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 like H and the latter would prefer the sum above.The post Hilbert transform as an infinite matrix first appeared on John D. Cook.