Analysis and harmonic synthesis (Q2914225)

The author gives a short but exhaustive description of the history of Fourier analysis, starting from the Ancient Greek's philosophy, and by focussing on the milestones of this theory. Moreover, he explains the developments of this theory from an epistemological point of view thus singling out the deep impact of Fourier analysis on some fundamental branches of mathematics and applications. With simple but rigorous words he explains us the basic elements of the Fourier theory and shows us its influence on harmonic analysis and wavelet theory. Thus without any mathematical background we can easily get acquainted with the hard concepts of convergence of trigonometric series, Lebesgue functional spaces, and about the complex methods of signal analysis and computational harmonic analysis. It is shown that the Fourier transform and the singular integrals are useful tools in all sciences (applied and theoretical), as well as the Fast Fourier transform and wavelet transform which is based on them.NEWLINENEWLINE In the second part of this paper the author gives a detailed analysis not only from a historical point of view but also from the contemporary one. He shortly discusses not only the least square method but also the Candès-Romberg-Tao theorem on sampling. He talks and explains us both the Banach theory together with the Wiener algebra. In particular, the author gives an updated list of the fundamental steps of the Wiener algebra, focussing on the compressed sensing theory. In a few pages he teaches us how to reconstruct a signal from a deterministic and a probabilistic point of view.NEWLINENEWLINE For those who are familiar with Fourier methods this paper is an excellent synthesis of many pioneering fields of science, for the others scholars it is an attractive persuasion towards these exciting modern fields.NEWLINENEWLINEFor the entire collection see [Zbl 1230.00037].