Laplace operator, fuzzy partition, proximity

Laplace operator is an important but ambiguous concept throughout natural sciences. It appears e.g. in graph theory, computer science, or feld theory. Every research area however considers its own customized version of this operator, depending on further plans on its utilization. And even though they all can call it by the same name, each defnition and what is more important - respective underlying space - are different. Nowadays, this operator attracts attention mainly in image and signal processing applications. For this reason, we investigated this operator from a more general point of view. Laplacian acts as an operator on functions. And although these functions themselves consider the underlying space as just a set of points, the concept of the Laplace operator requires the notion of the structure of the space. In literature, you can observe two main possibilities of definition which differ one from another in the following way. The first one, the Laplace-Beltrami operator, defined by the divergence of the gradient, is well known in euclidean space and can be considered as a continuous variant of the Laplace operator. In the continuous setting, the structure of the space determines computation of the gradient. Secondly, in the discrete case, Laplacian, being a linear operator, can be represented by a matrix, commonly called the graph Laplacian. The space structure here means determining the weighted graph. As shown by the work of Belkin, these two concepts are connected, because under certain conditions, the graph Laplacian converges to the Laplace-Beltrami operator. In this contribution, we will discuss the various ways in which these operators can be introduced in relation to the corresponding space. Moreover, we will investigate the properties of both mentioned variants. Last, but not least, we propose the formal expression for the Laplace-Beltrami operator in the space whose structure is determined by a fuzzy partition.