In this paper, we propose a method for designing a set of pulses whose spectrum is efficiently contained in amplitude and bandwidth. Because these pulses are derived from and have shapes that are either equal or similar to the Hermite pulses, we name our proposed transmit pulses as spectrally efficient Hermite pulses. Given that the proposed set of pulses does not constitute an orthonormal one, we also propose a set of receive templates which permit orthonormal detection of the incoming signals at the receiver. The importance of our proposal is in the potential implementation of M-ary pulse shape modulation systems, for ultra wideband communications, with sets of pulses that are efficiently contained within a specific bandwidth and limited to a certain amplitude.