Frequency multiplier

This is technically not a multiplier - what you have is a very resonant tank that, when excited, wants to oscillate at it's natural frequency. Creating the decaying sinusoid observed at the collector

Yes this IS multiplier in one of most fundamental example.So we have first integral (an input wave) and we also have nonlinear element (transistor) whose derive many additional integrals from input wave by way of distortions and as those are created in the same circuit in one place therefore "mixing" whose is multiplication proces itself begin immediately betwen all derived new integrals creating their proportional products and next they all as one harmonic rich waveform leave transistor at collector. then an particular single harmonic that is product of multiplication of two lucky integrals that became available is ftransformed and filtered by resonant transformer composed from antiresonant LC circuit in this case. and as we know resonant circuit tends to couple power of nonresonant frequencies (making them attenuated) into a resonant frequency one so in ideal case it eats all unwanted harmonic frequencies and put it's power into one resonant one that we want when we calculate LC circuit parameters. Resonant circuit wil not oscillate at all if input signal not contain harmonics that lies in resonant frequency range so this is NOT a proces of waveform creation