Re: Is there a name for this type of function?

pi2o...@xxxxxxxxxxxx wrote:
> [...] The thing that I wondered about was if there is a way of
> classifying functions which are not L2(R), but for which:
> Lim x-> infinity { [ Integral from -x to x of f ] / (2x) } < infinity
>
> Do functions which satisfy this property have a mathematical name?

Sort of. In general, a function f is in Lp(R) if the integral from
-infinity to infinity of |f|^p is finite.

Note there is a difference between the integral of f from -infinity to
+infinity and Lim x-> infinity [ Integral from -x to x of f ]. You have
to calculate the first type by taking the limit of the integral from a
to b of f, as a -> infinity, and b -> +infinity, independently of each
other.
Lim x-> infinity [ Integral from -x to x of f ] is sometimes called the
Principle Value of the integral from -infinity to +infinity of f.

--- Christopher Heckman

.

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