With a little bit of effort, yes. Alternatively, once one has the Hausdorff-Young inequality for finite abelian groups, one can deduce the corresponding inequality on Euclidean spaces, basically by approximating the Euclidean Fourier transform by a discrete Fourier transform (assuming to begin with that the function belongs for instance to the Schwartz class, and removing this hypothesis with a limiting argument later).

]]>Thank your for this beautiful post. I can share another example of this method – Harald Bohr’s proof of the arithmetic-geometric mean (in a paper named “The Arithmetic and Geometric Means”). It goes as follows:

By expanding, one can show that for any :

Taking the ‘th root, the AM-GM follows by the following limit:

One doesn’t need Stirling’s approximation, merely the simple limit . Alternatively, one can say that is the largest binomial coefficient, hence , and after taking the ‘th power and , we’re done again. ]]>