User:Robinh/Sandbox

Hi. I'm reading a maths paper that discusses positive definite functions and am having difficulty interpreting the notation. Here is the quote:


 * $$ K(x,y)= f(||x-y||), x\in X$$

where $$||\cdot ||$$ is again the Euclidean norm on $${\mathbb R}^n$$ but $$ X=S^{n-1}$$ is the unit sphere in $${\mathbb R}^n, n\geq 2$$. As there is a simple 1-1 correspondence between $$||x-y||$$ and the standard inner product $$$$ for all $$x,y\in S^{n-1}$$, it is more convenient to consider $$K(x,y)=g(< x,y> )$$. Schoenberg proved that, for $$g\in C\left[0,1\right]$$, blah blah blah.

OK, three questions:


 * Is the "simple 1-1 correspondence" just $$\arccos (x\cdot y)$$?
 * What is $$C[0,1]$$?