For a wide class of nonlocal potentials the SCHRÖDINGER integro-differential equations for the radial waves may be reduced to integral equations containing L2-kernels. Therefore a number of analytic and functional-analytic properties of the resolvent of the radial wave equation may be deduced by means of the FREDHOLM method. Thence, the completeness of the physical radial wave functions, among other things, is proved by a known method. The positive-energy bound states occurring with nonlocal potentials are interpreted as resonances of vanishing width. We consider the non-imaginary “resonance poles” of the resolvent in a strip 0 > Im k > —α of the complex k-plane 1/α being the range of the potential. The structure of the principal parts is specified. The results will be used in the second part of this article.