Linear operators and construction of symmetric symbols satisfying sum rules of order p

We present a new method to construct symmetric symbols corresponding to solvable multiscaling equations using linear operators. First, we define a linear operator corresponding to a symbol. We demonstrate that the existence of a solution to a multiscaling equation depends on the 1-eigenspace of the corresponding linear operator. Furthermore, we show how to construct symmetric symbols satisfying the sum rules of orders 1 and 2 by appropriately defining the linear operator on a basis. We also construct symmetric symbols satisfying sum rules of order 3 with degree 3. Several examples are provided to illustrate our theoretical results.