$X(\omega)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \pi[\delta(x-1)+\delta(\lambda+1)]\left[\pi \delta(\omega-\lambda)+\frac{1}{\jmath^{(\omega-\lambda)}}\right] d \lambda$

Step 1 ：The problem is asking for the Fourier transform of a function. The function is a sum of two Dirac delta functions, one centered at x=1 and the other at x=-1.

Step 2 ：The Fourier transform of a Dirac delta function is a complex exponential function.

Step 3 ：The integral of a product of a Dirac delta function and another function is simply the value of the other function at the point where the Dirac delta function is centered.

Step 4 ：Therefore, the Fourier transform of this function can be calculated by substitifying the values of the Dirac delta functions into the other function and summing the results.

Step 5 ：Let's denote the function as X = \(\pi*(I^{(\lambda_ - \omega)} + \pi*\delta(-\lambda_ + \omega))*(\delta(\lambda_ - 1) + \delta(\lambda_ + 1))\)

Step 6 ：The Fourier transform of X is then X_transform = \(\pi^2*\delta(\omega - 1) + \pi^2*\delta(\omega + 1)\)

Step 7 ：Final Answer: The Fourier transform of the given function is \(\boxed{\pi^2[\delta(\omega-1)+\delta(\omega+1)]}\)