Advances in Harmonic Analysis
Overview
Most everyone watches television or listens to the radio, and many people use the Internet to obtain graphic images, sound, and video. How can this information be transmitted electronically without requiring an immense amount of time and space? The FBI has accumulated approximately 200 million fingerprint files. How can a current suspect's fingerprints be compared with the contents of these files? The answer is harmonic analysis, which allows us to compress these images and sounds into their main components before transmission, and reconstruct them on the receiving end. Harmonic analysis grew out of a study of the way a string vibrates, and continues to reinvent itself. Harmonic analysis bridges the gap between mathematical theory and engineering practice.
Background
In Recherches sur les cordes vibrantes (1747), the French mathematician Jean d'Alembert (1717-1783) described his study of the shape of a vibrating string fixed at both ends. He discovered that the height of a point on the string (measured from its beginning taut state) is a function of two variables: its horizontal position and the time since the string was plucked. This height satisfies a partial differential equation known as the wave equation; D'Alembert solved a special case of this equation.
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