Hello,
The common definition of the non-negative definite functions is as follows:
Definition 1: A continuous complex-valued function $f(x)$ is called non-negative definite, if for any real numbers $x_1,\dots,x_m$ and complex numbers $\xi_1,\dots,\xi_m$, one has
$$
\sum_{k,j=1}^m f(x_k-x_j) \xi_k \bar{\xi}_j \ge 0\:.
$$
See wikipedia for example.
For my use, I need the following more general definition:
Definition 2: A continuous complex-valued function $f(x,y)$ is called non-negative definite, if for any real numbers $x_1,\dots,x_m$ and complex numbers $\xi_1,\dots,\xi_m$, one has
$$
\sum_{k,j=1}^m f(x_k,x_j) \xi_k \bar{\xi}_j \ge 0\:.
$$
For the first definition, there is a Characterization by Bochner's theorem, which is discussed in the post.
My question is, for the second definition, is there a characterization like Bochner's theorem? A necessary condition is that $f(x,y)$ is Hermitean in the sense that
$$
f(x,y) = \bar{f}(y,x)\:.
$$
Clearly, we need more conditions to guarantee that $f(x,y)$ is non-negative definite. Does any one know a way to determine whether a given function $f(x,y)$ is non-negative definite or not?
Thank you very much for any hints!
Anand
EDIT:
Thanks Mikael de la Salle for the useful link. After reading, I still have some problems.
As one might know, Definition 1 above can be extended to distributions. For example, a distribution $F$ is positive-definite, if for every $\psi\in C_c^\infty(R^d)$,
$$
\left(F,\psi *\widetilde{\psi}\right)\ge 0,
$$
where $\widetilde{\psi}(x)=\psi(-x)$. So it is natural to think that Definition 2 can also be extended to, say, $C_c^\infty(R^d)\otimes C_c^\infty(R^d)$, or $C_c^\infty(R^{2d})$. The condition in the book Kazhdan's property (T) that the kernel function should be continuous seems too restrictive. Are there some more general statements?
Second EDIT:
Here is a more explicit statement of the problem:
Theorem: Every translation-invariant positive-definite Hermitian bilinear functional $B(\phi,\psi)$ on $C_c^\infty(R^d)$ has the form
$$
B(\phi,\psi)= \int \hat{\phi}(\lambda) \overline{\hat{\psi}(\lambda)} d \mu(\lambda)
$$
where $\mu$ is some positive tempered measure and $\hat{\psi}$, $\hat{\phi}$ are the Fourier transforms, respectively, of $\psi$ and $\phi$.
See Gel'fand Volumn 4 P.169 Theorem 6. . My problem is
What happens if we remove the "translation--invariant" condition?
Thanks a lot!
Anand
Best Answer
Such a function is called function of positive type. For characterizations, see for example appendix C in the book Kazhdan's property (T) by Bekka, de la Harpe and Valette, available freely on Bachir Bekka's webpage.