Turn spatial derivatives into frequency multipliers

Try

Fourier series assign coefficients to discrete frequencies on a periodic interval. On the whole line, frequency becomes continuous. For an integrable function f, define its transform by f̂(ξ)=∫ over the real line of e^(−ixξ)f(x)dx. The inverse convention will include 1/(2π).

Fixing this convention prevents stray factors in convolution and energy identities. Other conventions are valid, but mixing two of them within one calculation is not.

Predict

Differentiate e^(−ixξ) with respect to x. When integrating f′ by parts, does the transform acquire iξ or −iξ? Predict the sign after accounting for both the integration-by-parts minus sign and the derivative of the exponential.

Experiment

For a Schwartz function, meaning smooth with every derivative decaying faster than any inverse power, all boundary terms vanish.

Spatial operation	Frequency operation
f′	iξ f̂
f″	−ξ² f̂
f(x−a)	e^(−iaξ) f̂(ξ)
e^(iax)f(x)	f̂(ξ−a)

Derive the translation row by substituting y=x−a. Then derive the modulation row by combining exponents. Translation changes phase; modulation shifts the frequency location. These operations are not interchangeable.

Understand

The Gaussian provides an explicit transform. For f=e^(−ax²), a positive, let F be its transform. Since f′=−2axf and F′=∫(−ix)e^(−ixξ)f, the derivative rule gives F′=−ξF/(2a). Solving this ODE yields F=C exp(−ξ²/(4a)). The value at zero is the Gaussian integral √(π/a), fixing C. Rapid Gaussian decay justifies differentiation under the integral and integration by parts.

For integrable f and g, define convolution (f∗g)(x)=∫f(x−y)g(y)dy. The absolute double integral is bounded by ∥f∥₁∥g∥₁, so Fubini applies. Substituting z=x−y factors the transform integral into f̂(ξ)ĝ(ξ). Under this convention there is no extra factor in the forward convolution rule.

These formulas require classes in which their operations make sense. An arbitrary L² function need not belong to L¹ on the whole line, so its transform integral need not converge absolutely at each frequency. Plancherel extends the transform to L² by approximation, producing an equivalence class rather than guaranteed pointwise values. Distributions extend the framework further, but are not permission to manipulate divergent integrals without definitions.

The heat kernel’s transform is e^(−κtξ²). Its Gaussian spatial width and Gaussian frequency width move in opposite directions: spreading in space suppresses high frequencies. The exact constants can be checked from the Gaussian formula rather than guessed from this qualitative description.

Apply

Transform a translated Gaussian and then a modulated Gaussian, keeping the phase and frequency shift separate. Use the second-derivative rule to turn a heat equation into a scalar ODE at each ξ. State the function class you are using before invoking integration by parts or Fubini.

Continue

Inversion and Plancherel justify returning from frequency space and comparing norms. They also expose why backward heat is unstable even though every individual frequency equation can be solved.