Hilbert transform

2 Articles

Treating Functions As Vectors In Hilbert Space

December 25, 2025 by 29 Comments

Perhaps the most beautiful aspect of mathematics is that it applies to literally everything, even things that do not exist in this Universe. In addition to this there are a number of alternative ways to represent reality, with Fourier space and its related transforms being one of the most well-known examples. An alternative to Euclidian vector space is called Hilbert space, as a real or complex inner product space, which is used in e.g. mathematical proofs. In relation to this, [Eli Bendersky] came up with the idea of treating programming language functions as vectors of a sort, so that linear algebra methods can be applied to them.

Of course, to get really nitpicky, by the time you take a function with its arguments and produce an output, it is no longer a vector, but a scalar of some description. Using real numbers as indices also somewhat defeats the whole point and claim of working in a vector space, never mind Hilbert space.

As with anything that touches upon mathematics there are sure to be many highly divisive views, so we’ll leave it at this and allow our esteemed readers to flex their intellectual muscles on this topic. Do you think that the claims made hold water? Does applying linear algebra to every day functions make sense in this manner, perhaps even hold some kind of benefit?

A Quick Look At The Hilbert Transform

July 9, 2023 by 30 Comments

While the Fourier transform gets all the attention, there are other transforms that engineers and mathematicians use to transform signals from one form to another. Sometimes you use a transform to make a signal more amenable to analysis. Other times, you do it because you want to manipulate it, and the transform is easier to change than the original signal. [Electroagenda] explains the Hilbert transform, which is often used to generate single-sideband signals.

The math behind the transformation is pretty hairy. However, if you understand the Fourier transformer, you can multiply the Fourier transform by -i sgn(ω), but that isn’t really going to help you much in a practical sense. If you don’t want to bog down in the math, skip immediately to section two of the post. That’s where it focuses more on the practical effect of the transform. You can think of the transform as a function that produces a 90 degree phase shift with a constant gain. For negative frequencies, the rotation is 90 degrees and for positive frequencies, the shift is negative.

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