InverseTransformMethode

Inverse transform is the mathematical operation that recovers the original function from its transformed representation. In many settings, a forward transform converts a function f into a representation F by projecting onto a basis or by changing the domain. The inverse transform, when it exists, reconstructs f from F, and is denoted by T^{-1} in notation and defined on the appropriate function space.

Common examples include the inverse Fourier transform, which recovers f from its frequency-domain form F(ω) via f(t) = (1/2π) ∫ F(ω) e^{i ω t} dω; the inverse discrete Fourier transform, f[n] = (1/N) ∑ F[k] e^{i 2π k n / N}; the inverse Laplace transform, f(t) from F(s) through contour integrals or tables; and the inverse Z-transform, which recovers a time sequence from its Z-domain representation. Wavelet and other multiscale transforms also have corresponding inverse transforms that reconstruct the signal from coefficients.

Existence and stability of an inverse transform depend on the transform being invertible (or having a stable pseudo-inverse) on the chosen function space. In practice, numerical inversion may be sensitive to noise or discretization and can be ill-posed; regularization and careful basis selection are common remedies.

Applications span signal processing, communications, image reconstruction, scientific computing, and solving differential equations using transform methods. Understanding the inverse transform is essential for interpreting transformed data and for ensuring that reconstructed signals faithfully represent the original quantities.