p-Adic integral operators in wavelet bases

A wide class of p-adic integral operators in bases of p-adic wavelets is considered and matrix elements of the corresponding matrices of these operators are shown to be nonzero only on a finite number of main diagonals. A method is described for approximating real integral operators by p-adic ones in wavelet bases. This approach is based on the existence of a natural one-to-one correspondence between bases of real and p-adic wavelets but is not necessarily defined by an almost every where one-to-one map. The action of these operators in the space of mean zero test functions is considered. The space can be treated as the linear span of the wavelets, which correspond to balls. The matrix of the operator in a wavelet basis is found to be finite-diagonal, which shows that its elements are nonzero on a finite number of main diagonals.