A Quick Question About Orthornomal Systems of Functions and Fourier Series

[TaylorM0192]

Two questions, actually. These just come from me doing a couple of problems on Fourier analysis from Rudin's text (I haven't actually taken a full course in the subject; we just spent about a week studying the topic in my real analysis class).

(1) The Weierstrass Approximation Theorem guarantees the existence of a sequence of trigonometric polynomials which converge to f uniformly if f is continuous (no restriction on the type of continuity; e.g. Lipschitz, Holder, C^n, etc.) and periodic. But we know there exists functions which are continuous with point-wise diverging Fourier series (and certainly ones which are not uniformly convergent), and these Fourier series are a indeed a special kind of trigonometric polynomial. To which sequence of trigonometric polynomials would this approximation theorem be referring if the Fourier series diverges? I realize the proof is non-constructive; is this a generally impossible question to answer? There are of course other approximation theorems too; for example the analogous result with polynomials on compact subsets and continuous functions being dense in L^2.

(2) Consider the Hilbert Space of square Lebesgue integrable functions L^2. When I think of Fourier series, I think of taking some arbitrary function in L^2, and then projecting its "components" onto the infinite sequence of orthonormal functions {e^inx}. Indeed, the Fourier coefficients are nothing more than the orthogonal projections onto these "basis" vectors. There in lies a subtlety I'm concerned about. If the Fourier series converges, does this in some sense mean that the set {e^inx} is a basis for L^2? Certainly this is impossible though, since there are many functions with divergent Fourier series. So does that mean functions with convergent Fourier series occupy a certain "subspace" of L^2 spanned by {e^inx}?

Furthermore, two inequalities have caused me some conceptual headaches: Plancheral's (identity) and Bessel's inequality. Both relate the l^2 and L^2 norms; i.e. the sequence of a function's projections onto an orthonormal sequence of functions is square summable, and therefore has a norm in l^2. Plancheral says that if these projections are onto the sequence {e^inx}, then the l^2 norm of that sequence of projections (i.e. sequence of Fourier coefficients) is equal to the L^2 norm of the function itself; whereas Bessel says that if the projections are onto an arbitrary orthonormal sequence of functions, the l^2 norm of that sequence of projections is less than or equal to the L^2 norm of the function which generated the sequence of projections. Btw, when I say sequence of projections, what I really mean are the scalar projections (i.e. the magnitudes of the orthogonal projections).

Assuming I have that straight, what makes the system of orthonormal functions {e^inx} so exceptionally special that equality holds in the Bessel inequality? Yet they have such glaring deficiencies, in particular, that they do not span the space L^2.

 

[TaylorM0192]

I think I answered my first question after proving Fejer's Theorem. Evidently, even if the Fourier series diverges at a point, its Cesaro means will still converge, and this is of course a trigonometric polynomial.

So that's interesting...the standard proof of Weierstrass' Trigonometric Approximation Theorem, which is nonconstructive and uses quite a bit of algebra (at least the one given in Rudin), yields a relatively simple construction (just Cesaro sum the Fourier coefficients!). The proof of this I thought was quite interesting too; basically, the Fourier coefficients are generated by convolutions with the Dirichlet Kernel, and the Cesaro Fourier coefficients are generated by convolutions with the Fejer Kernel (which itself is just a mean of Dirichlet Kernels). Then the proof that the Cesaro Fourier coefficients converges (uniformly in fact) to f is given by the approximation to identity lemma regarding kernels with certain properties; Dirichlet Kernels evidently do not have some of the required properties (for example, their absolute values diverge when integrated), hence convergence of Fourier series is not so simple.

Anyway, proving this theorem has given me even more out-there questions; but I think I'm just going to forget about them for now and hopefully things will come together in a clearer general picture after I take a full course in this subject, and in general continue taking analysis courses.

 

[math771]

1) The Weierstrass Approximation Theorem refers to a sequence of trigonometric polynomials that converge to a continuous and periodic function uniformly. This does not necessarily mean that the Fourier series of the function diverges, as there are functions with both convergent and divergent Fourier series that are continuous and periodic. The theorem guarantees the existence of a sequence of trigonometric polynomials that converge, but it does not specify which sequence it is referring to. It is generally impossible to determine which sequence of trigonometric polynomials the theorem is referring to, as the proof is non-constructive.

2) The set {e^inx} is not a basis for L^2, as there are functions with divergent Fourier series. However, functions with convergent Fourier series do occupy a subspace of L^2 spanned by {e^inx}. This subspace is known as the space of square integrable functions, which is a closed subspace of L^2.

Plancheral's and Bessel's inequalities both relate the l^2 and L^2 norms of a function and its projections onto an orthonormal sequence of functions. Plancheral's identity only holds for the specific case of the orthonormal sequence {e^inx}, which is why equality holds in this case. Bessel's inequality holds for any orthonormal sequence, but it may not necessarily hold as an equality.

The system of orthonormal functions {e^inx} is special because they are eigenfunctions of the Fourier transform. This means that when a function is projected onto this sequence, the resulting coefficients are the Fourier coefficients of that function. This relationship between the function and its Fourier coefficients is what allows for equality in Bessel's inequality. However, as mentioned before, this sequence does not span the entire space of L^2.