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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
2 See research was published in his Th´orie analytique de la chaleur in 1822.
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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- Winter '14