Convolution equations in spaces of holomorphic functions on convex domains \(\Omega\subset \mathbb{C}^N\) have been studied for a long time, starting with Ehrenpreis and Malgrange (for \(\Omega= \mathbb{C}^N\)). Here the author is interested in the case that \(\Omega\) is bounded and convex and that the convolution operator \(M_\mu\) is induced by a nonzero analytic functional \(\mu\) which is carried by a convex compact set \(K\subset \mathbb{C}^N\). By the theory of representing systems [see Yu. F. Korobejnik, Russian Math. Surveys 36, No. 1, 75-137 (1981; Zbl 0483.30003)] every holomorphic function \(f\) on \(\Omega\) can be represented as a Dirichlet series \(f(z)= \sum_{k=1}^\infty c_ke^{\langle\lambda^k,z\rangle}\) with \(\Lambda= (\lambda^k)_k\subset \mathbb{C}^N\), but this representation is not unique. Therefore, for solvable convolution equations \(M_\mu[f]=g\), both the right-hand side \(g\) and the solution \(f\) can be represented in the form of Dirichlet series in different ways, and the question arises which is the ‘best’ representation. The aim of the present paper is to solve this question in the context of Hilbert spaces \(X_\beta (\Lambda,\Omega)\) of holomorphic Dirichlet series \(f\) as above, where \(\lim_{k\to\infty} \log k/|\lambda^k|=0\), \(\beta= (\beta_k)_k\) is a sequence of real numbers, and \(f\) belongs to \(X_\beta (\Lambda, \Omega)\) if \(\sum_{k=1}^\infty|c_k|^2 e^{2\beta_k} <+\infty\). The author does not directly work with the function spaces \(X_\beta (\Lambda, \Omega)\), but with the corresponding sequence spaces \(X_\beta\) of the coefficients \(c_k\) of the Dirichlet series \(f\). The point here is to choose the sequence of coefficients in the representation by Dirichlet series (in a uniquely determined way) such that it has minimum Hilbert norm.