The problem of the continuation of positively definite functions $k(x)$ of the finite integral on all axis was examined in the works of M.G. Crein and Y.M. Berezanskyy. O.V. Lopotko using the method Y.M. Berezanskyy, the necessary and sufficient conditions of simple continuation of even positively definite functions at intervals at all axis, were obtained. In present article the sum of M.G. Crein and Y.M. Berezanskyy is considering from j-theory's position. The advantage of such approach is that j-theory gives the only universal method of solution the sums such as the problem of moments both discrete and continental. V.P. Potapov expressed a fruitful concession according to which all information of sum about of integral representation of functions is contained in some correlation, which is constructed according to the facts of sum, and so called 'Basic Matrix Inequality' (BMI). With the help of j-theory, using BMI the integral representations of functions of $P, K_{a} $ classes were integrated in the works of V.P. Potapov, I.V. Kovalyshyna.In this present work, using the j-theory the integral representation even positively definite functions $k(x)$ which are belong to $K_{b}$ class has been investigating. For this aim the 'associational' $w(z)$ function is connected with $k(x)$ function and the BMI is building. In the BMI we find the aspect of the integral representation for $k(x), w(z)$, then we are coming to find a solution of our sum. As the unification BMI outwardly doesn't differ from principal matrix inequalities of such well-known classic discrete interpolation sums such as Nevolinno-Picka, Caratheodora's problem, the moments and such continental sums as the M.G. Crein's sum about the integral representation of hermitian-positive functions, naturally, the possibility of using the V.P. Potapov's scheme for the construction of solution of BMI and our sum, is arised. Analysing BMI, firstly, we are answering the question: Does though one solution of BMI exist? Or does one representation of function $K_{b's}$ class exist? Answering these questions we have to consider two cases: the case of particular information of $k>0$ block and the case of not particular information $k>0$ block. In the first case we are directly calculating the solution of BMI. In the second case we are approximating our sum by discrete analogy. That's why we are proving that BMI always has a solution. Now, let's choose two essentially different solutions on the condition that the information block is $k>0$ and let's prove that BMI has multitude solutions, and its full totality has been describing by fractional-linear transformation with the help of Nevanlinovsky's not special pair. At last, we are proving the criterion of the significant integral representation of functions of $K_{b}$ class.

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