In mathematics – more specifically, in functional analysis and numerical analysis – Stechkin's lemma is a result about the ℓq 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 q=2.
Statement of the lemma
Let \( {\displaystyle 0<p<q<\infty } \) and let I be a countable index set. Let \( {\displaystyle (a_{i})_{i\in I}} \) be any sequence indexed by I, and for \( {\displaystyle N\in \mathbb {N} } \) let \( {\displaystyle I_{N}\subset I} \) be the indices of the N largest terms of the sequence \({\displaystyle (a_{i})_{i\in I}} \) in absolute value. Then
\( {\displaystyle \left(\sum _{i\in I\setminus I_{N}}|a_{i}|^{q}\right)^{1/q}\leq \left(\sum _{i\in I}|a_{i}|^{p}\right)^{1/p}{\frac {1}{N^{r}}}} \)
where
\( {\displaystyle r={\frac {1}{p}}-{\frac {1}{q}}>0}. \)
Thus, Stechkin's lemma controls the ℓq norm of the tail of the sequence \( {\displaystyle (a_{i})_{i\in I}} \) (and hence the ℓq norm of the difference between the sequence and its approximation using its N 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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