Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Question: Is there a shift-invariant submultiplicative seminorm $||\cdot||$ of $\ell^\infty$ which satisfies the following property?

If $f:\mathbb{N}\rightarrow\mathbb{N}$ is an increasing function bounded below by some exponential function $cr^n$ for real numbers $c>0,r>1$, then the (bounded) sequence $||(\frac{f(0)}{f(1)},\frac{f(1)}{f(2)},\frac{f(2)}{f(3)},\dots)||<1$

More generally, if $f,g$ are two increasing functions on $\mathbb{N}$ such that $g(n)\le f(n)$ then $||(\frac{f(0)}{f(1)},\frac{f(1)}{f(2)},\frac{f(2)}{f(3)},\dots)||<||(\frac{g(0)}{g(1)},\frac{g(1)}{g(2)},\frac{g(2)}{g(3)},\dots)||$

(This, for example, excludes $\lim\sup$, if we take $f(n)=4^{\lfloor\frac{n}{2}\rfloor+1}$ and $g(n)=2^n$)


This comes up when estimating the asymptotic growth of certain "sparse" infinite binary trees. If there were such a norm, then the number of nodes at depth $n$ in such trees could not be bounded below by any such exponential function. By the way, I have no knowledge of functional analysis.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.