Stechkin's lemma

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 $$0 < p < q < \infty$$ and let $$I$$ be a countable index set. Let $$(a_{i})_{i \in I}$$ be any sequence indexed by $$I$$, and for $$N \in \mathbb{N}$$ let $$I_{N} \subset I$$ be the indices of the $$N$$ largest terms of the sequence $$(a_{i})_{i \in I}$$ in absolute value. Then


 * $$\left( \sum_{i \in I \setminus I_{N}} | a_{i} |^{q} \right)^{1/q} \leq \left( \sum_{i \in I} | a_{i} |^{p} \right)^{1/p} \frac{1}{N^{r}}$$

where


 * $$r = \frac{1}{p} - \frac{1}{q} > 0$$.

Thus, Stechkin's lemma controls the ℓq norm of the tail of the sequence $$(a_{i})_{i \in I}$$ (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.

Proof of the lemma
W.l.o.g. we assume that the sequence $$(a_{i})_{i \in I}$$ is sorted by $$|a_{i+1}| \leq |a_{i}|, i \in I$$ and we set $$I= \mathbb{N}$$ for notation.

First, we reformulate the statement of the lemma to
 * $$\left( \frac{1}{N} \sum_{i \in I \setminus I_{N}} | a_{i} |^{q} \right)^{1/q} \leq \left( \frac{1}{N} \sum_{j \in I} | a_{j} |^{p} \right)^{1/p}. $$

Now, we notice that for $$d \in \mathbb{N}$$
 * $$ |a_i| \leq |a_{dN}|, \quad \text{for} \quad  i=dN+1, \dots, (d+1)N;  $$
 * $$ |a_{dN}| \leq |a_j|, \quad \text{for} \quad  j=(d-1)N+1, \dots, dN;  $$

Using this, we can estimate
 * $$ \left( \frac{1}{N} \sum_{i \in I \setminus I_{N}} | a_{i} |^{q} \right)^{1/q} \leq \left( \frac{1}{N} \sum_{d \in \mathbb{N}} N | a_{dN} |^{q} \right)^{1/q} = \left( \sum_{d \in \mathbb{N}} | a_{dN} |^{q} \right)^{1/q} $$

as well as
 * $$ \left( \sum_{d \in \mathbb{N}} | a_{dN} |^{p} \right)^{1/p} = \left( \frac{1}{N} \sum_{d \in \mathbb{N}} N | a_{dN} |^{p} \right)^{1/p} \leq \left( \frac{1}{N} \sum_{i \in I \setminus I_{N}} | a_{i} |^{p} \right)^{1/p}. $$

Also, we get by $ℓ^{p}$ norm equivalence:
 * $$ \left( \sum_{d \in \mathbb{N}} | a_{dN} |^{q} \right)^{1/q} \leq \left( \sum_{d \in \mathbb{N}} | a_{dN} |^{p} \right)^{1/p}. $$

Putting all these ingredients together completes the proof.