Examples of generalized elementary solutions in linear inverse problems

One fragment from the theory of inverse problems for second-order abstract differential equations is presented in detail. We consider a spectral problem which related to a linear inverse problem with parameters in the final condition. It is shown that eigenvalues of the spectral problem is expressed through zeros of an elementary entire function. For certain combinations of the parameters, some zeros can be multiples with multiplicity two. Then it is possible to appear generalized elementary solutions for the inverse problem. Explicit formulas for such solutions are given. We provide specific examples of inverse problems where general ideas are put into the practice.