In mathematics – more specifically, in functional analysis and numerical analysis – Stechkin's lemma is a result about the âÂÂ<sup>q</sup> norm of the tail of a sequence, when the whole sequence is known to have finite âÂÂ<sup>p</sup> 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 .
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 âÂÂ<sup>q</sup> norm of the tail of the sequence (and hence the âÂÂ<sup>q</sup> norm of the difference between the sequence and its approximation using its largest terms) in terms of the âÂÂ<sup>p</sup> norm of the full sequence and a rate of decay.
W.l.o.g. we assume that the sequence is sorted by and we set for notation.
First, we reformulate the statement of the lemma to
Now, we notice that for
Using this, we can estimate
as well as
Also, we get by norm equivalence:
Putting all these ingredients together completes the proof.