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I am looking for a (Hilbert) space of (real-valued) functions on $\mathbb{R}^n$ where the following map defines an inner product: $$ (f,g) \mapsto \lim_{T\to\infty} \frac{1}{\mathrm{Vol}[-T,T]^n}\int_{[-T, T]^n}f(x)\overline{g(x)} dx. $$ This looks similar to the space of almost periodic functions. However, I want my space to contain all of the indicator functions of periodic sets (which Bersicovitch space does not).

The best I can come up with is to just take the set of all functions $f$ for which the mean of squares $$ \lim_{T\to\infty} \frac{1}{\mathrm{Vol}[-T,T]^n}\int_{[-T, T]^n}|f(x)|^2 dx $$ is well-defined and finite. To make it into a Hilbert space, I just take the quotient by everything that has mean-value zero. Is this a well-known space? I did not find anything in the literature.

Bram
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    the website suggests this as related: https://math.stackexchange.com/questions/929500/inner-product-in-besicovitch-space?rq=1 which seems to be the space you are looking at – daw Feb 05 '24 at 15:40
  • @daw Thanks for the reply. I am aware of this space, this is what I call the space of almost periodic functions, which is not big enough for my purposes. I will edit the question to clarify a little. – Bram Feb 06 '24 at 07:19
  • @daw I'm sorry, I thought a bit more about it and I think you are right. I'll write a proper answer to this question myself. Thanks! – Bram Feb 07 '24 at 10:39

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As per daw's comment, the answer is to use Besicovitch space, described here. This space contains all periodic functions, which can be seen for example by using a Fourier series.

A thing that I overlooked is that the space of all functions for which the mean value of squares $$ \lim_{T\to\infty}\frac{1}{\mathrm{Vol}[-T,T]^n}\int_{[-T,T]^n}|f(x)|^2dx $$ is finite, is not a vector space. If it were, then $(f_1 + f_2)^2$ should have a finite mean value for all $f_1$, $f_2$ in the space, which implies that $f_1f_2$ should have finite mean value. A counter example of this is as follows. Let $f_1 = 1$, the constant $1$ function, and $S$ a set such that its indicator function $\chi_S$ is not in the space. Then define $h(x) = 1$ for a $x \in S$ and $h(x) = -1$ otherwise. Clearly, $h$ is in the space, since $h^2 = 1$, but the mean value of $1 h = h$ is not well-defined.

Beside Besicovitch space, there are other, much larger, spaces, that contains all functions for which $$ \limsup_{T\to \infty}\frac{1}{\mathrm{Vol}[-T,T]^n}\left(\int_{[-T,T]^n}|f(x)|^pdx\right)^{1/p} $$ is finite, for fixed $1 \leq p < \infty$. This then defines a seminorm. These spaces are known as Marcinkiewicz spaces.

Bram
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