Can a signal be considered as a superposition of other common waveforms with various frequencies?

Besides the sine wave, the sawtooth wave, square wave, triangular wave and trapezoidal wave are common waveforms as well. We obtain an important relation between sine wave and these common waveforms, and successfully represent a signal as a superposition of common waveforms with various frequencies (or periods). This is a new and practical generalization of Fourier´s idea. According to Fourier analysis, a periodic signal can be considered as a superposition of sine waves with various frequencies. Now that sawtooth wave, square wave, triangular wave and trapezoidal wave are common waveforms as well, a natural question is: can a signal be considered as a superposition of other common waveforms with various frequencies? In order to answer this practical question, we study the relation between the sine wave and common waveforms, and present a new generalization of the Fourier series