Let $K\colon [a,b]\times[a,b]\to\mathbb{R}$ be a continuous function (kernel), such that for any quadratically integrable function $x\colon[a,b]\to\mathbb{R}$ the following condition holds:

$$+\infty>\int_a^b\int_a^bK(s,r)x(s)x(r)\,ds\,dr \ge0$$

Question. How can one check this property, knowing only the function $K$ itself?

Motivation. If we define functions $K$ and $x$ over a finite set and replace integrals with the summation, then such $K$ will be a positive-semidefinite matrix. One can check if the matrix is positive-(semi)definite using the Sylvester's criterion. It is interesting for me if there exists something similar to this criterion in a continuous case.

  • $\begingroup$ Yes, it is called positive-(semi)definite kernel. See Positive-definite kernel, or Understanding positive definite kernel. $\endgroup$ – user539887 May 3 '18 at 11:24
  • $\begingroup$ @user539887, thank you very much! Although, I haven't found information about the existence of some direct analogue of Sylvester's criterion for positive-definiteness of a kernel. Could you please help me out with that? $\endgroup$ – Zeekless May 3 '18 at 20:11
  • $\begingroup$ Sorry, I am no expert on that. $\endgroup$ – user539887 May 4 '18 at 8:29

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