Fourier Transform

In this chapter, we are going to consider notions and problems similar to those in the previous chapter but in the non-compact and, correspondingly, non-periodic setting, on the real axis ℝ or on its half-axis ℝ₊= [ 0, ∞). Here the harmonics form a continuum, and the natural function spaces decompose in a continuous way. Historically, Fourier, who, in 1811, replaced the series representation of a solution by an integral representation, and thereby initiated the study of Fourier integrals, was the pioneer of this subject. It is not by accident that the whole topic is frequently called Fourier Analysis for its discoverer.