ak cos kt cos ktdt bj sin

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Unformatted text preview: ns which led to the theory of Fourier series were motivated by an attempt to understand heat flow.1 Nowadays, the notion of dividing a function into its components with respect to an appropriate “orthonormal basis of functions” is one of the key ideas of applied mathematics, useful not only as a tool for solving partial differential equations, as we will see in the next two chapters, but for many other purposes as well. For example, a black and white photograph could be represented by a function f (x, y ) of two variables, f (x, y ) representing the darkness at the point (x, y ). The photograph can be stored efficiently by determining the components of f (x, y ) with respect to a well-chosen “wavelet basis.” This idea is the key to image compression, which can be used to send pictures quickly over the internet.2 1 Fourier’s 2 See research was published in his Th´orie analytique de la chaleur in 1822. e St´phane Mallat, A wavelet tour of signal processing , Academic Press, Boston, 1998. e 62 We turn now to the ba...
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