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

Question

Consider the following integral over the whole space

$$I = \int \sqrt{h^{ij}(x) A_i(x) A_j(x)} d^3 x$$
where $A_i$ are the components of a highly localized vector field, and $h^{ij}$ 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 $\sigma$. In the limit $\sigma \to 0$, or when the vector field gets localized to almost a delta peak around a certain point $x_0$, we can write an approximation to the integral as

$$I \approx \sqrt{ h^{ij}(x_0) \int A_i (x) d^3 x \int A_j(y) d^3 y }$$

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 $\int (x^j - x^j_0) A_i dx, \int (x^j - x^j_0)(x^k - x^k_0) A_i dx, ...$ (without having to refer to $\sigma$ or anything similar) and the metric structure evaluated only at $x_0$. 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 $A_i(x)$, its derivatives and the coordinates, while only referring to $h^{ij}$ and its derivatives at some point $x_0$ which is chosen so that it is very close to some centroid of the $A_i$s.

How can I do that?

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

Comments

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

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The correction terms would have order $O(\sigma)$. 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 $x_0$. Thus the metric at $x_0$ alone is not sufficient to give better accuracy.

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@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 $x_0$, and the expansion would then refer only to curvature and other covariant geometric tensors at $x_0$.