Approximation in variation for nonlinear Mellin integral operators in multidimensional setting LAURA ANGELONI University of Perugia Dipartimento di Matematica e Informatica Via Vanvitelli 1, 06123 Perugia ITALY laura.angeloni@unipg.it GIANLUCA VINTI University of Perugia Dipartimento di Matematica e Informatica Via Vanvitelli 1, 06123 Perugia ITALY gianluca.vinti@unipg.it Abstract: In this paper we study convergence results for a family of nonlinear integral operators of Mellin type defined as (T w f )(s)=  IR N + K w (t,f (st)) dt 〈t〉 , s ∈ IR N + ,w> 0; here {K w } w>0 is a family of kernel functions, 〈t〉 :=  N i=1 t i , t =(t 1 ,...,t N ) ∈ IR N + , and f is a function of bounded variation on IR N + . The interest about approximation results in BV −spaces in the multidimensional setting of IR N + is due, apart from a mathematical point of view, also to the important applications of such results in image reconstruction and in other fields. For this reason, in order to treat our problem, we use a multidimensional concept of variation in the sense of Tonelli, adapted from the classical definition to the present setting of IR N + , equipped with the log-Haar measure. Key–Words: nonlinear Mellin integral operators, multidimensional variation, absolutely continuous functions 1 Introduction The importance of the study of the bounded variation functions is well known, not only in the mathemat- ical literature, but also in several applied fields, due to the important applications of some families of in- tegral operators acting on this space. An example is in the field of image reconstruction, where the setting of BV −spaces is suitable in order to describe jumps of grey-levels of the image that correspond, from a mathematical point of view, to discontinuities. An important role in this sense is played by the Mellin type integral operators. Mellin operators are widely studied in literature (see, e.g., [30, 25, 26], while for other asymptotic results about Mellin oper- ators, the reader can see, e.g. [15, 16, 17]), also be- cause of their applications. For example they reveal to be useful in optical physics and engineering (see, e.g., [27, 24]), in situations where, in order to reconstruct a signal, the samples are not uniformly spaced, as in the classical Shannon Sampling Theorem, but expo- nentially spaced. The above mentioned problems are naturally framed in multidimensional setting; hence, in order to treat such a theory, it is important to consider a multidimensional concept of variation and, as con- cerns the Mellin operators, the most suitable approach is to develop an approximation theory in the setting of IR N + equipped with the Haar measure μ(A) :=  A 〈t〉 −1 dt, where A is a Borel subset of IR N + . In view of these considerations, in this paper we study the asymptotic behaviour of the family of non- linear Mellin-type integral operators defined as (T w f )(s)=  IR N + K w (t,f (st)) dt 〈t〉 , (I) w> 0, s =(s 1 ,...,s N ) ∈ IR N + :=]0, +∞) N , where st := (s 1 t 1 ,...,s N t N ) and 〈t〉 :=  N i=1 t i , for f be- longing to the space of functions of bounded variation on IR N + (see [11] for results about the linear version of (I)). The literature about approximation results by means of integral operators in BV −spaces is quite extensive: several concepts of variation were studied (see, e.g., [33, 20, 21, 31, 19, 35, 18, 23, 5, 6, 8, 9, 10]) also in the multidimensional case ([14, 5, 7, 1, 2, 3, 12]). In this paper we will prove that, under suitable assumptions on the kernel functions, if f is absolutely continuous, then V [T w f − f ]→0,w→ + ∞.