Jump to content

Stechkin's lemma

From Wikipedia, the free encyclopedia

In mathematics – more specifically, in functional analysis and numerical analysisStechkin'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

[edit]

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.

Proof of the lemma

[edit]

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 p norm equivalence:

Putting all these ingredients together completes the proof.

References

[edit]
  • 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.