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Maximally flat (MAXFLAT) finite-impulse response digital filters, including the Hilbert transformers (HTs), have the smoothest magnitude responses among the available types of FIR digital filters. However, the transition bands of MAXFLAT filters are relatively wider, and this makes them unattractive in certain applications. We present new designs of even and odd length HTs by transforming a design of digital differentiators satisfying maximal linearity constraints at omega = pi/2. The resultant even and odd length HTs have highly smooth magnitude responses around omega = pi/2 and omega = (pi/4,3pi/4) respectively, and have relatively narrow transition bands compared to the existing MAXFLAT designs.