SUMMARY
The Fourier transform of \(f'(ax)\) is derived using the scaling theorem and the derivative rule. The correct formulation is \(H(\omega) = \frac{i \omega}{a^2} F\left(\frac{\omega}{a}\right)\), where \(H(\omega)\) is the Fourier transform of \(h(x) = f'(ax)\). The discrepancy arises from the application of the scaling theorem and the derivative rule, which must be carefully combined to avoid additional factors. The key takeaway is that the scaling factor is \( \frac{1}{a^2} \) when considering both transformations.
PREREQUISITES
- Understanding of Fourier Transform properties and definitions
- Familiarity with the scaling theorem in Fourier analysis
- Knowledge of differentiation in the context of Fourier Transforms
- Experience with integration techniques, particularly integration by parts
NEXT STEPS
- Study the implications of the scaling theorem in Fourier analysis
- Explore the application of the derivative rule in Fourier Transforms
- Learn about common pitfalls in Fourier Transform calculations
- Investigate advanced topics in Fourier analysis, such as convolution and its properties
USEFUL FOR
Mathematicians, physicists, and engineers working with signal processing, particularly those involved in Fourier analysis and its applications in differential equations.