Integral representations of even positive-definite functions on nuclear spaces (Q1066046)

An even-positive function is an even function k given on a nuclear space \(\Phi\) and satisfying the following condition: \[ \sum^{n}_{i,j=1}\frac{k(t^{(i)}+t^{(j)})+k(t^{(i)}- t^{(j)})}{\quad 2}\xi_ i{\bar \xi}_ j\geq 0,\quad \forall n,\quad \forall t^{(1)},...,t^{(n)}\in \Phi,\quad \forall \xi_ 1,...,\xi_ n\in {\mathbb{C}}. \] Any such function, if it is continuous and bounded, may be uniquely represented in the form \[ k(t)=\int_{\Phi '}\cos (\lambda,t)d\rho (\lambda) \] where \(\Phi\) ' is the dual of \(\Phi\), and conversely. The spectral integral representations for operator functions of cosine type are obtained as applications.