The awardee extended their pioneering Adaptive Fourier Decomposition (AFD or CoreAFD) originally defined on the unit circle and the real axis to a broad class of Hilbert spaces using innovative approaches, thereby establishing the concept of Pre-Orthogonal Adaptive Fourier Decomposition (POAFD). For practical applications, the methodology was extended to stochastic signals by incorporating probability and statistics, leading to the development of the Stochastic Pre-Orthogonal Adaptive Fourier Decomposition (SPOAFD) theory. Several variants of the method were developed within different mathematical spaces, including sparse solutions to unwinding AFD (in single and multivariate Hardy spaces), the existence and algorithms for n-parameter best approximations (n-Best Weighted Hardy Spaces, Granular Algorithm), and sparse representations on typical domains in higher-dimensional complex and Euclidean spaces. This special line of research has set a trend in mathematical studies and attracted the interest and collaborative research of renowned mathematicians such as R. Coifman, D. Baratchart, D. Alpay, I. Sabadini, and L. Cohen. Several results have been named after the awardee, such as "Qian’s Theorem" for the convergence of the Unwinding Blaschke Expansion and the "Fueter-Sce-Qian Theorem" for inducing Clifford monogenic functions. The functional analysis methods created by the awardee are based on interdisciplinary mathematical techniques and have been meaningfully applied in fields such as system identification, image and signal processing, biomedical signal analysis, and financial mathematics. Over a hundred related papers have been published in high-ranking international journals, including several in AMS Transactions, AMS Proceedings, ACHA, Science China Mathematics, AMC, Automatica, IEEE Transactions on Signal Processing, and IEEE Transactions on Information Theory, among others.

Fig 1 Singular Integrals and Fourier Theory on Lipschitz Boundaries

Fig 2 Trans of AMS: Hardy-Hodge Dcomposition in Clifford Analysis

Fig 3 (1)Named by Academician of the American Academy of Sciences, R. Coifman: Qian's Theorem on Unwinding Blaschke Decomposition (2) The Fueter-Sce-Qian theorem