Re: smoothing of functions through expectation operator
- From: David C. Ullrich <dullrich@xxxxxxxxxxx>
- Date: Fri, 16 Jan 2009 05:13:31 -0600
On Thu, 15 Jan 2009 17:14:43 -0800 (PST), simundza@xxxxxxxxx wrote:
Hi all,
For a continuous but not differentiable real valued function f: R -->
R and t distributed Normally with mean mu and variance sigma^2 ,
does the derivative of
E[ f ( x + t ) ]
with respect to x exist? Do I need to assume countable kink points?
That expected value need not exist. If you assume mild growth
conditions on f then it exists and yes, it defines an infinitely
differentiable function.
What area of math should I look in to learn more about this, and what
are the relevant theorems?
Real analysis, maybe to be more specific harmonic analysis.
The relevant theorem would be a result on differentability
of convolutions.
For example, if f , g, and g' are Lebesgue integrable then
the convolution f*g is differentiable, with derivative
given by the convolution f * (g').
Thanks very much,
Dan
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.
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