If the filter has a symmetric unit sample response, then we know that

|E(ω)|≥δ for some frequencies on the dense set, then a new set of frequencies corresponding to the L+2 largest peaks of |E(ω)| are selected and computation is repeated. Since the new set of L+2 extremal frequencies are selected to increase in each iteration until it converges to the upper bound, this implies that when |E(ω)|≤δ for all frequencies on the dense set, the optimal solution has been found in terms of the polynomial H(ω).

If |E(ω)|≥δ for some frequencies on the dense set, then a new set of frequencies corresponding to the L+2 largest peaks of |E(ω)| are selected and computation is repeated.

In general, the filter designs that contain more than L+2 alternations or ripples are called as Extra ripple filters.

According to Alternation theorem, a necessary and sufficient condition for P(ω) to be unique, best weighted chebyshev approximation, is that the error function E(ω) must exhibit at least L+2 extremal frequencies in S.

If the filter has a anti-symmetric unit sample response, then we know that

The chebyshev approximation problem is viewed as an optimum design criterion on the sense that the weighted approximation error between the desired frequency response and the actual frequency response is spread evenly across the pass band and evenly across the stop band of the filter minimizing the maximum error. The resulting filter designs have ripples in both pass band and stop band.

In general, the filter designs that contain maximum number of alternations or ripples are called as maximal ripple filters.

The weighted approximation error is defined as E(ω) which is given as

The error function E(ω) alternates in sign between two successive extremal frequency, Hence the theorem is called as Alternative theorem.