This work has been aimed to decompose a linear integral operator on univariate functions by using high dimensional modelling. The basic idea is to use Enhanced Multivariance Products Representation (EMPR) which has been recently proposed and developed. It was based on another approach, High Dimensional Model Representation (HDMR), which has been proposed by Sobol and then further developed by Rabitz, Demiralp, and other scientists. EMPR, in contrast to HD MR, uses univariate support functions to decompose a multivariate function. The representation introduced here is not based on the general EMPR. On the contrary, it is a specific EMPR version constructed for bivariate function decomposition. We call this decomposition "Tridiagonal Kernel Enhanced Multivariance Products Representation (TKEMPR)". It uses EMPR bivariate function decomposition consecutively such that in each step the remainder term is expanded again to a bivariate EMPR but with different support functions. Even though the skeleton and the purpose of the work is rather conceptual, we give certain confirmative examples too. (C) 2017 Elsevier B.V. All rights reserved.