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Algebra with differential operators is often used as a shortcut in calculus problems. I have previously asked about manipulations such as these: $$\int x^5e^xdx =\frac1Dx^5e^x=e^x\frac{1}{1+D}x^5=e^x(1-D+D^2+...)x^5 + C$$ After reading some replies to my post, I have become interested in learning the abstract algebra that makes this magic work. Strangely enough, although the manipulation of $D$ is commonly taught as a method of solving differential equations, all the material (under the umbrella of real analysis) I have found about its usage did not touch on why or how it worked. I have come to the conclusion that if I am to understand the true mechanics of $D$, I am to look in texts about abstract algebra and not calculus.

However, the only area in math I have really worked with is calculus. In fact, the reason why I am interested is so I can apply $D$ to its fullest extent in calculus and make full usage of its properties, while feeling comfortable that I'm not breaking algebraic rules and abusing notation. Thus, with no background at all in abstract algebra, I really don't know what area to start learning, what texts to read, and what direction I need to go in.

As someone with no background in algebra, what should the itinerary of my mathematical journey be? Are there any introductory, prerequisite topics I should cover? After that, what specific topics relating to $D$? References to textbooks, lecture notes or even relevant wikipedia articles would be very much appreciated.

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2 Answers2

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I would say what you're interested in is Fourier multipliers.

The Fourier transform has the convenient property that for suitable functions, $\widehat{Df}(\xi)=\xi\hat f(\xi)$ where I've used the French convention $D=\partial/i$. This suggests that a factor of $\xi$ represents one derivative, so a more complicated function of $\xi$ represents a more complicated differential operator. In general, we write $$a(D)f=\mathcal F^{-1}(a(\xi)\hat f(\xi))$$ and call $a$ the symbol of the operator. Lots of useful things can be written in terms of these symbols. For example:

  • Constant coefficient differential operators are polynomials in $\xi$: $$a_0f+a_1f'+a_2f''+\cdots+a_nf^{(n)}=(a_0+a_1D+a_2D^2+\cdots+a_nD^n)f.$$

  • The Hilbert transform (among other singular integral operators, which are important in harmonic analysis) has symbol $sgn(\xi)/i$.

  • Even more broadly, there's the idea of a pseudo-differential operator, where you now allow the symbol to depend on both $\xi$ (frequency) and $x$ (space). As a simple example, polynomials in $\xi$ with coefficients in $x$ give differential operators with non-constant coefficients. These come up all the time in PDE. As a more interesting example, the operator which maps the Dirichlet data on a suitable region to the Neumann data can be expressed in this way, and it very nearly looks like what you would call $|D|$.

The utility of these things largely comes from the fact, as you observed, that algebra of operators is equivalent to algebra of symbols, which often simplifies things.

Funktorality
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It sounds like you are interested in operator theory. The fact that you are asking this question suggests that you are pretty advanced for where you are but to really understand this, you need a background in functional analysis. I'm also not sure that abstract algebra is going to offer you any answers. Operator algebras generally fall under the umbrella of functional analysis. I'd recommend this text, though it may be overkill for your purposes.

Also just for the sake of motivating the development of operator theory a bit with some history, you might find this helpful.

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