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the following question is in second edition of "Fourier Series and Boundary Value Problems" by Churchill.

If a function $f$ has a representation $f(x) = \sum_{n = 1}^{\infty} A_n \ g_n(x)$ on a fundamental interval $(a,b)$ which the set $\{ g_n \}$ is orthogonal, but not normalized, $0 < || g_n || \neq 1$. Use the inner products to show formally that $A_n = \frac{(f,g_n)}{||g_n||^2}$.

$\textbf{My attempt:}$

$f(x) = \sum_{n = 1}^{\infty} A_n \ g_n(x)$

$\Longrightarrow \langle f,g_m \rangle = \langle \sum_{n = 1}^{\infty} A_n \ g_n(x),g_m(x) \rangle$

$\Longrightarrow \langle f,g_m \rangle = \sum_{n = 1}^{\infty} \langle A_n \ g_n(x),g_m(x) \rangle$

$\Longrightarrow \langle f,g_m \rangle = \sum_{n = 1}^{\infty} A_n \langle g_n(x),g_m(x) \rangle$

$\Longrightarrow \langle f,g_m \rangle = A_m$

I would like to know if my attempt is wrong or if the statement of the question is wrong. Thanks in advance!

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George
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1 Answers1

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Your conclusion is wrong. As the problem states, the $\{g_n\}$ are not orthonormal, just orthogonal. So $$\sum_{n=1}^\infty A_n \langle g_n(x), g_m(x)\rangle= A_m \|g_m\|^2.$$

Now you can conclude.

David Bowman
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