Both the essence of homogeneous NMR line broadening as well as the fact that a short monochromatic radiofrequency pulse exhibits a polychromatic spectral bandwidth and is consequently associated with a broad off-resonance NMR excitation band are often rationalized in terms of a superficial and/or misguided use of the ''Uncertainty Principle.'' The problem is of quite fundamental nature, and stems from several deeper ''issues'' that reach beyond the world of NMR. To explore this topic, we need to go back to some basic concepts including the Fourier transform, time-frequency analysis, different versions of the ''uncertainty principle,'' and linear systems theory. My theme will be developed in four subsequent articles. First, by employing an approach that is unconventional in its thematic structure and applied nomenclature, I present an overview of some basic ideas (periodicity, basis functions, harmonic waves, density functions, etc.) that will then pave the way to discussing, in a somewhat avant-garde manner, the Fourier transform in Part II.