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From: OwlHoot on 21 Jul 2010 06:28
This may seem a bit of weird question, and perhaps the answer
is, or should be, obvious.

But can anyone think of a sequence of functions, say { g_k(x) }
for k = 1, 2, .. such that the definite integral between two values,
at least one >> 1, of g_k(x) * x^r is negligible except for a few
values around r and including r (ideally _only_ r, if that is
attainable).

As I said, one of the limits of integration must be large, possibly
\inf. In fact 0 and \inf would be good limits.

Cheers

John Ramsden


From: James Waldby on 21 Jul 2010 12:20
On Wed, 21 Jul 2010 03:28:32 -0700, OwlHoot wrote:
....
> But can anyone think of a sequence of functions, say { g_k(x) } for k =
> 1, 2, .. such that the definite integral between two values, at least
> one >> 1, of g_k(x) * x^r is negligible except for a few values around r
> and including r (ideally _only_ r, if that is attainable).
>
> As I said, one of the limits of integration must be large, possibly
> \inf. In fact 0 and \inf would be good limits.

I don't quite see the overall picture (about quasi-orthogonalizing 1,
x, x^2, ... for large x), but have you considered functions like
(x-r) * e^(-(x-r)^(2*k)) ? (Ie, like u*exp(-u*u) when u=x-r and k=1)

--
jiw
From: Stephen Montgomery-Smith on 21 Jul 2010 15:07
OwlHoot wrote:
> This may seem a bit of weird question, and perhaps the answer
> is, or should be, obvious.
>
> But can anyone think of a sequence of functions, say { g_k(x) }
> for k = 1, 2, .. such that the definite integral between two values,
> at least one>> 1, of g_k(x) * x^r is negligible except for a few
> values around r and including r (ideally _only_ r, if that is
> attainable).
>
> As I said, one of the limits of integration must be large, possibly
> \inf. In fact 0 and \inf would be good limits.
>
> Cheers
>
> John Ramsden
>
>


I would look for functions of the form g_k(r) = p_k(x) e^(-x), where
p_k(x) is a polynomial of degree k. Use linear combinations of Laguerre
polynomials.

http://en.wikipedia.org/wiki/Laguerre_polynomials
From: Stephen Montgomery-Smith on 21 Jul 2010 18:19
Stephen Montgomery-Smith wrote:
> OwlHoot wrote:
>> This may seem a bit of weird question, and perhaps the answer
>> is, or should be, obvious.
>>
>> But can anyone think of a sequence of functions, say { g_k(x) }
>> for k = 1, 2, .. such that the definite integral between two values,
>> at least one>> 1, of g_k(x) * x^r is negligible except for a few
>> values around r and including r (ideally _only_ r, if that is
>> attainable).
>>
>> As I said, one of the limits of integration must be large, possibly
>> \inf. In fact 0 and \inf would be good limits.
>>
>> Cheers
>>
>> John Ramsden
>>
>>
>
>
> I would look for functions of the form g_k(r) = p_k(x) e^(-x), where
> p_k(x) is a polynomial of degree k. Use linear combinations of Laguerre
> polynomials.
>
> http://en.wikipedia.org/wiki/Laguerre_polynomials


You know, I didn't think this through. So maybe my suggestion is a load
of rubbish.
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