A construction of multiresolution analysis by integral equations

In this paper we present a versatile construction of multiresolution analysis of two variables by means of eigenvalue problems of the integral equation, for $\lambda =2$. As a consequence we show that if $\phi (x)$ is the solution of the equation $\phi (x) = \lambda \int _{\mathbb{R}} h (2x-y) \phi (y)dy$ with $supp \hat h(\omega ) = [-\pi , \pi ]$, then $V_{j} =span \{ \phi (2^{j} x_{1} -k_{1} )$ $\phi (2^{j} x_{2} -k_{2} ) \vert k_{1} , k_{2} \in \mathbb{Z} \}$ constructs a two-variable multiresolution analysis.