Géza Freud, orthogonal polynomials and Christoffel functions. A case study. (English) Zbl 0606.42020

Nevai, Paul

J. Approximation Theory 48, 3-167 (1986).

This voluminous paper treats one aspect of the work done by Geza Freud: the consistent use of the concept of Christoffel functions in the theory of orthogonal polynomials is highlighted. Not only did Freud himself contribute greatly to the development of the theory, but he was also the initiator and stimulator of an avalanche of publications on the subject. It is not possible to pay the respect due to all these mathematicians who have built the beautiful structure as is known nowadays (the list of references contains - apart from 72 papers due to Freud alone - some 445 references!), therefore only a paragraph-wise treatment of the ”monograph” under review will be given.
After some history, notation and philosophy - in that order - the author turns his attention to the subject of polynomials orthogonal on finite intervals and on the unit circle (approx. 72 pages, crammed with information). Using as tool the Christoffel function \(\lambda_ n(d\alpha,x)\) which is nothing else but the minimum of the integrals \(\int_{{\mathbb{R}}}| P(t)|^ 2d\alpha (t)\) taken over all polynomials of degree \(\leq n-1\) satisfying \(P(x)=1\), the subjects of Tauberian theorems with remainder terms, (absolute) convergence of orthogonal Fourier series and strong Cesáro summability are looked into. This is followed by quite a number of results on the asymptotic behaviour of the \(\lambda_ n\), starting with the most recent results and some discussion on historical developments. After applications of Christoffel functions to quadrature sums, interpolation (Lagrange, Hermite-Fejer), Szegö’s theory, zeros/asymptotics for orthogonal polynomials and equiconvergence of Fourier series, the stage is set (in a section called ”farewell to orthogonal polynomials on finite intervals”) for the second part of the paper (approx. 60 pages) on orthogonal polynomials on infinite intervals.
The attention is, at first, focused on so-called Freud-weights \(d\alpha =w(x)dx\) with \(w(x)=\exp (-Q(x))\), \(x\in {\mathbb{R}}\), where \(Q>0\) is an even \(C^ 1\) function on \({\mathbb{R}}\) such that xQ’(x) increases for \(x>0\) and Q’(x)\(\to \infty\) as \(x\to \infty\). It is in this field, that over the past four years enormous progress (on Freud weights and generalizations) has been made by - in alphabetical order - Bauldry, Bonan, Levin (A. L.), Lubinsky, Magnus (Alphonse), Màté, Mhaskar, Nevai, Rahmanov, Saff, Sheen, Totik and Ullman.
The results that are treated in the second part (covering aspects from Christoffel functions, Fourier series, Cesaro and de la Vallée Poussin means via quadrature, Lagrange interpolation to Plancherel-Rotach asymptotics) is a typical example of what happens in leaving the shelter of the compact support: the reader gets, on one hand, the impression that everything changes to quite an extend and, on the other hand, that at least 50 % stays true (albeit in a slightly modified form sometimes). Again the reader must suffer the cold shower of the speed with which a group of prolific mathematicians cranked out results; nevertheless, one gets the impression that it is doubtful whether the author has missed any of the relevant publications up to August 1985! The paper ends with a ”note added in proof” (the writing on the wall) which states that one of Freud’s conjecture on the coefficients in the recurrence relations for the orthogonal polynomials w.r.t. exponential weights has been proved between the submission of the manuscript and the revision 7 months later. There is in my opinion only one dangerous point in this case study under review: the enormous amount of material and the seemingly over increasing speed in which new developments follow, might frighten the novice who wants to enter the field of orthogonal polynomials. One thing is sure: it will not be easy to catch up with the top-specialists, but if one really seriously wants to try, this paper by Nevai sure will be as indispensable as the books on orthogonal polynomials by Freud and Szegö.

