The analysis and processing of massive amount of multivariate data and high-dimensional functions have become a basic need in many areas of science and engineering. To reduce the dimensionality for compact representation and visualization of high-dimensional information appear imperative in exploratory research and engineering modeling. Since D. Hilbert raised the 13th problem in 1900, the study on possibility to express high-dimensional functions via composition of lower-dimensional functions has gained considerable success. Nonetheless, no methods of realization are ever indicated, and not even all integrable functions can be treated this way, a fortiori functions in L2(W). The common practice is to expand high-dimensional functions into a convergent series in terms of a chosen orthonormal basis with lower dimensional ones. However, the length and rapidity of convergence of the expansion heavily depend upon the choice of basis. In this paper an attempt is made to seek an optimal basis for a given function provided with fewest terms and rapidest convergence. All elements of the optimal basis turned out to be products of single-variable functions taken from the unit balls of ingredient spaces. The proposed theorems and schemes may find wide applications in data processing, visualization, computing, engineering simulation and decoupling of nonlinear control systems. The facts established in the theorems may have their own theoretical interests.
We recall that L2(W) and l2, the space of square-summable sequences of reals, are isometrically isomorphic. Each element of L2(W) has its spectral image in l2 according to bases chosen in each spaces. If one identifies the square of norm of F(x) Î L2(W) with the energy or information it carries, in terminology of physics, the outcome of Theorems presented in this paper is to assert that for any given F there exists an optimal basis in L2(W) which furnishes the element with an image sharply concentrated on a few of spectrum-lines in l2, if the latter is equipped with canonical basis. This may be in conspicuous contrast with a flat spread of spectral lines with respect to a casually chosen basis for spectral analysis as it happens in many cases of practices.
Biography Dr Song Jian is a distinguished scientist in research and engineering, and a Science and Technology Policy-maker of China. Over the last four decades, he has made significant contributions to a diverse array of disciplines including optimal control, distributed parameter systems control, engineering cybernetics, and population control theory. He lead the program for the system design, launching and positioning of the country?s first telecommunication satellite. He initiated and successfully carried out the 'Sparks Program' aiming at alleviating rural poverty and developing rural/township enterprises throughout China. He has also initiated guided and implemented the 'Torch Program' , which spearheaded the development of high-tech industries in China.
Dr Song is currently an academician of both the Chinese Academy of Sciences and the Chinese Academy of Engineering Sciences; Professor of Tsinghua University Fudan University, and Harbin University of Technology; Research Professor, Institute of Information and Control; council member, China Association of Automation and China System Engineering Society; and a member of the Editorial Board, Encyclopaedia of China.
The posts that he has previously held include Head, Laboratory of Cybernetics, Institute of Mathematics, Academia Sinica (1960-70): Designer-Scientists and Vice Chief Designer-Scientist (1960-80), Head, Space Science Division (1971-78), and Vice President (1978-81) and Vice President (1978-81), Academy of Space Technology, Seventh Ministry of Machine-Building Industry; Vice Minister and Chief Commander of Communication Satellite Launching Operations, Ministry of Astronauts (1980-84) and Vice President, China Society of Demographic Science (1982-86) and China System Engineering Society (1985-87). Minister of Science and Technology and Sate Councillor (Vice Premier) (1984-1988).
Dr Song's international experience includes Visiting Professor at MIT, Harvard University, University of Minnesota, University of Texas, distinguished Honorary Professor of Washington University (St. Louis) and Doctor of Humane Letters, University of Houston in the United States. Dr Song is an active foreign member of the Russian Academy of Sciences, the Royal Swedish Academy of Engineering Sciences and Foreign Associate of the National Academy of Engineering, USA.
He has authored, co-authored, or edited 11 books and has written and published 160 scientific articles. He has received numerous awards including the Albert Einstein Award (1987) for signal accomplishment in science
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