Task: Summarization / Target language
User instruction:
Summarize the text below in 3 sentences in Japanese

1. Properties of the Fourier Transform  
 ㅇ Preserves the fundamental properties of related operations in both transform domains (time and frequency)  

2. Main characteristics of the Fourier Transform  
1) Linearity - a x₁(t) + b x₂(t) ↔ a X₁(f) + b X₂(f)  
2) Duality (also called symmetry) - In a Fourier transform pair, the only difference is the sign in the exponential. - Therefore, from one transform pair you can easily infer the dual pair. X(t) ↔ x(−f) - e.g., rectangular pulse ↔ sinc function Π(t/τ) ⇔ τ sinc(fτ) ; τ sinc(τt) ⇔ Π(fτ)  
3) Time shift - A shift by t₀ in the time domain corresponds to multiplication by the complex exponential exp(−j2πft₀) in the frequency domain. x(t−t₀) ↔ X(f) exp(−j2πft₀)  
4) Frequency shift (also called modulation) - Multiplying by the complex exponential exp(−j2πf₀t) in the time domain corresponds to a shift by f₀ in the frequency domain. exp(j2πf₀t) x(t) ↔ X(f−f₀) - Multiplying by a sinusoid in the time domain causes a frequency shift. cos(2πf₀t) x(t) ↔ ½ [ X(f−f₀) + X(f+f₀) ] ; sin(2πf₀t) x(t) ↔ (1/2j) [ X(f−f₀) − X(f+f₀) ]  
5) Time scaling - x(at) ↔ 1/|a| X(f/a)  

 ㅇ Convolution - x(t)*y(t) ↔ X(f)Y(f) - x(t)y(t) ↔ X(f)*Y(f)  
 ㅇ Time differentiation - d/dt x(t) ↔ j2πf X(f) - dⁿ/dtⁿ x(t) ↔ (j2πf)ⁿ X(f)  
 ㅇ Frequency differentiation - t x(t) ↔ (j/2π) d/df X(f) - tⁿ x(t) ↔ (j/2π)ⁿ dⁿ/dfⁿ X(f)  

6) Time integration ∫_{−∞}^{t} x(τ) dτ ↔ 1/(j2πf) X(f) + ½ X(0) δ(f)  

7) When the time function x(t) is complex, the Fourier transform pair becomes the conjugate with the sign reversed: x*(t) ↔ X*(−f)  

8) When the time function x(t) is real, the spectrum has symmetry: X(−f) = X*(f) or X(f) = X*(−f). In other words, the Fourier transform exhibits Hermitian symmetry (conjugate symmetry). Features of Hermitian symmetry: Amplitude and phase … Amplitude → even symmetry, i.e., |X(f)| = |X(−f)| … Phase → odd symmetry, i.e., arg[X(f)] = −arg[X(−f)] Real and imaginary parts … Real part Re{X(f)} → even symmetry, Re{X(f)} = Re{X(−f)} … Imaginary part Im{X(f)} → odd symmetry, Im{X(f)} = −Im{X(−f)} * Meaning: If you know X(f) for positive frequencies, you can determine X(f) for negative frequencies; you need only one side. (Application example: see SSB modulation)  

9) Rayleigh’s energy theorem (or Parseval’s theorem) E = ∫_{−∞}^{∞} |x(t)|² dt = ∫_{−∞}^{∞} |X(f)|² df

Evaluation criteria:
The summary must be written in Japanese.
The summary must be three sentences long.
The summary must include all key properties of the Fourier transform presented in the text.