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Approximation theory and functional analysis

1991 Journal of Approximation Theory  
Articles on algorithms for diverse types of approximation processes. C. K. CHUI, ED., Approximation Theory and Functional Analysis, Academic Press, 1990, 241 pp.  ...  At first a tool for investigating rational and special irrational numbers, continued fractions play an important role in function theory and approximation theory.  ... 
doi:10.1016/0021-9045(91)90045-c fatcat:yyf3fsulcbfnlecadwefy43doe

Linear dynamic approximation theory

Jan Mycielski
1979 Journal of Approximation Theory  
We prove the convergence of a statistical learning algorithm. The possibility of applications of this theory to biology is discussed.  ...  We estimate the sum of the squares of the absolute reiative errors of some general linear learning algorithms and the sum of the squares of the coefficients obtained by the perceptron algorithm.  ...  Dynamic approximation theory (a term chosen here because of the analogy of this theory with dynamic programming [3] ), which is a chapter of learning theory or prediction theory, looks at a slightly different  ... 
doi:10.1016/0021-9045(79)90023-6 fatcat:o7gmrza77jcwrfxumeoydzleay

Weierstrass and Approximation Theory

Allan Pinkus
2000 Journal of Approximation Theory  
), what he contributed to approximation theory (and why), and some of the consequences thereof. We start this story by relating a little about the man and his life.  ...  We discuss and examine Weierstrass' main contributions to approximation theory. Academic Press WEIERSTRASS This is a story about Karl Wilhelm Theodor Weierstrass (Weierstra?  ...  In 1854, Weierstrass published the paper On the theory of Abelian functions in Crelle's Journal fu r die Reine und Angewandte Mathematik (the first mathematical research journal, founded in 1826, and now  ... 
doi:10.1006/jath.2000.3508 fatcat:2myzxxd3kfhvdk2h3v7vhzh57m

Publications of Géza Freud in approximation theory

1986 Journal of Approximation Theory  
PUBLICATIONS OF GkZA FREUD IN APPRO~MA~~N THEORY  ...  A lower estimate in the theory of spline approximations (with V. Popov), Studia Sci. Math. Hungar. 6 (1971), 387-391. MR 46, No. 5908. [Russian] Hungar. 6 (1955), 467473. MR 17-1083. 18a.  ... 
doi:10.1016/0021-9045(86)90080-8 fatcat:gaeacmw34jhsjgc6zhvghapfl4

Exact constants in approximation theory

Rolf J Nessel
1992 Journal of Approximation Theory  
This book, which constitutes Volume 38 of the Encyclopedia of Mathematics and its Appiications, for the first time outlines a general approach to one of the classical problems in approximation theory,  ...  various Jackson inequalities, and with the approximation of classes of functions, determined by abstract moduli of continuity.  ...  Auf Ahlberg, Nilson, und Walsh geht die Theorie der Splines zuriick, die auf Typ, und Basisdarstellungen bis hin zu B-Splines haben ein Pendent im Reellen.  ... 
doi:10.1016/0021-9045(92)90073-w fatcat:nvwh63n4uzdcpfp5wq6uehlxna

Sparse representations and approximation theory

Allan Pinkus
2011 Journal of Approximation Theory  
We try to illustrate how the theory of sparse representation leads to new and interesting problems in approximation theory, while the results and techniques of approximation theory can further add to the  ...  In particular, we consider in detail ℓ m 1 -approximation, which is fundamental in the theory of sparse representations, and the connection between the theory of sparse representations and certain nwidth  ...  Hopefully, however, the ideas, results, and techniques of approximation theory can further add to the theory of sparse representations.  ... 
doi:10.1016/j.jat.2010.10.007 fatcat:nixnvt4lhvdydlbpdexxyrmdca

The Theory of Best Approximation of Functions

V.L. Goncharov
2000 Journal of Approximation Theory  
Approximation theory was no exception.  ...  The results of Poncelet just mentioned are stated (other than in lithographed lecture notes not available to us) in his work on approximate radical values published in Crelle's Journal in 1835 [64] .  ... 
doi:10.1006/jath.2000.3476 fatcat:zkljqosl5jfjnjj7i7vqkdcuua

Approximation theory on SU(2)

