In this paper, we study a kind of interpolation problems on a given nodal set by trigonometric polynomials of order n and entire functions of exponential type according as the nodal set is $$\left\{ {\frac{{2k\pi }}{n}} \right\}_{k = 0}^{n - 1} or \left\{ {\frac{{2k\pi }}{\sigma }} \right\}_{k = - \infty }^{ + \infty } $$ respectively. We established some equivalent conditions and found the explicit forms of some interpolation functions on the interpolation problems. As a special case, the explicit forms of fundamential functions of (0,m)-interpolat on by trigonometric case or entire functions case (in B2 σ) respectively, if they exist, may follow from our results. Besides, we also considered the convergence of the interpolation functions at above stated.