For \(a>0\) and nonnegative integers m and n the classes \(\mathcal {A}^a_{m,n}\) of entire functions are introduced. An entire function \(\omega \) belongs to \(\mathcal {A}^a_{m,n}\) if \(\omega \) is real on the imaginary axis and has the form \((i\mu )^{n-m+1}\omega (\mu ) =\chi _1(\mu )\cos \mu a+ i\chi _2(\mu )\sin \mu a+\Psi (\mu )\) , where \(\chi _1\) is a polynomial of degree \(n+1\) with leading coefficient \(i^{n+1}\) , \(\chi _2\) is a polynomial of degree at most n, and \(\Psi \) is small with respect to the other terms. When \(m=0\) , these functions are sine type functions. The functions in \(\mathcal {A}^a_{m,n}\) have infinitely many zeros with explicitly given asymptotic behaviour, for which the first \(n+1\) terms in the asymptotics are determined by the coefficients in the polynomials \(\chi _1\) and \(\chi _2\) . Conversely, any sequence of complex numbers with such an asymptotic representation determines a unique function from a class \(\mathcal {A}^a_{m,n}\) whose zeros are this given sequence. Some explicit formulas for the direct and the inverse problem are provided.
