Hardy notation

In complexity theory and mathematics, the Hardy notation, introduced by G. H. Hardy, is used for asymptotic comparison of functions, equivalently to Landau notation (also known as "Big O notation").

It is defined in terms of Landau notation by

:$flesssim\; g\; iff\; f\; in\; O(g)$ and $fll\; g\; iff\; fin\; o(g).$

(Similar symbols are used, like $preceq$ resp. $prec!!!!!!!!prec$.)

The Hardy notation is commonly abused similarly to the Landau notation.For example, where $h^2\; =\; O(h^3)$ is the shortened/abused Landau notation of $hmapsto\; h^2\; in\; O(hmapsto\; h^3)$,the expression $h^2\; lesssim\; h^3$ is the shortened/abused Hardy notation of $hmapsto\; h^2\; lesssim\; hmapsto\; h^3$.

For more examples and applications, see Landau notation and references therein.

ee also

* Big O notation: more explicit definitions, explanations, properties and related notations for real valued functions
* Taylor's theorem, maybe the most important application of Landau notation in mathematical analysis
* Asymptotic expansion: approximation of functions generalizing Taylor's formula.
* Nachbin's theorem: a precise way of bounding complex analytic functions so that the domain of convergence of integral transforms can be stated.