2 results
Differential-free Characterisation of Smooth Mappings with Controlled Growth
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- Journal:
- Canadian Mathematical Bulletin / Volume 61 / Issue 3 / 01 September 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 628-636
- Print publication:
- 01 September 2018
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- Article
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In this paper we give some generalizations and improvements of the Pavlović result on the Holland–Walsh type characterization of the Bloch space of continuously differentiable (smooth) functions in the unit ball in
${{\text{R}}^{m}}$.
UNCERTAINTY PRINCIPLES CONNECTED WITH THE MÖBIUS INVERSION FORMULA
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 3 / December 2013
- Published online by Cambridge University Press:
- 25 January 2013, pp. 460-472
- Print publication:
- December 2013
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- Article
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Two arithmetic functions
$f$ and 
$g$ form a Möbius pair if 
$f(n)= {\mathop{\sum }\nolimits}_{d\mid n} g(d)$ for all natural numbers 
$n$. In that case, 
$g$ can be expressed in terms of 
$f$ by the familiar Möbius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members 
$f$ and 
$g$ of a Möbius pair are both finitely supported, then both functions vanish identically. Here we prove two significantly stronger versions of this uncertainty principle. A corollary is that in a nonzero Möbius pair, one cannot have both 
${\mathop{\sum }\nolimits}_{f(n)\not = 0} 1/ n\lt \infty $ and 
${\mathop{\sum }\nolimits}_{g(n)\not = 0} 1/ n\lt \infty $.
