Recover functions and propagate frequency energy

Try

A frequency formula is useful only if it corresponds to a function and preserves the intended information. Under the forward convention f̂(ξ)=∫e^(−ixξ)f(x)dx, inversion for Schwartz functions is f(x)=(1/(2π))∫e^(ixξ)f̂(ξ)dξ.

Check the factor using a Gaussian. Applying the transform twice produces 2πf(−x), not simply f(x). Reflection and normalization both matter.

Predict

For the heat equation, high-frequency amplitudes are multiplied by numbers below 1. Predict whether its L² norm can increase. For backward heat, predict whether small high-frequency errors remain small at a fixed positive time.

Experiment

Transform uₜ=κuₓₓ to obtain ∂ₜû=−κξ²û. Thus û(ξ,t)=e^(−κξ²t)f̂(ξ).

Evolution	Multiplier	Modulus behavior
Forward heat	e^(−κξ²t)	At most 1
Backward heat	e^(κξ²t)	Unbounded as frequency grows
Translation	e^(−ictξ)	Exactly 1

Use these rows to predict contraction, instability or norm preservation before calculating a spatial solution. The assertions concern positive time and the specified L² framework.

Understand

One inversion proof inserts a Gaussian factor e^(−εξ²) into the inverse integral. For ε positive, Fubini and the Gaussian transform identify the result as convolution with a Gaussian approximate identity. As ε decreases to zero, the convolution approaches f. On the frequency side, dominated convergence applies because the Schwartz transform decays rapidly. These two limits establish inversion. The decay of the transform follows by repeated integration by parts and multiplication rules, not by assuming inversion in advance.

Inversion then yields the inner-product identity ∫f̂ conjugate(ĝ)=2π∫f conjugate(g) for Schwartz functions. Setting g=f gives Plancherel: ∥f̂∥₂²=2π∥f∥₂². The normalized transform is therefore an isometry. Density of Schwartz functions in L² lets us define the transform of general L² data as the limit of transforms of approximations. The isometry proves independence of the approximation. Density and inversion also give surjectivity of the extension. These density facts are stated analysis dependencies.

For forward heat, dominated convergence applied to |e^(−κξ²t)−1|²|f̂|² proves L² initial recovery as t tends to zero. The multiplier also proves contraction immediately. Inversion identifies the solution with heat-kernel convolution when both descriptions are available.

For backward heat, choose transforms concentrated near larger and larger frequencies and scale their initial norms down. The exponentially growing multiplier can restore a fixed output norm at time T. Thus no bounded evolution operator exists on all L². This does not mean every specially chosen datum lacks a backward solution; it means arbitrary small perturbations are uncontrolled in this data class.

Apply

Compare two initial profiles whose difference has known L² norm. Bound the later forward-heat difference without evaluating a convolution. Then design a high-frequency backward-heat perturbation showing failure of continuous dependence. Identify the fixed observation time, norm and normalization in your argument.

Continue

Point sources and discontinuities require distributions. The next lesson defines their derivatives by test-function pairings so that delta identities have precise meaning.