1. Statement of the lemma

2. References

In mathematics — more specifically, in functional analysis and numerical analysis — Stechkin's lemma is a result about the ℓ^q norm of the tail of a sequence, when the whole sequence is known to have finite ℓ^p 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 ℓ^q norm of the tail of the sequence (and hence the ℓ^q norm of the difference between the sequence and its approximation using its largest terms) in terms of the ℓ^p norm of the full sequence and an rate of decay.

References

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