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#### TaylorM0192

##### New member

- Mar 6, 2012

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(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.