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Introduction to Linear Algebra with Mathematica

Glossary

Preface

Definition of Fourier Series

 

Product of Fourier Series

Suppose we have two Fourier sum-functions f(x) and g(x). When these functions are defined by trigonometric series \eqref{EqFourier.5}, there is no suitable formula to determine the Fourier coefficients for their product (f g)(x). However, when these functions are expanded into complex Fourier series
\[ f(x) \,\sim\, \sum_{k=-\infty}^{\infty} \alpha_k e^{k{\bf j} \pi x/\ell} , \qquad g(x) \,\sim\, \sum_{k=-\infty}^{\infty} \beta_k e^{k{\bf j} \pi x/\ell} , \]
then their product is
\[ f(x)\,g(x) \,\sim\, \sum_{k=-\infty}^{\infty} \gamma_k e^{k{\bf j} \pi x/\ell} , \]
where
\[ \gamma_n = \sum_{j=-\infty}^{\infty} \alpha_j \beta_{n-j} \]
is known as the convolution of two Fourier series.
Theorem 17: If two trigonometric series S and T have coefficients o(1/n) and O(1/n), respectively, and converge at x0 to sums s and t, then the product ST converge at x0 to sum st.

The condition of the theorem is false if both S and T have coefficients O(1/n). For if

\[ S = T = \sum_{n\ge 1} \frac{\sin (nx)}{n} = \begin{cases} \phantom{-}\frac{\pi -x}{2} , & \ \mbox{for } 0< x < \pi , \\ 0, & \ \mbox{for } x=0, \\ - \frac{\pi +x}{2} , & \ \mbox{for } -\pi < x < 0 , \end{cases} \qquad \mbox{on interval } \ -\pi < x < \pi , \]
both series converge at x = 0 to sum 0, whilst ST, which is the Fourier series of
\[ S^2 = \left( \frac{\pi -x}{2} \right)^2 -\frac{\pi^2}{12} = \sum_{n\ge 1} \frac{\cos (nx)}{n^2} , \qquad 0 \le x < 2\pi , \]
converges at x = 0 to π²/6.

Note that little o notation is used to describe an upper bound that cannot be tight. In other words, we say that g(n) ∈ o(f(n)) when

\[ \lim_{n\to \infty} \frac{f(n)}{g(n)} = 0 . \]
When the ratio f(n)/g(n is bounded, we denote with big-oh notation.

 

 

 

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