Note to self: this set of notes requires some fixing.

Theorem (Carathéodory–Toeplitz/Bochner)(CT for , B for ) If is a positive measure on then

for all and all . Conversely if is a (hermitian symmetric) continuous function on then is the Fourier transform of a positive measure when all matrices are positive semi-definite

*Proof:* If then for all and all

I.e., the matrix with entries is positive semi-definite. (Not that since is a real measure, so the matrix is Hermitian).

For the converse, we use an idea of Féjer. By taking Riemann sums

and so if we define

then because

Now observe that positive measure converges weak- to for some (uniqueness of weak- limit points follows from ). Now,

which converges as to c where .

We can do better than

via

Note that . It follows immediately from the following theorem that the Fourier transform extends continuously from to when , yielding a function in a (thin subset) of , where .

Theorem (Hausdorff-Young)For all ,

where and .

*Proof:* This assertion is true when by Minkowski’s inequality and when by Plancherel’s theorem. The result follows from the Riesz–Thorin interpolation theorem (cf below).

Theorem (Riesz–Thorin Interpolation)Suppose is a -linear transformation andwith and . Then

where

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