Large peak to average power ratios (PAPRs) are problematic for communication systems. One possible approach to control the PAPR is the tone reservation method. We analyze the tone reservation method for general complete orthonormal systems, and consider two solvability concepts: strong solvability and weak solvability. Strong solvability requires a rather strong control of the peak value of the transmit signal by the energy of the information signal, and thus might be to restrictive for practical applications. Therefore, the concept of weak solvability was introduced, which only requires the boundedness of the transmit signal. In this paper we prove that weak solvability and strong solvability are equivalent for arbitrary complete orthonormal systems.