When is the ordinary continuous Fourier transform integral well-defined versus when generalized functions are required?

Q: When is the ordinary continuous Fourier transform integral well-defined versus when generalized functions are required?

The ordinary integral $\int_{-\infty}^{\infty} g(t)e^{-2\pi i ft} dt$ is well-defined if $g(t)$ is absolutely integrable (i.e., $\int |g(t)| dt < \infty$). However, ideal sinusoids sustained indefinitely are not absolutely integrable because their energy spreads over infinite time. For such periodic signals, the Fourier transform requires generalized functions (distributions), specifically Dirac delta spikes, rather than converging to a finite ordinary number.

The ordinary integral

\int_{-\infty}^{\infty} g(t)e^{-2\pi i ft} dt

is well-defined if

g(t)

is absolutely integrable (i.e.,

\int |g(t)| dt < \infty

). However, ideal sinusoids sustained indefinitely are not absolutely integrable because their energy spreads over infinite time. For such periodic signals, the Fourier transform requires generalized functions (distributions), specifically Dirac delta spikes, rather than converging to a finite ordinary number.

Conditions

Considering the limit as the time window $T \to \infty$ .
Analyzing either decaying transient signals or sustained periodic signals.

Reasoning, step by step

Check if $\int_{-\infty}^{\infty} |g(t)| dt$ converges.
If yes, the standard Riemann/Lebesgue integral defines the transform normally.
If no (e.g., $g(t) = \sin(2\pi f_0 t)$ ), the integral diverges in the ordinary sense.
Invoke distribution theory to represent the spectrum as delta functions $\delta(f-f_0)$ .
Recognize that practical recordings are finite, so they use finite-window integrals which approximate these deltas with broadened peaks.

Example

The script explains: 'absolute integrability is a sufficient condition for an ordinary integral. An indefinitely sustained ideal sinusoid is not absolutely integrable: its delta spikes use generalized functions... Actual recordings have finite-window peak widths and leakage.'

Common misconceptions

Believing that the Fourier transform of a pure sine wave is a regular function with infinite height at one point; it is a distribution.
Assuming all physical signals are perfectly periodic and infinite in duration.

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When is the ordinary continuous Fourier transform integral well-defined versus when generalized functions are required?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

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Related questions

When is the ordinary continuous Fourier transform integral well-defined versus when generalized functions are required?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

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