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Approximation theory and functional analysis

1991
*
Journal of Approximation Theory
*

Articles on algorithms for diverse types

doi:10.1016/0021-9045(91)90045-c
fatcat:yyf3fsulcbfnlecadwefy43doe
*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*. ...##
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Linear dynamic approximation theory

1979
*
Journal of Approximation Theory
*

We prove the convergence

doi:10.1016/0021-9045(79)90023-6
fatcat:o7gmrza77jcwrfxumeoydzleay
*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 ...##
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Weierstrass and Approximation Theory

2000
*
Journal of Approximation Theory
*

), what he contributed to

doi:10.1006/jath.2000.3508
fatcat:2myzxxd3kfhvdk2h3v7vhzh57m
*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 ...##
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Publications of Géza Freud in approximation theory

1986
*
Journal of Approximation Theory
*

PUBLICATIONS

doi:10.1016/0021-9045(86)90080-8
fatcat:gaeacmw34jhsjgc6zhvghapfl4
*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. ...##
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Exact constants in approximation theory

1992
*
Journal of Approximation Theory
*

This book, which constitutes Volume 38

doi:10.1016/0021-9045(92)90073-w
fatcat:nvwh63n4uzdcpfp5wq6uehlxna
*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. ...##
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Sparse representations and approximation theory

2011
*
Journal of Approximation Theory
*

We try to illustrate how the

doi:10.1016/j.jat.2010.10.007
fatcat:nixnvt4lhvdydlbpdexxyrmdca
*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. ...##
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The Theory of Best Approximation of Functions

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] . ...

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Approximation theory on SU(2)

1968
*
Journal of Approximation Theory
*

Here, where the requisite knowledge

doi:10.1016/0021-9045(68)90034-8
fatcat:6dxea6h4onc6pi3j3wrzlwquui
*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] ). ...##
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On some open problems of approximation theory

1980
*
Journal of Approximation Theory
*

Besides the problem

doi:10.1016/0021-9045(80)90138-0
fatcat:kjgvefw7g5agdbo7grvsxhd3ja
*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*. ...##
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Approximation theory and functional equations (II)

1973
*
Journal of Approximation Theory
*

Th e corresponding

doi:10.1016/0021-9045(73)90101-9
fatcat:vhyt22ns6zejhk3i2f7dqbdkwi
*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. ...##
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On approximation theory and functional equations

1972
*
Journal of Approximation Theory
*

Thus q~ = B is an

doi:10.1016/0021-9045(72)90016-0
fatcat:yohh75sxszbkpcfzurmytcgcue
*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. ...##
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A geometric problem in approximation theory

1982
*
Journal of Approximation Theory
*

We use some results in

doi:10.1016/0021-9045(82)90051-x
fatcat:dalmv4llejekpcfstbglifuleu
*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. ...##
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On the foundations of best approximation theory

1980
*
Journal of Approximation Theory
*

In this paper we excavate the foundations

doi:10.1016/0021-9045(80)90081-7
fatcat:rs37f3mmgbhcpii4kjcjgcgcqy
*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*. ...##
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Polynomials of binomial type and approximation theory

1978
*
Journal of Approximation Theory
*

We show that there is a one-to-one correspondence between these operators and basic sets

doi:10.1016/0021-9045(78)90105-3
fatcat:czhzvluiazbdjecfmifcmmgcze
*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. ...##
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A constructive development of Chebyshev approximation theory

1980
*
Journal of Approximation Theory
*

In this case, the classical

doi:10.1016/0021-9045(80)90012-x
fatcat:mfe7f53vmfeldoubmhnindbnhm
*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 ...
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