Harmonic Analysis

Harmonic analysis strives to understand the behavior of functions, using analytic tools. In particular, it works to understand oscillating functions and waveforms. It employs tools from functional analysis, linear algebra, and its cousin Fourier analysis. The connection between harmonic analysis and functional analysis takes place via Hilbert spaces.

Hilbert Space: An inner-product space is said to be a Hilbert space if it is complete with respect to the norm induced by the inner-product.

Banach Space: A Banach space is a complete normed space.

Note: Not every norm is induced by an inner-product. This means that the two definitions above describe different spaces. To say that a space is complete means that every Cauchy sequence converges, and that value lies in the space.

At an elementary level, Fourier analysis tries to represent a function $f$ as a series of sines and cosines: \[ \frac{1}{2}a_0 + \sum_{n=1}^{\infty} a_n \cos(n\theta) + b_n \sin(n\theta) \] where the coefficients are given by: \[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx \] and \[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx \]

There are some readily natural questions to be answered:

1. What are the conditions under which a Fourier series is convergent?
2. Is every trigonometric series convergent? Is it a Fourier series?
3. Does every function have a representation as a Fourier series?

Parseval's Theorem: Suppose that $f(x)$ is some square integrable function on $[-\pi, \pi]$, that is $\int_{-\pi}^{\pi} [f(x)]^2 \, dx < \infty$ with Fourier series \[ \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(n\theta) + b_n \sin(n\theta). \] One can then conclude \[ \frac{1}{\pi} \int_{-\pi}^{\pi} [f(x)]^2 = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2). \] This establishes a link between square integrable functions and square summable sequences, $L^2[-\pi, \pi] \rightarrow \ell^2$.

The techniques of harmonic analysis appear in a wide array of mathematical topics: number theory, topological groups, as well as science and engineering. Understanding the oscillatory nature of waves often uses the tools of harmonic analysis. This has lead to various applications in Electrical Engineering. In particular, compression techniques have been developed via the Fourier transform, and its discrete cousin, to compress images and music.

Hardy spaces: One can pursue two theories simultaneously. That which takes place in the unit disk and that which occurs in the upper-half-plane, which we will denote by $\mathbb{C}_+$. For an analytic function, in the unit disk $f$, we say that $f \in \text{H}^p(\mathbb{D})$ if \[ \sup_{0 < r < 1} \int_0^{2\pi} \vert f(re^{i \theta}) \vert^p \, \frac{d\theta}{\pi} \] is finite. This defines a norm $\Vert \cdot \Vert_p$, on the space. We have the norm, for when $p = \infty$ as well, $\Vert f \Vert_{\infty} = \sup_{z \in \mathbb{D}} \vert f(z) \vert$ defining the space \[ \text{H}^{\infty}(\mathbb{D}) = \left\{f \in \mathbb{A}(\mathbb{D}) \mid \sup_{\stackrel{0 < r < 1}{\theta \in \mathbb{R}}} \vert f(re^{i\theta}) \vert < + \infty \right\}. \] One can then make analogous definitions for the upper-half-plane.

Some Cornerstones of $\text{H}^p$ Theory:

1. The subharmonicity of $\vert f \vert^p$, $p > 0$, for an analytic function $f$, in the unit disk, and the subharmonicity of $\log \vert f \vert$ and $\log^+ \vert f \vert$.
2. Nontangential and radial maximal functions, which describe the behavior at the boundary.
3. Use of Blaschke products to factor out all zeros of an analytic function.
4. Use of singular integrals, understood in the Cauchy Principal-Value sense.

There are two distinct parts of my mathematical life. Before I knew what harmonic analysis was and then after I became comfortable with the basic techniques. They are extremely interesting techniques, and in particular, provide intellectually stimulating paths of inquiry. These ideas found a way into my Linear Algebra courses that I've taught. This was mostly driven by my students' interests. I never would have been comfortable enough, to make this happen, without encountering these topics in class and via my colleagues own inquiries.