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The concept of Fourier synthesis is heavily employed in both consumer electronic products and fundamental research. In the latter, pulse shaping is key to dynamically initialize, probe and manipulate the state of classical or quantum systems. In nuclear magnetic resonance, for instance, shaped pulses have a long-standing tradition and the underlying fundamental concepts have subsequently been successfully extended to optical frequencies and even to implement quantum gate operations. Transferring these paradigms to nanomechanical systems requires tailored nanomechanical waveforms. Here, we report on an additive Fourier synthesizer for nanomechanical waveforms based on monochromatic surface acoustic waves. As a proof of concept, we electrically synthesize four different elementary nanomechanical waveforms from a fundamental surface acoustic wave at $ f_1 \sim 150$ MHz using a superposition of up to three discrete harmonics $f_n$. We employ these shaped pulses to interact with an individual sensor quantum dot and detect their deliberately and temporally modulated strain component via the opto-mechanical quantum dot response. Importantly, and in contrast to the direct mechanical actuation by bulk piezoactuators, surface acoustic waves provide much higher frequencies (> 20 GHz) to resonantly drive mechanical motion. Thus, our technique uniquely allows coherent mechanical control of localized vibronic modes of optomechanical crystals, even in the quantum limit when cooled to the vibrational ground state., 18 pages - final manuscript and supporting material