Abstract
With the preceding explanations we have not yet said anything about the remainder, and therefore the convergence, of the Fourier series, whereupon I am going to speak now. I observe at first that, as far as I know, we cannot find, in the literature, an easy formula for the remainder of the finite trigonometric series that makes use of finite equidistant ordinates. Yet, it is this series that one uses the most in applications when one is dealing with periodic phenomena that are to be represented as functions of time. I mention here some disciplines, which make use of these means: Meteorology (representation of the change of temperature and of other meteorological phenomena as functions of time), sound analysis, theory of geomagnetism (dependence of the magnetic elements on the circles of latitude from the geographical longitude), and electrical engineering (change in the intensity of alternating currents over time). It is in fact a result coming from meteorological practice that suggests paying particular attention to estimating the remainder; I want to describe it here in some detail, because it fits the basic ideas of these lectures.
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Klein, F. (2016). IV. Further Discussion About the Trigonometric Representation of Functions. In: Elementary Mathematics from a Higher Standpoint. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49439-4_5
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DOI: https://doi.org/10.1007/978-3-662-49439-4_5
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