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Representing a particular function with a sum of other functions

A Taylor expansion approximates the behavior of a suitably defined function, $f(x)$ in the neighborhood of a point, $x_o$ , with a bunch of functions, $p_i (x)$ , defined by the set of powers:

$$p_i \equiv \vec{p} = (x^0 , x^1 , \ldots , x^j , \ldots )$$ (17-5)

The polynomial that approximates the function is given by:

$$f(x) = \vec{A} \cdot \vec{p}$$ (17-6)

where the vector of coefficients is defined by:

$$A_i \equiv \vec{A} = (\frac{1}{0!} f(x_o) , \frac{1}{1!} \ensurem... ...{1}{j!} \ensuremath{\left.\frac{d^{j}{f}}{d{x}^{j}}\right\vert _{x_o}}, \ldots)$$ (17-7)

The idea of a vector of infinite length has not been formally introduced, but the idea that as the number of terms in the sum in Eq. 17-6 gets larger and larger, the approximation should converge to the function. In the limit of an infinite number of terms in the sum (or the vectors of infinite length) the series expansion will converge to $f(x)$ if it satisfies some technical continuity constraints.

However, for periodic functions, the domain over which the approximation is required is only one period of the periodic function--the rest of the function is taken care of by the definition of periodicity in the function.

Because the function is periodic, it makes sense to use functions that have the same period to approximate it. The simplest periodic functions are the trigonometric functions. If the period is $\lambda$ , any other periodic function with periods $\lambda/2$ , $\lambda/3$ , $\lambda/N$ , will also have period $\lambda$ . Using these "sub-periodic" trigonometric functions is the idea behind Fourier Series.