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  How to approximate integrals of a function by a function of integrals?

+ 3 like - 0 dislike
1112 views

Consider the following integral over the whole space

I=hij(x)Ai(x)Aj(x)d3x


where Ai are the components of a highly localized vector field, and hij is some well behaved Riemannian metric. One can imagine the magnitude of the components of the vector field to scale roughly with a normalized spatial Gaussian with a variance length σ. In the limit σ0, or when the vector field gets localized to almost a delta peak around a certain point x0, we can write an approximation to the integral as

Ihij(x0)Ai(x)d3xAj(y)d3y

I believe there are similar higher order corrections to this expression when the field becomes very slightly non-localized. These would involve terms such as (xjxj0)Aidx,(xjxj0)(xkxk0)Aidx,... (without having to refer to σ or anything similar) and the metric structure evaluated only at x0. However, I do not know how to methodically search for these corrections.

In other words, I am looking for an approximation scheme which would express the integral I in terms of functionals of Ai(x), its derivatives and the coordinates, while only referring to hij and its derivatives at some point x0 which is chosen so that it is very close to some centroid of the Ais.

How can I do that?

(I have posted a related version of this problem at Math Stackexchange.)

asked Mar 1, 2018 in Mathematics by Void (1,645 points) [ no revision ]
recategorized Mar 2, 2018 by Dilaton

It seems likely that given the set of moments of two functions with smooth Fourier transforms, you can recover their inner product.

The correction terms would have order O(σ). But by expanding the metric around zero one sees already that one gets at first order a term depending on the derivative of the metric at x0. Thus the metric at x0 alone is not sufficient to give better accuracy.

@ArnoldNeumaier I agree, and this is also mentioned in the question. Of course, when the dust settles, I believe the expansion should even be possible to state in terms of momenta of the vector field computed with respect to derivatives of the geodesic exponential around x0, and the expansion would then refer only to curvature and other covariant geometric tensors at x0

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