Optimization of Approach and Recovery - Sun Yongsheng Collection

Author: Sun Yongsheng
Publisher:
Publish Date: 2005-08-01
Features: This collection of essays is selected from the papers I published between 1958 and 2002, some of which were co-authored with the editor and the students of the editor. It basically presents the historical trajectory of my studies in function approximation theory. In the mid-1950s, the editor was a graduate student at the Mechanics and Mathematics Department of Moscow State University, studying under Professor S.Stechkin in function approximation theory. Under his enthusiastic guidance, the editor completed his doctoral dissertation, which marked the beginning of the editor's work in approximation theory. The main results of this dissertation, as well as some subsequent related results, are included in the first few papers of this collection.
After returning to China, the editor's scientific research career took a winding path. Between 1958 and 1976, continuous political movements forced the suspension of teaching and research work for 13 years. After the Cultural Revolution, scientific research began to resume in 1978. At that time, I was nearly 50 years old, and my prime research years had passed. The little knowledge I had acquired in the 1950s had almost been completely lost, and the editor had to start from scratch. Despite my mediocre abilities, I still had a passion and drive for learning new mathematical knowledge. The immediate task was to quickly find a direction to move forward. Function approximation theory is like a century-old tree with old trunks and new branches, dense and complex. The editor was deeply confused, feeling as if "walking around the tree three times, how can one choose a branch to lean on?"
In addition to studying the recent works of Stechkin and Nikolskii, the editor successively chose to study three internationally renowned schools: the Soviet school's extremal theory in approximation theory, primarily width theory, led by Professors N.Korneievich and V.Tikhomirov; the American school's spline theory, width theory, and -optimality recovery theory led by Professors Karlin, Micchelli, and Pinkus; and the information complexity theory of Professors Traub and Wozniakowski. The learning process lasted for several years, and by the 1980s, the editor had undergone a continuous process of "shedding feathers," gradually approaching the convergence point of three directions: classical approximation theory, computational mathematics, and the mathematical theory of computer science. This is a new research field characterized by multidirectional cross-penetration.
In this field of research, the editor is a latecomer, but I am determined to carve out a place for Chinese mathematicians in this vast domain. The other groups of papers in this collection are merely steps the editor (and the editor's co-authors) have taken on this long journey. The core content of these studies is the transition of width and -optimality recovery research from T to R. Although the steps are small, the editor feels that they are solid. They witness the editor and the co-authors' perseverance, embody our efforts, and are permeated with our sweat. As the saying goes, "A broomstick is precious to its owner." These four words perfectly describe this collection of essays.