Mhaskar’s area of research is approximation theory and harmonic analysis. He has done pioneering work in the theory of weighted polynomial approximation on the real line, in particular, making deep contributions in the areas of orthogonal polynomial expansions and applications of potential theory to the study of orthogonal polynomials, now known as Freud polynomials. This work is recognized through such terms as Mhaskar-Rahmanov-Saff number and Mhaskar-Saff functional. Since 1990, he has been interested in machine learning and signal processing, making pioneering contributions to the theory of approximation capabilities (expressive power) of shallow and deep neural networks, kernel-based methods, and manifold learning. He has published two books, five edited volumes, and over 150 refereed papers. His research is supported currently by the National Science Foundation, and previously by the U.S. Air Force, U.S. Army, and Office of the Director of National Intelligence (U.S.A.).

Mhaskar serves on the editorial boards of Applied and Computational Harmonic Analysis, Journal of Approximation Theory, Frontiers in Applied Mathematics and Statistics, Jaen Journal of Approximation, and Mathematical Foundations of Computing. He has given a number of plenary talks in international conferences as well as a number of colloquia all over the world. He has held visiting positions at many universities in the U.S., Germany, India, and Australia. In 2012, he was a consultant at Qualcomm. Currently, he has an affiliation with the University of California, Santa Barbara. His honors include the Alexander v. Humboldt fellowship (5 times), John von Neumann distinguished professorship at Technical University in Munich in 2011, and August-Wilhelm Scheer visiting professor at TUM (postponed due to the pandemic). An international conference was held in 2016 to celebrate his 60th birthday year.