Abstract Linear Algebra 19 | Fourier Coefficients

Orthogonal projections in inner-product spaces turn “finding coefficients” into simple inner-product calculations—no linear systems required. For a finite-dimensional subspace U with an orthonormal basis B = {b1, …, bK}, any vector x in the ambient inner-product space V projects onto U as

x|U = Σ_{j=1..K} ⟨x, b_j⟩ b_j.

Those scalars ⟨x, b_j⟩ are the coefficients of the linear combination, and when the basis is orthonormal they come directly from inner products. The payoff is practical: even when x already lies in U, the same formula reproduces x exactly, and the method avoids solving a system of equations. In this setting, the linear combination is often called the Fourier expansion of x with respect to the orthonormal basis, and the scalars are called Fourier coefficients—terminology that later connects to classical Fourier analysis.

The lesson then becomes concrete through a function-space example built inside a finite-dimensional span of specific functions on [−π, π]. The subspace V is spanned by four functions: a normalized constant function (1/√2), cos(x), cos(2x), and sign(x). An inner product is defined using an integral:

⟨f, g⟩ = (1/π) ∫_{−π}^{π} f(x)g(x) dx.

With this inner product, the chosen functions behave like an orthonormal set. The constant and cos(x) are normalized so their inner products with themselves equal 1, and symmetry arguments determine orthogonality: cos(x) is even, sign(x) is odd, so their product is odd; integrating an odd function over a symmetric interval yields 0. Similar checks across the remaining pairs establish an orthonormal basis (denoted O and B in the transcript).

Using that orthonormal basis, the Fourier coefficients of a target function u(x) = sin(x^2) are computed by evaluating four inner products—one per basis element. The coefficient for the constant basis element comes out to 1/2. The coefficient for cos(x) is 0, because the integrand reduces to a form that evaluates to the same value at −π and π. The coefficient for cos(2x) is −1/2, coming from a longer integral that yields −π/2 before the normalization cancels. Finally, the coefficient for sign(x) is 0 again due to odd-function symmetry.

With only two nonzero coefficients, the Fourier expansion collapses to a simple identity:

sin(x^2) = (1/2)·(1/√2) + (−1/2)·cos(2x)

which the transcript rewrites in its final simplified form as

sin(x^2) = 1/2 − (1/2)cos(2x).

The broader takeaway is that once an orthonormal basis is in place, Fourier coefficients are just inner products, and the resulting expansion expresses the function as a linear combination of basis elements—an idea that generalizes beyond this toy example and connects to how orthonormal bases can be constructed (e.g., via the Gram–Schmidt procedure) in later material.