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

Having the solution for P(ω), we can now compute the error function E(ω) from

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

We know that we can have at most L-1 local maxima and minima in the open interval 0<ω<π. In addition, ω=0 and π are also usually extrema. It is also maximum at ω for pass band and stop band frequencies. Thus the error function of a low pass filter has at most L+3 extremal frequencies.

FX denotes an array of maximum size 10 that specifies the weight function in each band.

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.

Let us consider the design of a low pass filter with the pass band edge frequency ωP and the ripple in the pass band is δ1, then from the general specifications of the chebyshev filter, in the pass band the filter frequency response should satisfy the condition

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

The weighting function on the approximation error allows to choose the relative size of the errors in the different frequency bands. In particular, it is convenient to normalize W(ω) to unity in the stop band and set W(ω)=δ2/δ1 in the pass band.

|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).