Deconvolution ($g=h\ast f⟼f$)

Deconvolution is the inverse problem of recovering an input $f$ from a known response kernel $h$ and measured output $g=h\ast f$. The convolution theorem turns it into $G=HF$, but division by a small $H$ amplifies noise and a zero of $H$ loses information in the corresponding mode. A unique stable inverse requires assumptions on the operator and admissible inputs; a formal Fourier transform quotient alone does not establish either property.

Tikhonov regularization of deconvolution can minimize $|HF-G{|}^2+\lambda |F{|}^2$ separately at each frequency. Completing the square gives $F_{\lambda }=H^{\ast }G/(|H{|}^2+\lambda )$ for $\lambda >0$. The regularization parameter trades reduced noise amplification against bias: the reconstruction multiplies the true noiseless component by $|H{|}^2/(|H{|}^2+\lambda )$ and the noise by $H^{\ast }/(|H{|}^2+\lambda )$.