of the best n-term approximation. It is also shown that if a dictionary does not satisfy the ULI property, then there exists an arbitrarily small compact perturbation of this dictionary for which the elements of the best n-term approximation need not exist. The obtained results are applied to frames.
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This research was partially supported by the Polish Ministry of Science and Higher Education Grant No. N N201 269335.
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Bechler, P. Existence of the best n-term approximants for structured dictionaries. Arch. Math. 99, 61–70 (2012). https://doi.org/10.1007/s00013-012-0406-y