The exact answer is that there is no general, simple formula for the Fourier transform of a composite function—often referred to as a "function of a function"—such as $h(x) = g(f(x))$, directly in terms of the individual Fourier transforms of $f(x)$ and $g(x)$. While the Fourier transform has elegant properties for many operations, function composition is a uniquely complex case.
Understanding the Fourier Transform
The Fourier transform is a fundamental mathematical tool that decomposes a function into its constituent frequencies. It converts a function from its original domain (e.g., time or space) into the frequency domain, revealing the spectrum of frequencies present in the function.
The Fourier transform of a function $f(x)$ is commonly denoted as $F(k)$ or $\hat{f}(k)$, where $k$ represents the spatial or angular frequency. This transformation allows for easier analysis of signals and images, particularly in fields like signal processing, optics, and quantum mechanics. The inverse transform can convert the function back from the frequency domain to the original domain.
Why Composition is Different: The Challenge of $g(f(x))$
When we talk about a "function of a function," we typically refer to a composite function, denoted as $h(x) = g(f(x))$. This means that the output of one function, $f(x)$, becomes the input for another function, $g(y)$.
Unlike other common operations like addition, multiplication, or convolution, function composition is a highly non-linear operation. The Fourier transform, by its nature, is a linear operator that interacts predictably with linear combinations and certain non-linear operations (like multiplication and convolution, through the convolution theorem). However, this linearity does not extend to function composition.
Fourier Transform Properties vs. Function Composition
To illustrate this, let's compare how the Fourier transform handles common operations with the complexity of function composition:
| Operation on Functions $f(x)$ and $g(x)$ | Description | Fourier Transform Property | Result in Frequency Domain |
|---|---|---|---|
| Sum: $f(x) + g(x)$ | Adding two functions. | Linearity: The transform of a sum is the sum of the transforms. | $\hat{f}(k) + \hat{g}(k)$ |
| Product: $f(x) \cdot g(x)$ | Multiplying two functions. | Convolution Theorem: A product in one domain is a convolution in the other. | $\frac{1}{2\pi} (\hat{f} \hat{g})(k)$ (where $$ denotes convolution) |
| Convolution: $(f * g)(x)$ | A mathematical operation representing the blending of two functions. | Convolution Theorem: A convolution in one domain is a product in the other. | $\hat{f}(k) \cdot \hat{g}(k)$ |
| Composition: $g(f(x))$ | Applying one function's output as the input to another. | No General Theorem: No simple, direct algebraic relationship exists. | No simple, direct formula involving $\hat{f}(k)$ and $\hat{g}(k)$ |
This table clearly highlights that while addition, multiplication, and convolution have elegant and widely applicable Fourier transform properties, function composition does not. There is no universally applicable formula that links $\widehat{g(f(x))}(k)$ to $\hat{f}(k)$ and $\hat{g}(k)$ in a straightforward manner.
Why No Simple Formula Exists for $g(f(x))$
The core reason for the absence of a general formula lies in the fundamental difference between the operations:
- Non-linearity of Composition: Function composition transforms the domain of $g$ based on the values of $f(x)$, which is a highly non-linear distortion. Fourier transforms excel at linear operations and operations that can be reduced to convolutions (which are also linear in a different sense).
- Information Loss/Obscurity: The frequency content of $g(f(x))$ depends critically on how $f(x)$ stretches, compresses, or distorts the input to $g(y)$. This complex interplay makes it impossible to derive a simple algebraic relationship in the frequency domain.
Specific Scenarios and Approximations
While a general formula is elusive, there are a few specific cases or approximation strategies:
- Phase Modulation ($FT[e^{if(x)}]$): In optics and signal processing, functions like $e^{if(x)}$ (where $g(y) = e^{iy}$) are common. The Fourier transform of such a function represents its diffraction pattern or spectrum. Even for this specific type of composition, there isn't a simple general formula in terms of $\hat{f}(k)$. Often, numerical methods or approximations (e.g., Fresnel or Fraunhofer approximations in optics) are used.
- Polynomial Composition: If $g(y)$ is a polynomial, say $g(y) = y^n$, then $g(f(x)) = (f(x))^n$. The Fourier transform of $(f(x))^n$ involves repeated convolutions of $\hat{f}(k)$ with itself in the frequency domain. For example, $FT[(f(x))^2] = \frac{1}{2\pi} (\hat{f} \hat{f})(k)$. This becomes increasingly complex as $n$ increases and is still a specific instance, not a general rule for any* $g(y)$.
- Taylor Series Approximation: One common approach is to approximate $g(y)$ using its Taylor series expansion around a point $y_0 = f(x_0)$:
$g(f(x)) \approx g(y_0) + g'(y_0)(f(x) - y_0) + \frac{g''(y_0)}{2}(f(x) - y_0)^2 + \dots$
Taking the Fourier transform of this expansion involves transforms of powers of $f(x)$ (which are convolutions of $\hat{f}(k)$) and linear terms. While this can provide an approximation, it is not a direct formula for the composite function's transform itself. - Numerical Computation: In practical applications, if the Fourier transform of a composite function $g(f(x))$ is needed, it is most often computed numerically. The function $h(x) = g(f(x))$ is first calculated point-wise in the original domain, and then its Fourier transform is computed using a Fast Fourier Transform (FFT) algorithm.
Conclusion
In summary, while the Fourier transform is a powerful tool with well-defined properties for many operations, it does not offer a general, simple formula for the transform of a composite function $g(f(x))$. The non-linear nature of function composition makes it fundamentally different from operations like addition, multiplication, or convolution, which have direct counterparts in the frequency domain. Researchers and practitioners typically rely on specific approximations, numerical methods, or direct computation for such cases.
