Optimum criteria are derived for the feasibility of winding L-fold symmetric magnetic coils whose angular densities contain a finite number of parameters and have, at worst, jump discontinuities but are not prescribed beyond their initial harmonic coefficients. This is the first of three theoretical papers addressing specifically constraints: (1) that specify the angular domains where current reversals are forbidden; (2) that state whether the winding is continuous, discontinuous, or discrete there; and (3) that prescribe whether the winding density is monotonic and/or has stated lower or upper bounds. It is shown that the curve fitting problem for arbitrary L, under any of the above constraints, can be transformed into the corresponding problem of a generic density that is non-negative, two-fold symmetric, and cosinusoidal (i.e. a horizontal deflection winding), the essential ingredient being a one-to-one correspondence between the harmonic coefficients for the original density and those of the generic density. The criteria for feasibility of constructing this generic density are derived from first principles, and those for the primitive densities under study are then inferred by use of the above correspondences. Each such set of criteria consists of inequalities among the harmonic coefficients, the different sets being thus related to each other in a fundamental way. The derivation takes advantage of the theory of power moments by exploiting the linear connection between the harmonic coefficients and the power moments of a certain transform of the generic density.