When an image is compressed, it not only becomes slightly blurred: wherever there is strong contrast, its discontinuities are accentuated. This is known as the Gibbs phenomenon, and understanding it requires knowing how a wave can be decomposed into harmonics.
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The introduction of series of functions, followed by theorems establishing when continuity or differentiability is preserved in the limit, made it possible to construct objects with unusual, counterintuitive properties. Some famous mathematicians had tremendous fun with them!
Until the early 19th century, everything seemed continuous. Trigonometric series brought discontinuous functions onto the scene. They prompted Riemann to generalize the notion of the integral, led Weierstrass to clarify the notion of continuity, and spurred Darboux to study derivatives.
How do we go from discrete digital signals to the light and color waves emitted by our screens, and how can we reduce incoming data streams as much as possible without overly degrading the images? Converting signals to a spectral representation allows them to be compressed selectively to suit our visual system.
What could a quadratic polynomial possibly have in common with a vector in three-dimensional space? At first glance, nothing: they are different kinds of objects. Yet both have the same form—each is described by a triple of numbers. Better still, calculations with one correspond to calculations with the other!
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