SEMINARS
Fourier transform in Lp spaces (3)

2018-07-25　10:00 — 11:30

1106, Math Building

Yue-Jun Peng

Université Clermont-Auvergne, France

Yachun Li

The Fourier transform is a very useful tool in mathematical analysis. It can be formally defined as
an improper Riemann integral, making it an integral transform. However, the Fourier transform may
not be defined even for smooth functions. In this course, I will present the Fourier transform in
Lp spaces for $p \in [1,2]$. I will begin with the definition of Lp spaces by recalling the Riesz-
Fischer theorem and the Riesz representation theorem. In these spaces, another important
mathematical concept is the convolution of functions. For this I will talk about a sufficient
condition so that two functions in Lp type spaces are well defined. I will also give the proof of
the Young inequality on the convolution. Based on these preparations, I introduce the Fourier
transform in L1 space. Usual properties will be presented such as the derivative of the Fourier
transform, the Fourier transform of the derivative of a function, the Fourier transform of the
convolution of two functions and the Fourier inversion theorem. They are discussed in a rigorous
mathematical framework. I will present two ways to define the Fourier transform in L2. One is
through the L1 space and another is through the Schwartz space. Both methods are rigorous due to the
Plancherel theorem. Finally, I will talk about the Fourier transform in Lp spaces for all $p \in [1,2]$ and the Hausdorff-Young inequality. The proof of this inequality is a consequence of Riesz-
Thorin theorem.