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- #1

$$

\int_{-\pi}^{\pi}te^{xt}\cos(yt)g(t)dt \quad\text{and}\quad

-\int_{-\pi}^{\pi}te^{xt}\sin(yt)g(t)dt

$$

define the fourier transform of g as

$$

G(z) = \int_{-\pi}^{\pi}e^{zt}g(t)dt

$$

We know t, e^{xt}, sine, and cosine are continuous which means their products are continuous but we don't know about g(t).