Can this book be called a textbook? To a certain extent the answer is yes, but not exactly. It is, so to say, a schematic textbook. More precisely, what is presented is a summary of my experience after reading textbooks, monographs and basic papers as well as working in mathematics. The key word is in the second part of the title:“a path”. This means that the main objective of these notes is very modest: to present a collection of basic, well-known and some lesser-known results that may serve as a background for future research in certain topics of harmonic analysis on the real line. Many of these results are also a necessary basis for multivariate extensions. The latter can be found in [34]: the present notes are an outgrowth of the first chapter of that book and much of the first part was generalized to the multivariate case in the second part of that book. Anticipating a natural question, I would like to say that the subject has by no means been exhausted, and there are enough open problems both in dimension one and, even more, in several dimensions. Honestly, the text before you is more an essay than a textbook, though it can be used as a skeleton for constructing certain special courses, and if not completely then for outlining the instructive matter.