Hilbert transform as an infinite matrix

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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

f(t) = sum_{n=1}^infty left{ a_n sin(nt) + b_ncos(nt) right}

then the Fourier series of its Hilbert transform is

f_H(x) = sum_{n=1}^infty left{ -b_n sin(nx) + a_ncos(nx) right}

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.

left[ begin{array}{cc|cc|cc|c} 0 & -1 & 0 & 0 & 0 & 0 & cdots \ 1 & 0 & 0 & 0 & 0 & 0 & cdots \ hline 0 & 0 & 0 & -1 & 0 & 0 & cdots \ 0 & 0 & 1 & 0 & 0 & 0 & cdots \ hline 0 & 0 & 0 & 0 & 0 & -1 & cdots \ 0 & 0 & 0 & 0 & 1 & 0 & cdots \ hline vdots & vdots & vdots & vdots & vdots & vdots & ddots end{array} right] left[ begin{array}{c} a_1 \ b_1 \ hline a_2 \ b_2 \ hline a_3 \ b_3 \ hline vdots end{array} right]

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 like H and the latter would prefer the sum above.

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