The new originally capital general solution of determining the prototype filter function as the response that satisfies the specifications of all pole low-pass continual time filter functions of odd and even order is presented in this paper. The approximation problem of filter function was solved mathematically, most directly applying the summed Christoffel-Darboux formula for the orthogonal polynomials. The starting point in solving the approximation problem is the direct application of the Christoffel-Darboux formula for the initial set of continual Jacobi orthogonal polynomials in the finite interval [−1, +1] in full respect to the weighting function with two free real parameters. General solution for the monotonic and non-monotonic filter functions is obtained in a compact explicit form, which is shown to enable generation of the Jacobi filter functions in a simple way by choosing the values of the free real parameters. Moreover, the proposed solution with the same criterion of approximation is used to generate the appropriate best known classical approximation functions for particular specifications of free parameters: the Gegenbauer, Legendre and Chebyshev filter functions of the first and second kind as well. The approximation is shown to yield a very good compromise solution in view of the filter frequency characteristics (both magnitude and phase characteristics). The paper proposes new class filter functions with an excellent approach to ideal filter characteristic.