Theory of self-reproducing kernel and dispersion inequalities

The theory of the self‐reproducing kernel by Aronszajn has been investigated for a Hilbert space with norm $${\mathrm{\Vert f\Vert }}^2\text{=(1/2π)}{\displaystyle {\int }_0^{\text{2π}}}{\mathrm{d(\theta |f(e}}^{\mathrm{i\hspace{0.17em}\theta }}{\mathrm{)|}}^2\text{+(1/π)}{\displaystyle {\int }_{\text{−1}}^1}{\mathrm{dx\hspace{0.17em}\lambda (x)|f(x)|}}^2$$ , where f(z) is a H² function and λ(x) is a nonnegative summable function. The self‐reproducing kernel of this space satisfies an integral equation. The dispersion inequalities for various problems in the high energy physics can be treated in unified and generalized manner by this theory.