In the following we give the outline of this PhD thesis.In Chapter 1, we present the basic mathematical notations and concepts which are necessary to understand the work presented in this thesis. Section 1.3 in this chapter discussing the Jacobi ellipsoidal coordinate system might be of great interest for the reader who is not familiar with the spheroidal geometry. Not only the definitions of the harmonic functions, harmonic basis system and separable harmonic functions are given but also a detailed discussion on the necessary and sufficient conditions for a harmonic function to be separable in Jacobi ellipsoidal coordinates is given. Some recurrence formulae for the associated Legendre functions of the first kind on [-1, 1] andon0= u = b are also added in this chapter. Also, a formula for the series expansion of the associated Legendre functions of the first kind ~ Pn,j where n ¿ N0, j¿{0, ..., n} is calculated. Numerically stable recursive relations for the associated Legendre functions of the second kind are also briefly discussed. In the last section of this chapter, very interesting results on a reproducing kernel of a Hilbert space are recapitulated.