Planchereltétel

Planchereltétel is a mathematical term that refers to a specific type of integral, often encountered in the field of Fourier analysis and related areas of mathematics. It is named after the French mathematician Jules-Henri Plancherel, who made significant contributions to functional analysis and Fourier series.

The Plancherel theorem relates the integral of the square of a function to the integral of the square of its Fourier transform. More formally, if f(x) is a function, and its Fourier transform is denoted by $\hat{f}(\xi)$, then the Plancherel theorem states that the integral of $|f(x)|^2$ over all real numbers is equal to the integral of $|\hat{f}(\xi)|^2$ over all real numbers, up to a constant factor that depends on the normalization convention used for the Fourier transform. This relationship is a fundamental result in $L^2$ spaces, which are function spaces where the integral of the square of the function's magnitude is finite.

The theorem establishes an isometry between a function space and its Fourier transform space. This means that the Fourier transform preserves the "length" or "norm" of functions in $L^2$. This property is crucial for many applications of Fourier analysis, including signal processing, quantum mechanics, and the study of differential equations. It allows one to translate problems about functions into problems about their Fourier transforms, and vice versa, while preserving important structural information.