In Greek, the word *analysis* means "dissolving into pieces". The reversed process, called *synthesis*, assembles the analyzed pieces after they have been individually studied. This procedure is well known e.g. from chemistry, and it is the heart of harmonic analysis.

This already becomes apparent in classical Fourier analysis, which can be described as the root of harmonic analysis.

This course starts with an introduction to *Fourier series*. Here we will see how a periodic function can be represented as a superposition of waves of fixed frequencies. Next we move on to the *Fourier transform*. Among other important results we will prove Plancherel's theorem and the Hausdorff-Young inequality. Along the way we will encounter convolutions, approximate identities, interpolation, and tempered distributions.

In the second part of the course we will examine *Fourier multiplier operators*. These naturally lead us to the study of *singular integral operators*, such as the Hilbert transform. As an important tool we introduce the Calderon-Zygmund decomposition.

Harmonic analysis is a beautiful subarea of mathematics, which is of large historical importance and has much ongoing research. In addition it is an important tool in many branches of mathematics, such as: differential equations, mathematical physics, probability theory, image processing, and signal processing.

# References

Loukas Grafakos, Classical and Modern Fourier Analysis, Pearson Education, 2004.

Yitzhak Katznelson, An introduction to harmonic analysis, Dover Publications, 1976.

Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces,

Princeton University Press, 1971.

Elias M. Stein, Harmonic Analysis, Princeton University Press, 1993.

Elias M. Stein and Rami Shakarchi, Fourier Analysis, An Introduction,

Princeton University Press, 2003.

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Basic facts about Functional Analysis, Measure Theory, and L^p spaces