Most of the integration methods in numerical analysis have this form: Sum(aif(xi)). Now I'm trying to find a formula like: f(Sum(aixi)). Indeed, one function evaluation of an average of mesh points instead of a weighted average of function values. It is clear that from mead point theorem, there exists a point that the integral will concentrate at it. Please share you ideas if interested.
The trouble is that you can only get second order accuracy, at best, with the suggested formula. So the conventional form Sum(aif(xi)) is the only viable alternative.
Actually, there is the mean value theorem of integral calculus that expresses the integral of f(x) for x in (a,b) as (b-a) f(a+theta (b-a)) for some theta in (0,1) for sufficiently smooth functions f. The difference to your idea is that in your formula you do not have the interval length (b-a).. For example. for f=1 identically, the integral should equal the length of the interval. (in case of infinite intervals of integration, one may introduce weight functions, for instance exp(-x^2) for integrals over the whole real line.)
For me the formula f(Sum(aixi)) looks strange. For positively definite f it reminds me something related to the expected value of x in domain of f (i.e. ) rather than an integral.
An interesting idea. There might be some restricted functions with this property. As an example say integral a*cosx for x= [ 0 pi/2 ] = a is a*cos(0)=a It seems this might be possible with eigen functions of selected differential equations.. just a conjecture..
Cheers.
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