The author constructs an embedding of the space \({\mathcal B} (\mathbb{T})\) of hyperfunctions on the unit circle \(\mathbb{T}\) in a differential algebra \({\mathcal H}(\mathbb{T})\) whose elements are called generalized hyperfunctions. This allows one to define the product of two hyperfunctions without any restriction. Point values of a hyperfunction are also defined. These point values are elements of an algebra \({\mathcal C}\) whose set of invertible elements is denoted by \({\mathcal C}^* \). In Section 2 of the paper, some results on classical spaces of functions on \(\mathbb{T}\) are proved. In Section 3, the set \({\mathcal H}^*(\mathbb{T})\) of invertible elements of \({\mathcal H}(\mathbb{T})\) is characterized. Finally, the Cauchy problem \[ u'+ fu+gu^2=0; \quad u(z_0)=\mu, \] where \(f,g\in{\mathcal H}(\mathbb{T})\), \(z_0 \in \mathbb{T}\) and \(\mu\in {\mathcal C}^*\) is considered, and an existence theorem for a solution \(u\in{\mathcal H}^*(\mathbb{T})\) is proven.