There are some related tricks that I used for functions with rotational symmetry and functions on domains with rotational symmetry. Here is where I write them down to remember.
Throughout this, I will use spheres and balls a lot. The
There are a lot of ways that we can show that the
1 Radial functions
A function
Put another way, consider the
Put another way again, let
These functions are important in, e.g. spherical distributions.
2 In dot-product kernels
Radial functions are connected to dot product kernels, in that dot product kernels have rotational symmetries in their arguments, i.e.
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3 Polynomial integrals on rotationally symmetric domains
I may have found this via John D. Cook, I cannot remember anymore.
Folland (2001) is an elementary introduction to integrating functions over a sphere. He explains: Let
Let
Uses: This is a good mnemonic for the volume of a ball or sphere when we can find by setting
Can we approximate the integral of an arbitrary radial function
OK, how about if we give up on radial functions per se and consider the class of even functions invariant to permutations of the axes,
This implies that the integrals over ball and sphere both go to zero as
But what is happening on the sphere?
4 Radial integrals on rotationally symmetric domains
Suppose we have an arbitrary function
On the ball it is slightly more complicated,
The rationale for this latter one is given in the next section, although I should probably clarify at some point. Anyway, this is essentially a univariate integral, you will note.
What can we say about these integrals? For one
I might come back to this and talk about something about the rate of growth of
An easy one: Suppose
Let us look at some bounding curves for various values of
Unintuitively (for me), the functions need to be more tightly controlled to keep that integral bounded in higher dimensions. Ultimately it approaches a constant
5 Generating random points on balls and spheres
How to generate uniformly random points on n-spheres and in n-balls lists a few methods.
Of use to me is the Barthe et al. (2005) method, (which can be generalised to balls that are based on arbitrary
From that same page we learn that for
6 Directional statistics
Apparently a whole field? See Pewsey and García-Portugués (2020).
7 Random projections
Closely related. See low-dimensional projections.
8 Transforms
How to work with radial functions.
8.1 Hankel transforms
A classic transform for dealing with general radial functions:
8.2 Integration algebra
A weird rabbit hole I fell down; it concerns a cute algebra over radial function integrals. Or at least, over nearly integrals and nearly derivatives. It turns out to be not that useful for the kinds of problems I face, which are computational. You can prove some cool things this way, and maybe with the right structure you could even compute things.
The rabbit hole is Robert Schaback and Wu (1996). They handle the multivariate Fourier transforms and convolutions of radial functions through univariate integrals, which we think of as a kind of warped Hankel transform. This is a good trick if it works, because this special case is relevant to, e.g. isotropic stationary kernels. They tweak the definition of the radial functions. Specifically, they call function
Robert Schaback and Wu (1996) is one of those articles where the notation is occasionally ambiguous and it would have been useful to mark which variables are vectors and which scalars, and overloading of definitions. Also, they recycle function names: watch out for
Now if
So, in
Comparing it with the Hankel transform
With this convention, and the symmetry of radial functions, we get
Let
But what is this operator
If something can be made to come out nicely with respect to this integral operator
We have a sweet algebra over these
We have fixed points
We can use these formulae to calculate multidimensional radial Fourier transforms, in principle. With
We have some tools for convolving multivariate radial functions via their univariate representations. Consider the convolution operator on radial functions
For dimensions
For
Note that the operators
Now, how do we solve PDEs this way? Starting with some test function
For compactly supported functions, we proceed as follows: We now take the characteristic function
9 𝓁₁
What is the rotation equivalent for the