Help verifying a Fourier Transform identity I don't know why I keep getting stuck, but I'm having trouble verifying the following identity from one of my textbooks:
$$\chi(T(\lambda - \lambda_j)) = \frac{1}{T\pi}\int_{-\infty}^{\infty} \hat{\chi}(t/T) e^{it \lambda} \cos(t\lambda_j) \,dt + \chi(T(\lambda+\lambda_j))$$
I know that I need to separate the complex exponential, and use Euler's identity, but I'm having trouble getting the $\chi(T(\lambda + \lambda_j))$ term to work out. My computations keep coming up with $\chi(T(\lambda_j - \lambda))$.
I don't think it's important to the calculation, but $T>1$, $\lambda,\lambda_j \geq 0$, and $\chi$ is a Schwartz class function.
Thanks in advance!
 A: $$\chi(T(\lambda - \lambda_j)) = \frac{1}{T\pi}\int_{-\infty}^{\infty} \hat{\chi}(t/T) e^{it \lambda} \cos(t\lambda_j) \,dt + \chi(T(\lambda+\lambda_j))$$
$$\chi(T(\lambda - \lambda_j)) -\chi(T(\lambda+\lambda_j))= \frac{1}{T\pi}\int_{-\infty}^{\infty} \hat{\chi}(t/T) e^{it \lambda} \cos(t\lambda_j) \,dt $$
You just need to prove this :
$$\chi(T(\lambda + \lambda_j)) = \frac{1}{2\pi T}\int_{-\infty}^{\infty} \hat{\chi}(t/T) e^{it \lambda} e^{i(t\lambda_j)} \,dt $$
$$\chi(T(\lambda + \lambda_j)) = \mathscr {F^{-1}}\left \{\frac{1}{ T} \hat{\chi}(t/T)  e^{i(t\lambda_j)} \right \}  $$
$$\mathscr{F}\left \{\chi(T(\lambda + \lambda_j)) \right \}  =  \frac{1}{ T} \hat{\chi}(t/T)  e^{i(t\lambda_j)}  $$
And also:
$$\mathscr{F}\left \{\chi(T(\lambda - \lambda_j)) \right \}  =  \frac{1}{ T} \hat{\chi}(t/T)  e^{-i(t\lambda_j)}  $$
You first scale then you shift. Thats correct.
Maybe there is a typo in the book. Because the sign on the LHS should be a minus sign
$$\chi(T(\lambda - \lambda_j)) = \frac{1}{T\pi}\int_{-\infty}^{\infty} \hat{\chi}(t/T) e^{it \lambda} \cos(t\lambda_j) \,dt \color {blue}{-\chi(T(\lambda+\lambda_j))}$$
