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Advanced Engineering Mathematics, Lecture 3.1: Fourier series and orthogonality
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Advanced Engineering Mathematics, Lecture 3.1: Fourier series and orthogonality.
In this lecture, we revisit how to define an inner product on the space of 2π-periodic functions so that the set consisting of all cos(nx) and sin(mx) functions are pairwise orthogonal. Analogously to how we decompose a vector in R^n into components, this allows us to decompose an arbitrary 2π-periodic function into an sum of sine and cosine waves, which we call its Fourier series. The so-called Fourier coefficients are simply the inner products of the function with these individual trigonometric functions.
In this lecture, we revisit how to define an inner product on the space of 2π-periodic functions so that the set consisting of all cos(nx) and sin(mx) functions are pairwise orthogonal. Analogously to how we decompose a vector in R^n into components, this allows us to decompose an arbitrary 2π-periodic function into an sum of sine and cosine waves, which we call its Fourier series. The so-called Fourier coefficients are simply the inner products of the function with these individual trigonometric functions.
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