Hilbert transform as an infinite matrix

The previous post linked to a post I wrote a few years ago about the Hilbert transform and Fourier series. That post says that if the Fourier series of a function is

then the Fourier series of its Hilbert transform is

When I looked back at that post I thought about how if you thought of the Fourier coefficients as elements of an infinite vector then the Hilbert transform can be represented as multiplying by an infinite block matrix.

I rarely see infinite matrices except in older math books. Apparently they were more fashionable a few decades ago than they are now. I suppose the notation falls between two stools, too concrete for some tastes and not concrete enough for others. The former folks would prefer something likeH and the latter would prefer the sum above.

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