Reviewer: M.G.de Bruin

MSC:

42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
42-02	Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
42A20	Convergence and absolute convergence of Fourier and trigonometric series
40G05	Cesàro, Euler, Nörlund and Hausdorff methods
40F05	Absolute and strong summability

Keywords:

Christoffel functions; Tauberian theorems; convergence of orthogonal Fourier series; strong Cesáro summability; Freud-weights

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[148]	Freud, G., On weighted polynomial approximation on the whole real axis, Acta Math. Acad. Sci. Hungar., 20, 223-225 (1969) · Zbl 0172.34103
[149]	Freud, G., Lagrangesche Interpolation über die Nullstellen der Hermiteschen Orthogonalpolynome, Studia Sci. Math. Hungar., 4, 179-190 (1969) · Zbl 0191.35901
[150]	Freud, G., Ein Beitrag zur Theorie des Lagrangeschen Interpolationsverfahrens, Studio Sci. Math. Hungar., 4, 379-384 (1969) · Zbl 0186.11401
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[152]	Freud, G., On two polynomial inequalities, I, Acta Math. Acad. Sci. Hungar., 22, 109-116 (1971) · Zbl 0219.41003
[153]	Freud, G., On an extremum problem for polynomials, Acta Sci. Math. (Szeged), 32, 287-290 (1971) · Zbl 0222.41003
[154]	Freud, G., On a class of orthogonal polynomials, Mat. Zametki, 9, 511-520 (1971), [In Russian] · Zbl 0215.13701
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[156]	Freud, G., On an inequality of Markov type, Soviet Math. Dokl., 12, 570-573 (1971) · Zbl 0221.41008
[157]	Freud, G., On approximation with weight \(( exp{−x^22}\) by polynomials, Soviet Math. Dokl., 12, 1837-1840 (1971) · Zbl 0254.41004
[158]	Freud, G., On a class of orthogonal polynomials, (Constructive Function Theory Proceedings, Internat. Conf.. Constructive Function Theory Proceedings, Internat. Conf., Varna, 1970 (1972), Publ. House Bulgarian Acad. Sci: Publ. House Bulgarian Acad. Sci Sofia), 177-182 · Zbl 0235.33012
[159]	Freud, G., A contribution to the problem of weighted polynomial approximation, (Butzer, P. L.; etal., Linear Operators and Approximation. Linear Operators and Approximation, ISNM, Vol. 20 (1972), Birkhäuser-Verlag: Birkhäuser-Verlag Basel), 431-447 · Zbl 0259.41004
[160]	Freud, G., On two polynomial inequalities, II, Acta Math. Acad. Sci. Hungar., 23, 137-145 (1972) · Zbl 0263.26013
[161]	Freud, G., On Hermite-Fejér interpolation sequences, Acta Math. Acad. Sci. Hungar., 23, 175-178 (1972) · Zbl 0256.41002
[162]	Freud, G., On Hermite-Fejér interpolation processes, Studia Sci. Math. Hungar., 1, 307-316 (1972) · Zbl 0299.41001
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[164]	Freud, G., On direct and converse theorems in the theory of weighted polynomial approximation, Math. Z., 126, 123-134 (1972) · Zbl 0222.41004
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[166]	Freud, G., On weighted \(L_1\)-approximation by polynomials, Studia Math., 46, 125-133 (1973) · Zbl 0254.41003
[167]	Freud, G., Investigations on weighted approximation by polynomials, Studia Sci. Math. Hungar., 8, 285-305 (1973) · Zbl 0291.41005
[168]	Freud, G., On polynomial approximation with the weight \(exp{−x^{2k}2}\), Acta Math. Acad. Sci. Hungar., 24, 363-371 (1973) · Zbl 0269.41004
[169]	Freud, G., On the converse theorems of weighted polynomial approximation, Acta Math. Acad. Sci. Hungar., 24, 389-397 (1973) · Zbl 0269.41005
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[171]	Freud, G., Error estimates for Gauss-Jacobi quadrature formulae, (Miller, J. J.H, Topics in Numerical Analysis (1973), Academic Press: Academic Press London/New York), 113-121 · Zbl 0285.65019
[172]	Freud, G., On estimations of the greatest zeros of orthogonal polynomials, Acta Math. Acad. Sci. Hungar., 25, 99-107 (1974) · Zbl 0276.42006
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[176]	Freud, G., An estimate of the error of Padé approximants, Acta Math. Acad. Sci. Hungar., 25, 213-221 (1974) · Zbl 0285.41011
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[193]	Freud, G.; Mhaskar, H. N., Weighted polynomial approximation in rearrangement invariant Banach function spaces on the whole real line, Indian J. Math., 22, 3, 209-224 (1980) · Zbl 0516.41004
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