Fourier Transform 2 | Trigonometric Polynomials [dark version]

Fourier series set up approximations of periodic functions by building them from sine and cosine waves, and the key move is to standardize everything to a 2π-periodic setting. Once a function repeats every 2 units, 4 units, or 6 units, the approximation strategy becomes clearer: represent the target as a linear combination of basic trigonometric building blocks that already repeat every 2π. That standardization matters because the familiar sine and cosine functions are naturally 2π-periodic, so stretching or compressing the original function to match that period simplifies the math without changing the essential periodic structure.

The construction starts by defining a function space: all real-valued functions on ℝ that satisfy f(x+2π)=f(x) for every real x. This set forms a real vector space, meaning sums and scalar multiples stay inside the same 2π-periodic world. From there, the transcript builds intuition with examples. Constant functions are trivially 2π-periodic. More interesting are functions like sin(x), which oscillate once per 2π, and sin(2x), which oscillates twice per 2π—an explicit way to control “frequency” by changing the multiplier inside the sine or cosine.

To approximate more complicated periodic functions, the approach collects an infinite family of simple waves into a linearly independent set U. The set includes all sine functions sin(kx) for k=1,2,3,… (odd functions that vanish at the origin) and all cosine functions cos(kx) plus the constant function (even functions that equal 1 at the origin). Linear independence is crucial: it guarantees that if a periodic function is expressed as a linear combination of these basis waves, the coefficients are uniquely determined. That uniqueness is the foundation for later approximation results.

With U in hand, the transcript defines what it calls a real trigonometric polynomial: a finite linear combination of sine and cosine terms. Such a polynomial has the form a0 plus a finite sum of cosine terms ak cos(kx) and a finite sum of sine terms bk sin(kx), with real coefficients. These polynomials are the approximating objects for 2π-periodic functions, analogous in spirit to Taylor polynomials but fundamentally different in their building blocks and approximation behavior.

Finally, the transcript introduces a complex version of trigonometric polynomials. Allowing complex coefficients and using the identity that links sine/cosine to complex exponentials makes the representation more compact: instead of writing separate sine and cosine sums, one can write a single sum over exponentials e^{ikx} (or equivalently e^{i k x} terms) with coefficients Ck. The complex formulation streamlines the algebra, setting up the next step—geometry in function space—where orthogonality becomes the organizing principle for Fourier analysis.