David L Ragozin
1968 Journal of Approximation Theory  
Here, where the requisite knowledge of Lie theory is minimal, we prove a special case of the relevant inequality (Theorem 4.1). Now we give a brief summary of the individual sections.  ...  We now adapt a technique of Korovkin for approximating on the circle group ( [2] , pp. 336-9 and [7] ).  ... 
doi:10.1016/0021-9045(68)90034-8 fatcat:6dxea6h4onc6pi3j3wrzlwquui

On some open problems of approximation theory

P Turán
1980 Journal of Approximation Theory  
Besides the problem of approximation by polynomials, Chebyshev was already interested in the theory of rational approximation.  ...  well-knowz role piayed by the function ' x I in the theory of polynomial approximation.  ... 
doi:10.1016/0021-9045(80)90138-0 fatcat:kjgvefw7g5agdbo7grvsxhd3ja

Approximation theory and functional equations (II)

R.C Buck
1973 Journal of Approximation Theory  
Th e corresponding theory parallels closely that for the simpler two variable case in which H = {all ,4(x) + B(y)}.  ...  We are interested in the approximation properties of H as a subspace of C[X x Y x Z] on special compact sets KC X X Y X Z.  ... 
doi:10.1016/0021-9045(73)90101-9 fatcat:vhyt22ns6zejhk3i2f7dqbdkwi

On approximation theory and functional equations

R.C Buck
1972 Journal of Approximation Theory  
Thus q~ = B is an approximate solution of the system (3). Conversely, suppose that (3) has arbitrarily good approximate solutions for any choice of the functions ui obeying restrictions (4).  ...  lies in HIK , and (2) has approximate solutions sp if Fiies in the closure of HIK. We now proceed with the proof of Theorem 1.  ... 
doi:10.1016/0021-9045(72)90016-0 fatcat:yohh75sxszbkpcfzurmytcgcue

A geometric problem in approximation theory

Leopold Flatto
1982 Journal of Approximation Theory  
We use some results in approximation theory to translate the property Hk into algebraic terms (Theorem 2).  ...  Theorem 1 below brings out the significance of inadmissible signature for approximation theory. The geometric characterization of inadmissible signatures and lifted signatures is a difficult problem.  ... 
doi:10.1016/0021-9045(82)90051-x fatcat:dalmv4llejekpcfstbglifuleu

On the foundations of best approximation theory

Douglas S Bridges
1980 Journal of Approximation Theory  
In this paper we excavate the foundations of best-approximation theory with the tools of Bishop's constructive analysis.  ...  The second part of the paper deals with the characterisation and existence of minimax polynomial approximations to elements of C [O, I], and with the pointwise continuity of the minimax approximation mapping  ...  theory.  ... 
doi:10.1016/0021-9045(80)90081-7 fatcat:rs37f3mmgbhcpii4kjcjgcgcqy

Polynomials of binomial type and approximation theory

Mourad E.H. Ismail
1978 Journal of Approximation Theory  
We show that there is a one-to-one correspondence between these operators and basic sets of binomial type {P"(x)},"p" with P,(x) > 0 for x > 0.  ...  This correspondence is achieved via inverting a family of bilateral Laplace transforms. SAC& t) = j-WA, t, u>f(u) du, -co aw X(u-t) -z-w, at p(t) A > 0, 1 m W(A, t, u) du = 1.  ...  The exponential operators are approximation operators, that is, S,,(f, t) +f(t) as h -+ co, for certain classes of functions5 The following are examples of exponential operators.  ... 
doi:10.1016/0021-9045(78)90105-3 fatcat:czhzvluiazbdjecfmifcmmgcze

A constructive development of Chebyshev approximation theory

Douglas S Bridges
1980 Journal of Approximation Theory  
In this case, the classical theory [IO, p. 651 tells us that there is a unique best approximant b of a in H, characterised by the property: there exists j E (0, 1) such that (-I)"+ (ab)(x,) = dist(a, H  ...  Among such subsets of [0, I], there are certain ones whose importance for our earlier characterisation and existence theory makes them obvious candidates for our attention: namely, sets of the form {x,  ...  ACI(NOWLEIXMENT The author wishes to thank the Department of Mathematics of the University of Edinburgh for welcoming him as a Visiting Postdoctoral Fellow during the period in which this work was carried  ... 
doi:10.1016/0021-9045(80)90012-x fatcat:mfe7f53vmfeldoubmhnindbnhm
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