Re: Smoothest function passing through n points

"XGR131" <XGR131@xxxxxxxxx> writes in article <1117032203.893719.167040@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> dated 25 May 2005 07:43:23 -0700:
>let points x_1,y_1 x_2,y_2 x_3,y_3 .. x_n,y_n be given
>let a=x_1 and b=x_n
>We are sseking the function f(x) , where f(x_1)=y1 , f(x_2)=y_2 ...
>f(x_n)=y_n and derivative of f exists on the interval (a,b)
>
>now if there is any other function g that also pass through all the
>given points then f'(x)=<g'(x) on the interval (a,b)

As somebody else pointed out, the average g'(x) on (a,b) must be equal to
the average f'(x) but I will assume you're looking for the one with the
lowest maximum absolute value. Then what you are describing is the function
you get by connecting the dots using straight line segments.

>But first : Does f always exist? if not what are the conditions for it
>to exist
>and for illustraion let points 0,0 1,1 2,4 be given , what is f if f
>passes through all the given points and also if g too passes through
>all the points 0,0 1,1, 2,4 then f'=<g'

The connect-the-dots function always exists and always has the least
variance in the first derivative, but most people wouldn't consider it
"smooth" because f''(x) has Dirac delta functions in it.

--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
.

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