Understanding Fourier Series

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Comparing Functions and vectors.

function f(x)

Finite dimensional Infinite dimensional

A vector can be written in the following different ways,


If the decomposition is along an orthogonal frame along the vectors and then the expression would be,

In general the dot product of two $n-$dimensional vectors and , can be written as,

It is useful to think of a real function over an interval
as a vector with infinite components. Here the argument serves
as an index and the function value as the vector component. Analogous to vector dot product, the dot product between two functions $f$ and $g$ defined over the same interval can be written as,

Using this definition of the dot product, one can show that the following functions
are orthogonal to each-other (mutual dot products are zero) on the interval
.

Thus in parallel with writing a vector in terms of it’s components, one can write any (finite, smooth and continuous on (I am not trying to be mathematically precise, the aim is to give an intuitive feel)) function in terms of
the above basis functions in the same manner,


Notice the similarity of the expression of a function in terms of it’s components and a vector in terms of it’s components. Hence decomposition of a function in its Fourier components is quite akin to decomposition of a vector in its Cartesian components.