Best Approximation by Linear Superpositions (approximate Nomography)This book deals with problems of approximation of continuous or bounded functions of several variables by linear superposition of functions that are from the same class and have fewer variables. The main topic is the space of linear superpositions D considered as a sub-space of the space of continous functions C(X) on a compact space X. Such properties as density of D in C(X), its closedness, proximality, etc. are studied in great detail. The approach to these and other problems based on duality and the Hahn-Banach theorem is emphasized. Also, considerable attention is given to the discussion of the Diliberto-Straus algorithm for finding the best approximation of a given function by linear superpositions. |
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Best Approximation by Linear Superpositions (approximate Nomography) S. I͡A. Khavinson No preview available - 1997 |
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A₁ According to Lemma approximation to f arbitrary sets assume assumptions of Theorem Banach space basis functions best approximation Borel measures Borel set bounded functions Chapter closed lightning bolts closed set coincide compact sets compact spaces Consider construct continuous function converges coordinates COROLLARY defined denote dense set equivalence class everywhere dense example F uniformly finite formula function f Gn(x h¹(x h₂ Hence Hn(y holds homeomorphism II(a inequality irreducible lightning bolts Kolmogorov's theorem Lemma Let Q Let us show levelled function levelling algorithm linear functionals linear superpositions mappings measure µ necessary and sufficient norm obtain p₁ parallelepipeds problem proof of Theorem Proposition proved proximinal Q contains quasi-all representation satisfying sequence subsets subspace BD Theorem 2.2 topology uniformly separates measures V₁ values vertically levelled X₁ Xi+1 y₁
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Page 1 - Then / (xi, z2. x2) = g [Ф (xi,x2) , x3] , and we obtain representation of an arbitrary function of three variables by a superposition of functions of two variables...