Mathematics 541: current and upcoming topics

Fourier Series (Sept 8 - 24)

Fourier series of functions in L¹(T): definition, basic properties, uniqueness ([K], Section I.1; [SS], Sections 2.1-2.2)
Convolution and summability kernels ([K], Section I.1 and I.2; [SS], Sections 2.3-2.4) Note that the terminology may vary: "good kernels" in [SS] are the same as "summability kernels" in [K]).
The Fejer kernel and Cesaro summability ([K], Sections I.2-I.3; [SS], Section 2.5)
L² theory ([K], Section I.5; [SS], Section 3.1). I mentioned briefly interpolation and the L^p theory; the interpolation theorem I used is due to Riesz and Thorin, see e.g. . We will return to this in more detail in connection with the Fourier transform on Rⁿ.
Fourier convergence for differentiable functions ([K], Section II.2; [SS], Section 3.2.1)
Continuous functions with divergent Fourier series ([SS], Section 3.2.2)
Sets of divergence ([K], Section II.3). This is interesting, but a bit technical and beyond the scope of a basic harmonic analysis course. I only sketched some of the main ideas of proofs.

Applications of Fourier series

Equidistribution of sequences ([SS], Section 4.2)

The Fourier transform on Rⁿ (Sept 27 - Oct 29)

In this part, I will mostly follow [W]. Note that Katznelson only covers the Fourier transform on the line and not in higher dimensions. Large parts of the basic theory are the same, but there are also significant differences between the 1-dimensional and higher-dimensional cases. This will be discussed in class.

Fourier transform on L¹ ([K], Section VI.1.1-VI.1.7; [SS], Section 5.1.2; [W], Chapter 1)
Basic properties: differentiation/multiplication formulas, behaviour under linear transformations ([SS], Section 5.1.4 and 6.2; [W], Chapter 1)
Fourier transform of Gaussians ([SS], Section 5.1.4 and 6.2; ([W], Chapter 1)
Schwartz space and the Fourier transform of Schwartz functions ([SS], Section 5.1.4 and 6.2; [W], Chapter 2)
Convolution, "good kernels" ([K], Section VI.1.9-VI.1.10; [SS], Section 5.1.4 and 6.2; [W], Chapter 3)
Duality and inversion formula ([K], Section VI.1.11-VI.1.12; [SS], Section 5.1.5 and 6.2; [W], Chapter 3). Note that Katznelson uses the Fejer kernel on the line, whereas in class we used Gaussians as in [SS] and [W].
Plancherel's theorem and the Fourier transform on L² ([K], VI.3.1; [SS], Section 5.1.6 and 6.2; [W], Chapter 3)
Fourier transform on L^p, 1 < p < 2, and the Hausdorff-Young inequality ([K], VI.3.2-VI.3.4; [W], Chapter 4)
Tempered distributions and the distributional Fourier transform
Uncertainty principle ([SS], Section 5.4. The proof of Benedicks's theorem was taken from this article by Ph. Jaming.)
Poisson summation formula ([K], VI.1.15; [SS], Section 5.3)

The Fourier transform of singular measures (November 1 - 17)

Stationary phase method ([W], Chapter 6; see also [S], Chapter VIII)
Application: decay estimates for the Schrodinger equation
Application: counting lattice points in large discs. (This expository article explains the basic method, and here is the article by Huxley with the best current bound. The second link works from UBC, but probably not from home unless you use a UBC proxy.)

The Hilbert transform: an introduction (November 17 - December 3)

This will follow mostly [D], Chapter 3.

Textbooks and resources

[D] J. Duoandikoetxea, Fourier Analysis, American Mathematical Society, 2001
[K] Y. Katznelson, An Introduction to Harmonic Analysis, Cambridge University Press, 2004.
[S] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993
[SS] E.M. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003
[W] T. Wolff, Lectures on Harmonic Analysis, American Mathematical Society, 2003
More will be added as needed.