The filtering operation removes scales associated with high frequencies, and the operation can accordingly be interpreted in Fourier space. For a scalar field ϕ ( x , t ) , {\displaystyle \phi ({\boldsymbol {x}},t),} the Fourier transform of ϕ {\displaystyle \phi } is ϕ ^ ( k , ω ) , {\displaystyle {\hat {\phi }}({\boldsymbol {k}},\omega ),} a function of k , {\displaystyle {\boldsymbol {k}},} the spatial wave number, and ω , {\displaystyle \omega ,} the temporal frequency. ϕ ^ {\displaystyle {\hat {\phi }}} can be filtered by the corresponding Fourier transform of the filter kernel, denoted G ^ ( k , ω ) : {\displaystyle {\hat {G}}({\boldsymbol {k}},\omega ):}