Posts tagged with harmonic analysis

Dave Rusin

(Source: math.niu.edu)

Transgressing boundaries, smashing binaries, and queering categories are important goals within certain schools of thought.

Reading such stuff the other week-end I noticed (a) a heap of geometrical metaphors and (b) limited geometrical vocabulary.

In my opinion functional analysis (as in, precision about mathematical functions—not practical deconstruction) points toward more appropriate geometries than just the `[0,1]` of fuzzy logic. If your goal is to escape “either/or” then I don’t think you’ve escaped very much if you just make room for an “in between”.

By contrast `ℝ→ℝ` functions (even continuous ones; even smooth ones!) can wiggle out of definitions you might naïvely try to impose on them. The space of functions naturally lends itself to different metrics that are appropriate for different purposes, rather than “one right answer”. And even trying to define a rational means of categorising things requires a lot—like, Terence Tao level—of hard thinking.

I’ll illustrate my point with the arbitrary function ƒ pictured at the top of this post. Suppose that ƒ∈𝒞². So it does make sense to talk about whether ƒ′′≷0.

But in the case I drew above, ƒ′′≹0. In fact “most” 𝒞² functions on that same interval wouldn’t fully fit into either “concave" or "convex”.

So “fits the binary” is rarer than “doesn’t fit the binary”. The “borderlands” are bigger than the staked-out lands. And it would be very strange to even think about trying to shoehorn generic 𝒞² functions into

• one type,
• the other,
• or “something in between”.

Beyond “false dichotomy”, ≶ in this space doesn’t even pass the scoff test. I wouldn’t want to call the ƒ I drew a “queer function”, but I wonder if a geometry like this isn’t more what queer theorists want than something as evanescent as “liminal”, something as thin as "boundary".

hi-res

In harmonic analysis and PDE, one often wants to place a function `ƒ:ℝᵈ→ℂ` on some domain (let’s take a Euclidean space `ℝᵈ` for simplicity) in one or more function spaces in order to quantify its “size”….

[T]here is an entire zoo of function spaces one could consider, and it can be difficult at first to see how they are organised with respect to each other.

For function spaces `X` on Euclidean space, two such exponents are the regularity `s` of the space, and the integrability `p` of the space.

—Terence Tao

Hat tip: @AnalysisFact

hi-res

Standing waves in 2-D via dhiyamuhammad.

Pretty amazing that if you simply add together oscillations = vibrations = waves = harmonics and constrain them within a box, that all these shapes emerge. (See this video for such waves being constructed in real life). By the way, mathematicians sometimes refer to these as “square drumhead” problems because a drumhead is a real-life 2-D surface that vibrates in exactly these kinds of ways to produce the sounds we associate with various drums.

In the link Muhammad points to—Harmonic Resonance Theory—the mathematics of standing waves are applied to the problem of Gestalt in psychology of sense experience.

hi-res