Fouriertunnukset

Fouriertunnukset, in Finnish mathematical terminology, refer to Fourier symbols—functions that describe how a linear operator acts on the frequency components of a function. They encode the frequency-domain modification the operator imposes, allowing the operator to be studied through its action on the Fourier transform of a function.

In the simplest setting of constant-coefficient operators on Euclidean space, an operator A is described by a symbol sigma(ξ) such that Af is obtained by multiplying the Fourier transform by sigma and then transforming back: Af(x) = F^{-1}[sigma(ξ) Ff(ξ)](x). On a torus or periodic setting, the transform becomes a sum over discrete frequencies, and the action is Af = ∑_{k∈Z^n} e^{ix·k} sigma(k) f̂(k). Thus a Fourier symbol specifies how each frequency k is scaled (and possibly phase-shifted) by the operator.

In the broader framework of pseudodifferential operators, a symbol is a function sigma(x, ξ) defined on space and frequency variables that satisfies specific growth and smoothness conditions, often described by symbol classes S^m. These conditions control how sigma behaves as |ξ| → ∞ and govern the mapping properties of the associated operator between function spaces.

Examples include sigma(ξ) = |ξ|^s, which corresponds to fractional differentiation; sigma(ξ) = (iξ_j)^m for partial derivatives; and more generally, symbols that define Fourier multipliers or smoothing operators. Fouriertunnukset thus provide a compact way to characterize a wide range of operators by their action in the frequency domain.

See also: Fourier transform, Fourier series, multipliers, pseudodifferential operators, symbol classes.