Stechkin's lemma

In mathematics more specifically, in functional analysis and numerical analysis Stechkin's lemma is a result about the q norm of the tail of a sequence, when the whole sequence is known to have finite ℓp norm. Here, the term "tail" means those terms in the sequence that are not among the N largest terms, for an arbitrary natural number N. Stechkin's lemma is often useful when analysing best-N-term approximations to functions in a given basis of a function space. The result was originally proved by Stechkin in the case .

Statement of the lemma

Let and let be a countable index set. Let be any sequence indexed by , and for let be the indices of the largest terms of the sequence in absolute value. Then

where

.

Thus, Stechkin's lemma controls the ℓq norm of the tail of the sequence (and hence the ℓq norm of the difference between the sequence and its approximation using its largest terms) in terms of the ℓp norm of the full sequence and an rate of decay.

References

  • Schneider, Reinhold; Uschmajew, André (2014). "Approximation rates for the hierarchical tensor format in periodic Sobolev spaces". Journal of Complexity. 30 (2): 56–71. CiteSeerX 10.1.1.690.6952. doi:10.1016/j.jco.2013.10.001. ISSN 0885-064X. See Section 2.1 and Footnote 5.
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