The Hausdorff−Young inequality is a foundational result in the mathematical field of
Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...
. As a statement about
Fourier series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
, it was discovered by and extended by . It is now typically understood as a rather direct corollary of the
Plancherel theorem
In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It is a generalization of Parseval's theorem; often used in the fields of science ...
, found in 1910, in combination with the
Riesz-Thorin theorem, originally discovered by
Marcel Riesz
Marcel Riesz ( ; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford alg ...
in 1927. With this machinery, it readily admits several generalizations, including to multidimensional Fourier series and to the
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
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
, define its "Fourier coefficients" by
:
for each integer
. The Hausdorff-Young inequality can be used to show that
:
Loosely speaking, this can be interpreted as saying that the "size" of the function
, 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
spaces and
Lebesgue integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the axis. The Lebesgue integral, named after French mathematician Henri L ...
.
The conjugate exponent
Given a nonzero real number
, define the real number
(the "conjugate exponent" of
) by the equation
:
If
is equal to one, this equation has no solution, but it is interpreted to mean that
is infinite, as an element of the
extended real number line
In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential ...
. Likewise, if
is infinite, as an element of the
extended real number line
In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential ...
, then this is interpreted to mean that
is equal to one.
The commonly understood features of the conjugate exponent are simple:
* the conjugate exponent of a number in the range