Fourier transform of a function f(x)is defined as
F(k) = Integrate<( f(x) Exp(-i k x ) ), {x, -Infinity,Infinity}> where the first braket is the integrand and the curly braket is the integration variable and the limits of integration. And the inverse tranform is f(x) = Integrate<( F(k) Exp(i k x) ), {k, -Infinity,Infinity}>
where the functional form of the two functions may be different.
Now my problem is when we apply this to any physical situation, where the f(x) and F(k) will be some physical quantities eg. electric field, magnetic field, current density, charge density etc.
According to dimensional analysis, both sides of the equation should have the same dimension. So if we take a physical situation where say our function is current density. Whether the fourier transform of current density also have the dimension of current density?
Let J(r) = Integrate<( J(q) Exp( ik.r ) ), {k,-Infinity,Infinity}>
Now the exponent is dimensionless and dq has the dimension of
(L)^(-1). So should J(r)will have the dimensional of J(k) multiplied by dimension of length.
Also can somebody please say what is the dimension of dirac delta function? whether \delta(x) has the dimension of 1/x ?
Thanks