Here is a problem that am I stuck and I appreciate any help. In essence, I am trying to show that the only solutions for the described problem are the ones provided below. Best..

**Setup:** In what follows, we consider the functions defined over the half-closed symmetric interval $\mathcal{I}\triangleq [-\frac{1}{2},\frac{1}{2}]$ with the usual inner product defined as
\begin{align}
\langle f \mid g \rangle = \int_{x\in\mathcal{I}} f(x)\, g(x) \, \mathrm{d}x
\end{align}
Consider orthonormal basis for the the even-symmetric functions $\{\varphi_k\}_{k=0}^\infty$ that are defined for all $x\in\mathcal{I}$ as
\begin{align}
\varphi_0(x)&=1\\
\varphi_k(x)&=\sqrt{2}\cos(2\pi x),\quad k=1,2,\ldots.
\end{align}

**Conjecture:** Among all real valued even-symmetric functions defined over the interval $\mathcal{I}$,
the following functional equation (for the given fixed real pair ($\alpha_0\neq 0, \alpha_1$),
\begin{align}
f(x) \circledast \ln(f(x)) = \alpha_0+\alpha_1\varphi_1(x)
\end{align}
is satisfied either by
\begin{align}
f_1 (x) = \beta_0+\beta_1\varphi_1(x),
\end{align}
or
\begin{align}
f_2 (x) = \gamma_0 \exp\left(\gamma_1 \varphi_1(x)\right),
\end{align}
for some appropriately selected constants $(\beta_n,\gamma_n)$, for $n=0,1$. Here, the convolutions are in the circular sense over the interval $\mathcal{I}$.

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to produce bold text. Mathjax is meant for mathematical displays alone. Thank you. $\endgroup$ – Ricardo Andrade Dec 17 '14 at 19:14onlysolutions to the functional equation? In any event you need to assume more properties of $f$ than that it is real-valued and even: if it takes negative values, is not measurable, or is measurable but very far from being integrable, the convolution written above does not make sense. $\endgroup$ – Ian Morris Dec 18 '14 at 10:45