We define the Fourier series description as $\mathfrak{F}$, and can be written as follows:
$$
\begin{aligned}
\mathfrak{F}(\hat{f}(k))&=
\sum_{k=-\infty}^{\infty}
\hat{f}(k)e^{ikt}
\\&=
\sum_{k=-\infty}^{-1}
\hat{f}(k)e^{ikt}
+\hat{f}(0)+
\sum_{k=1}^{\infty}
\hat{f}(k)e^{ikt}
\\&=
\mathfrak{F}(\hat{f}(k))_{-}
+\hat{f}(0)+
\mathfrak{F}(\hat{f}(k))_{+}
\end{aligned}
$$
where $\mathfrak{F}(\hat{f}(k))_{+}$ is summation in positive side and $\mathfrak{F}(\hat{f}(k))_{-}$ is negative side. These summations have a following relation:
$$
\mathfrak{F}(\hat{f}(k))_{-}=\mathfrak{F}(\hat{f}(-k))_{+}
$$
Hence these four series can be derived as follows:
$$
\begin{aligned}
\sum_{k=1}^{\infty}\hat{f}(k)[\cos(kt)+i\sin(kt)]
=& \mathfrak{F}(\hat{f}(k))_{+} \\
\sum_{k=1}^{\infty}\hat{f}(-k)[\cos(kt)-i\sin(kt)]
=& \mathfrak{F}(\hat{f}(-k))_{+} \\
\sum_{k=1}^{\infty} a_k \cos(kt)
=&
\cfrac{1}{2}\biggl[
\mathfrak{F}(\hat{f}(k)) + \mathfrak{F}(\hat{f}(-k))
\biggr]-\hat{f}(0)\\
\sum_{k=1}^{\infty} b_k \sin(kt)
=&
\cfrac{1}{2}\biggl[
\mathfrak{F}(\hat{f}(k)) - \mathfrak{F}(\hat{f}(-k))
\biggr]
\end{aligned}
$$
where example of derivation:
$$
\begin{aligned}
\sum_{k=1}^{\infty} a_k \cos(kt) =&
\sum_{k=1}^{\infty}
\biggl[
\cfrac{1}{\pi}\int_{0}^{2\pi}f(t)\cos(kt)dt
\biggr]
\cos(kt) \\=&
\sum_{k=1}^{\infty}
\biggl[
\cfrac{1}{2\pi}
\int_{0}^{2\pi}f(t)(e^{ikt}+e^{-ikt})dt
\biggr]
\frac{e^{ikt}+e^{-ikt}}{2}\\=&
\cfrac{1}{2}
\sum_{k=1}^{\infty}
\bigl(
\hat{f}(-k)+\hat{f}(k)
\bigr)
(e^{ikt}+e^{-ikt}) \\=&
\cfrac{1}{2}
\Biggl[
\sum_{k=1}^{\infty} \hat{f}(-k)e^{ikt} +
\sum_{k=1}^{\infty} \hat{f}(-k)e^{-ikt} +
\sum_{k=1}^{\infty} \hat{f}(k)e^{ikt} +
\sum_{k=1}^{\infty} \hat{f}(k)e^{-ikt}
\Biggr] \\=&
\cfrac{1}{2}\biggl[
\mathfrak{F}(\hat{f}(-k))_{+} +
\mathfrak{F}(\hat{f}(k))_{-} +
\mathfrak{F}(\hat{f}(k))_{+} +
\mathfrak{F}(\hat{f}(-k))_{-}
\biggr] \\=&
\cfrac{1}{2}\biggl[
\mathfrak{F}(\hat{f}(k)) + \mathfrak{F}(\hat{f}(-k))
\biggr]-\hat{f}(0).
\end{aligned}
$$
Owing to the problem's description, the function has finite norm, and Parseval's identity shows that the norm of Fourier series and function $f(t)$ are corresponded.
$$
\begin{aligned}
\|\mathfrak{F}(\hat{f}(k))\|_{2}^{2}=&
\|\mathfrak{F}(\hat{f}(-k))\|_{2}^{2}\\=&
\frac{1}{2\pi}\int_{0}^{2\pi}|f(t)|^2dt < \infty
\end{aligned}
$$
Therefore the Fourier series is convergent so that $\|\mathfrak{F}(\hat{f}(-k))_{+}\|_{2}^2$ and $\|\mathfrak{F}(\hat{f}(-k))_{-}\|_{2}^2$ are finite, and eventually above four series are also converged.
