Series Expansion of Function over Complete Orthonormal Set

Theorem

Let $\map f x$ be a real function defined over the interval $\openint a b$.

Let $\map f x$ be able to be expressed in terms of a complete orthonormal set of real functions $S := \family {\map {\phi_i} x}_{i \mathop \in I}$ for some indexing set $I$:

$\map f x = \ds \sum_{i \mathop \in I} a_i \map {\phi_i} x$

Then the coefficients $\family {a_i}_{i \mathop \in I}$ can be determined as:

$\forall i \in I: a_i = \ds \int_a^b \map f x \map {\phi_i} x \rd x$

$\ds \map f x$

$=$

$\ds \sum_{i \mathop \in I} a_i \map {\phi_i} x$

$\ds \leadsto \ \ $

$\ds \map f x \map {\phi_n} x$

$=$

$\ds \sum_{i \mathop \in I} a_i \map {\phi_i} x \map {\phi_n} x$

for arbitrary $n \in I$

$\ds \leadsto \ \ $

$\ds \int_a^b \map f x \map {\phi_n} x \rd x$

$=$

$\ds \int_a^b \sum_{i \mathop \in I} a_i \map {\phi_i} x \map {\phi_n} x rd x$

$\ds $

$=$

$\ds \sum_{i \mathop \in I} a_i \int_a^b \map {\phi_i} x \map {\phi_n} x \rd x$

$\ds $

$=$

$\ds \sum_{i \mathop \in I} a_i \delta_{i n}$

$\ds $

$=$

$\ds a_n$

$\blacksquare$

1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 8$. Orthonormal Sets of Functions: $(3)$