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Classes of linear-phase finite-impulse response (FIR) filters with a piecewise-polynomial impulse response are proposed for the four types of linear-phase FIR filters. In addition, very efficient recursive structures to implement these filters in a straightforward and consistent manner are proposed. The desired impulse response is created by using a parallel connection of several filter branches. Only one branch has an impulse response of the full filter length, whereas the impulse responses are shorter for the remaining branches but the center is at the same location. The arithmetic complexity of these filters is proportional to the number of branches and the common polynomial order for each branch, rather than the actual filter order. In order to generate the overall piecewise-polynomial impulse response the polynomial coefficients are found, with the aid of linear programming, by optimizing the responses in the minimax sense, for both narrowband conventional filters and narrowband differentiators. The generation of these structures is based on the use of accumulators so that after using an accumulator, the resulting impulse response is divided into two parts. The first part follows the desired polynomial form, and the second part is what is left after the division, i.e., the nonpolynomial part. This same procedure can be used for all the following accumulators. Several examples are included, illustrating the benefits of the proposed filters, in terms of a reduced number of unknowns used in the optimization and the reduced number of multipliers required in the actual implementation.