Hausdorff–Young inequality

The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered by William Henry Young (1913) and extended by Hausdorff (1923). It is now typically understood as a rather direct corollary of the Plancherel theorem, found in 1910, in combination with the Riesz-Thorin theorem, originally discovered by Marcel Riesz in 1927. With this machinery, it readily admits several generalizations, including to multidimensional Fourier series and to the Fourier transform on the real line, Euclidean spaces, as well as more general spaces. With these extensions, it is one of the best-known results of Fourier analysis, appearing in nearly every introductory graduate-level textbook on the subject.

The nature of the Hausdorff-Young inequality can be understood with only Riemann integration and infinite series as prerequisite. Given a continuous function ${\displaystyle f:(0,1)\to \mathbb {R} }$, define its "Fourier coefficients" by

${\displaystyle c_{n}=\int _{0}^{1}e^{-2\pi inx}f(x)\,dx}$

for each integer n. The Hausdorff-Young inequality can be used to show that

${\displaystyle \left(\sum _{n=-\infty }^{\infty }|c_{n}|^{3}\right)^{1/3}\leq \left(\int _{0}^{1}|f(t)|^{3/2}\,dt\right)^{2/3}.}$

Loosely speaking, this can be interpreted as saying that the "size" of the function f, as represented by the right-hand side of the above inequality, controls the "size" of its sequence of Fourier coefficients, as represented by the left-hand side.

However, this is only a very specific case of the general theorem. The usual formulations of the theorem are given below, with use of the machinery of L^p spaces and Lebesgue integration.