Parsevalegyenlségek

Parsevalegyenlőségek, also known as Parseval's identity or Parseval's theorem, are fundamental results in Fourier analysis that relate the energy of a signal to the energy of its spectral representation. The core idea is that the sum of the squares of the coefficients of a Fourier series, or the integral of the square of the magnitude of a Fourier transform, is equal to the integral of the square of the function itself.

For a periodic function f(x) with Fourier series coefficients a_n, the identity states that the average power of the signal over one period is proportional to the sum of the squares of these coefficients. Specifically, for a function with period 2π, the equation is often written as 1/π * ∫[−π, π] |f(x)|^2 dx = 2 * sum_{n=−∞ to ∞} |c_n|^2, where c_n are the complex Fourier coefficients. A simplified version for real-valued functions often appears as 1/π * ∫[−π, π] |f(x)|^2 dx = a_0^2/2 + sum_{n=1 to ∞} (a_n^2 + b_n^2), where a_n and b_n are the cosine and sine coefficients respectively.

For non-periodic functions, the analogous result comes from the Fourier transform. If F(ω) is the Fourier transform of f(t), then Parseval's theorem states that the integral of |f(t)|^2 over all time is equal to 1/(2π) times the integral of |F(ω)|^2 over all frequencies. This means that the total energy of a signal is conserved whether it is viewed in the time domain or the frequency domain. These identities have broad applications in signal processing, quantum mechanics, and other areas of applied mathematics and physics.