20 intro to fourier series - 18.03 Lecture#20 Oct 23 2009...

Unformatted text preview: 18.03 Lecture #20 Oct. 23, 2009: notes Topic for today is an introduction to Fourier series. What Fourier series do is express a (complicated) general function as an (infinite) sum of simple functions. In order to talk about the philosophical underpinnings this idea, begin with a theory that you have seen before: Taylor series. So suppose g is a function of the real variable x . The theory of Taylor series is concerned with the following pieces of information about g : 0) the value g (0) of g at the origin; 1) the first derivative g ′ (0) of g at the origin; 2) the second derivative g ′′ (0) of g at the origin; and so on. Taylor series use a collection of very special functions: 0) the constant function 1, whose value at the origin is 1, and whose higher derivatives are all 0 at the origin; 1) the function x , whose value at the origin is 0, whose first derivative at the origin is 1, and whose higher derivatives at the origin are all 0; 2) the function x 2 / 2!, whose value and first deriviative at the origin are 0; whose second derivative at the origin is 1; and whose higher derivatives at the origin are all 0; and so on. The Taylor series of g is the infinite series g (0) · 1 + g ′ (0) · x + g ′′ (0) · x 2 / 2! + ··· (Taylor) If you compare the two lists above, you’ll see that the Taylor series of g is designed to have the same values and derivatives as g at the origin . Of course it often (but not always!) happens that the Taylor series of g converges, and that the sum is equal to g . You may like to think of Taylor series as a tool for computing g , but that isn’t always the best point of view. What is more fundamental is that replacing g by its Taylor series (or by a few terms of that series) focuses attention on the derivatives of g at the origin, throwing away extraneous complications....
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