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](http://en.wikipedia.org/wiki/Positive-definite_function) 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](https://mathoverflow.net/questions/62949/positive-definite-functions-and-fourier-transforms). 


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.](http://books.google.com/books?id=dd35YQEACAAJ) . My problem is



```
   What happens if we remove the "translation--invariant" condition?

```

Thanks a lot!


Anand



Characterization of the non-negative definite functions $f(x,y)$ 
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](http://perso.univ-rennes1.fr/bachir.bekka/).


