关于可简化函数复合操作的积分变换的技术问询

Great question—totally get why you’re frustrated with vague or unsatisfying answers to this! Let’s break down what’s out there for turning functional composition (like f(g(x))) into simpler operations via integral transforms, since the usual suspects (Laplace, Fourier, Mellin) only handle linear operations like shifts or scaling, not general composition.

First, let’s set clear expectations: there’s no universal, widely used integral transform that turns arbitrary functional composition into multiplication or convolution the way Fourier turns convolution into multiplication. But there are specialized approaches and transforms that work for specific cases or in more abstract mathematical settings:

• Linear/scaling compositions (the "easy" restricted cases): The standard transforms you already know do handle limited forms of composition. For example:

  • The Fourier transform simplifies linear compositions like f(ax + b) to a scaled, shifted version of its transform: $\mathcal{F}\{f(ax + b)\} = \frac{1}{|a|} e^{2\pi i b \xi / a} \hat{f}(\xi/a)$
  • The Mellin transform handles scaling compositions f(kx) by converting it to a multiplicative factor: $\mathcal{M}\{f(kx)\} = k^{-s} \tilde{f}(s)$
    But these only work when the inner function g(x) is linear—hardly the general composition you’re curious about.

• Functional Fourier Transform: This is a more niche, abstract transform used in harmonic analysis on function spaces. The core idea is to represent functions in a basis where the composition operator acts diagonally, meaning composing two functions translates directly to multiplying their basis coefficients. It’s not as accessible or widely applied as the standard transforms, but it’s explicitly designed to tackle composition-related operations.

• Group-theoretic Fourier transforms: If you’re working within a formal group structure (where composition corresponds to group multiplication), the Fourier transform defined on that group can turn composition into pointwise multiplication. For example, on the additive group $\mathbb{R}$, this is just the standard Fourier transform handling linear composition, but for more complex groups (like matrix groups), this extends to non-linear compositions tied to group operations.

• Convolution-based workarounds: In some specific practical scenarios, you can rephrase composition as a convolution in a transformed domain. For instance, if g(x) is a probability density function, composing f with g (in the context of random variable transformations) can be linked to convolution via characteristic functions (a specialized type of Fourier transform). But this is a very targeted use case, not a general solution.

The reason prior answers might feel unsatisfying is that this problem doesn’t have a neat, all-purpose solution like the transforms you’re familiar with. Most tools here are either limited to specific composition types or require diving into more advanced mathematical frameworks that aren’t as widely taught or used in applied settings.

备注：内容来源于stack exchange，提问作者Daigaku no Baku