Phi1FhatY
Phi1FhatY is a symbolic notation used in some areas of harmonic analysis, nonlinear signal processing, and related fields to denote the first-stage transform obtained by applying a Phi-type functional to the Fourier-domain representation of a quantity Y, modulated by another function F. The exact definition of Phi1FhatY varies with the author and context, but it is commonly interpreted as a nonlinear map that operates on the product of the Fourier transform of F and Y. In many treatments, the expression is written as Phi1FhatY = Phi1( hat{F} · Y ), where hat{F} is the Fourier transform of F and Y is a signal or dataset.
Formal definition can be given, in a typical setup, as follows: Let F belong to a suitable
Applications and interpretation: The construction appears in nonlinear spectral analysis and adaptive filtering, where the first
See also: Fourier transform, nonlinear transforms, spectral methods, thresholding operators.