g[x_,y_]:=(1/(2*π*(σ^2)))*(E^(-(((x^2)+(y^2))/(2*(σ^2)))));
G[x_,y_]=FourierTransform[g[a,b],{a,b},{x,y}];

animateGaussFrequency[σControl_]:=Column[{
Plot3D[{g[x,y]/.{σ->σControl},G[x,y]/.{σ->σControl}},{x,-3,3},{y,-3,3},
PlotStyle->{
Directive[Specularity[,3],,Lighting->{{"Ambient",},{"Directional",,ImageScaled[{0,2,2}]},{"Directional",,ImageScaled[{2,2,2}]},{"Directional",,ImageScaled[{2,0,2}]}},Opacity[0.5]],
Directive[Specularity[,3],,Lighting->{{"Ambient",},{"Directional",,ImageScaled[{0,2,2}]},{"Directional",,ImageScaled[{2,2,2}]},{"Directional",,ImageScaled[{2,0,2}]}},Opacity[0.5]]
},
PlotRange->All,
PlotLegends->{"spatial","frequency"},
AxesLabel->{"x,(Subscript[ω, 1])","y,(Subscript[ω, 2])","(Subscript[g, σ]),(Subscript[G, σ])"},
PlotLabel->"σ = "<>ToString[σControl],
ImageSize->Large,
AxesStyle->Medium,
BaseStyle->{FontSize->14}
],

Plot[{g[x,0]/.{σ->σControl},G[x,0]/.{σ->σControl}},{x,-3,3},
PlotRange->All,
PlotLegends->{"spatial","frequency"},
AxesLabel->{"x,(Subscript[ω, 1])","(Subscript[g, σ])(x,0),(Subscript[G, σ])((Subscript[ω, 1]),0)"},
ImageSize->Large,
AxesStyle->Medium,
BaseStyle->{FontSize->14}
]
}];

Manipulate[
animateGaussFrequency[σ]
,{{σ,1.936},0.1,3}]

(*Export[FileNameJoin[{NotebookDirectory[],"frames/sigma=00.png"}],Table[animateGaussFrequency[σ],{σ,0.136,3.036,0.1}],"VideoFrames",Antialiasing->True];*)

Simplified versions of the Gaussian function and the corresponding Fourier transform.

Simplify[g[x,y],σ≥0]

Simplify[G[(Subscript[ω, 1]),(Subscript[ω, 2])],σ≥0]

Applying two Gaussian functions (here in the frequency domain) leads to...

2π*(G[(Subscript[ω, 1]),(Subscript[ω, 2])]/.{σ->(Subscript[σ, s])})*(G[(Subscript[ω, 1]),(Subscript[ω, 2])]/.{σ->(Subscript[σ, x])})//Simplify

...one Gaussian with adjusted σ

G[(Subscript[ω, 1]),(Subscript[ω, 2])]/.{σ->(Sqrt[(Subscript[σ, s]^2)+(Subscript[σ, x]^2)])}

Test this in the spatial domain (with concrete numbers and on the special Dirac delta function)

Convolve[g[x,y]/.{σ->10},DiracDelta[x,y],{x,y},{a,b}]

Convolve[g[x,y]/.{σ->2},((E^((1/200) (-(x^2)-(y^2))))/(200 π)),{x,y},{a,b}]

Convolve[g[x,y]/.{σ->(Sqrt[(10^2)+(2^2)])},DiracDelta[x,y],{x,y},{a,b}]

g[x,y]/.{σ->(Sqrt[(10^2)+(2^2)])}

And in the frequency domain

FourierTransform[DiracDelta[x,y],{x,y},{(Subscript[ω, 1]),(Subscript[ω, 2])}]

((1/(2 π))*2π*(G[(Subscript[ω, 1]),(Subscript[ω, 2])]/.{σ->10}))*2π*(G[(Subscript[ω, 1]),(Subscript[ω, 2])]/.{σ->2})

InverseFourierTransform[((E^(-52 ((Subscript[ω, 1]^2)+(Subscript[ω, 2]^2))))/(2 π)),{(Subscript[ω, 1]),(Subscript[ω, 2])},{x,y}]//Simplify

(1/(2 π))*2π*G[(Subscript[ω, 1]),(Subscript[ω, 2])]/.{σ->(Sqrt[(10^2)+(2^2)])}

The used Gaussian is normalized

NIntegrate[g[x,y]/.{σ->2},{x,-100,100},{y,-100,100}]