# On discrepancy theorems with applications to approximation theory

Banach Center Publications (1995)

- Volume: 31, Issue: 1, page 115-123
- ISSN: 0137-6934

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topBlatt, Hans-Peter. "On discrepancy theorems with applications to approximation theory." Banach Center Publications 31.1 (1995): 115-123. <http://eudml.org/doc/262757>.

@article{Blatt1995,

abstract = {We give an overview on discrepancy theorems based on bounds of the logarithmic potential of signed measures. The results generalize well-known results of P. Erdős and P. Turán on the distribution of zeros of polynomials. Besides of new estimates for the zeros of orthogonal polynomials, we give further applications to approximation theory concerning the distribution of Fekete points, extreme points and zeros of polynomials of best uniform approximation.},

author = {Blatt, Hans-Peter},

journal = {Banach Center Publications},

keywords = {zeros of polynomials; best uniform approximation},

language = {eng},

number = {1},

pages = {115-123},

title = {On discrepancy theorems with applications to approximation theory},

url = {http://eudml.org/doc/262757},

volume = {31},

year = {1995},

}

TY - JOUR

AU - Blatt, Hans-Peter

TI - On discrepancy theorems with applications to approximation theory

JO - Banach Center Publications

PY - 1995

VL - 31

IS - 1

SP - 115

EP - 123

AB - We give an overview on discrepancy theorems based on bounds of the logarithmic potential of signed measures. The results generalize well-known results of P. Erdős and P. Turán on the distribution of zeros of polynomials. Besides of new estimates for the zeros of orthogonal polynomials, we give further applications to approximation theory concerning the distribution of Fekete points, extreme points and zeros of polynomials of best uniform approximation.

LA - eng

KW - zeros of polynomials; best uniform approximation

UR - http://eudml.org/doc/262757

ER -

## References

top- [1] H.-P. Blatt, E. B. Saff and V. Totik, The distribution of extreme points in best complex polynomial approximation, Constr. Approx. 5 (1989), 357-370. Zbl0705.41030
- [2] H.-P. Blatt and R. Grothmann, Erdős-Turán theorems on a system of Jordan curves and arcs, ibid. 7 (1991), 19-47. Zbl0764.30003
- [3] H.-P. Blatt, On the distribution of simple zeros of polynomials, Approx. Theory 69 (1992), 250-268. Zbl0757.41011
- [4] H.-P. Blatt, Verteilung der Nullstellen von Polynomen auf Jordanbögen, in: P. Ganzinger (ed.), Informatik, Festschrift zum 60. Geburstag von G. Hotz, Teubner, Stuttgart, 1992.
- [5] H.-P. Blatt and H. N. Mhaskar, A general discrepancy theorem, Ark. Mat., to appear. Zbl0797.30032
- [6] P. Erdős and P. Turán, On the uniformly dense distribution of certain sequences of points, Ann. Math. 41 (1940), 162-173. Zbl66.0347.01
- [7] P. Erdős and P. Turán, On the uniformly dense distribution of certain sequences of points, ibid. 51 (1950), 105-119.
- [8] T. Ganelius, Sequences of analytic functions and their zeros, Ark. Mat. 3 (1953), 1-50.
- [9] R. Grothmann, Interpolation points and zeros of polynomials in approximation theory, Habilitationsschrift, Kath. Universität Eichstätt, 1993.
- [10] M. I. Kadec, On the distribution of points of maximum deviation in the approximation of continuous functions by polynomials, Amer. Math. Soc. Transl. (2)26 (1963), 231-234.
- [11] W. Kleiner, Sur l'approximation de la représentation conforme par la méthode des points extrémaux de M. Leja, Ann. Polon. Math. 14 (1964), 131-140. Zbl0128.07004
- [12] N. S. Landkof, Foundations of Modern Potential Theory, Springer, New York, 1972.
- [13] H. N. Mhaskar, Some discrepancy theorems, in: Approximation Theory, Tampa, E. B. Saff (ed.), Lecture Notes in Math. 1287, Springer, New York, 117-131.
- [14] M. Mignotte, Remarque sur une question relative à des fonctions conjuguées, C. R. Acad. Sci. Paris, to appear. Zbl0773.31002
- [15] Ch. Pommerenke, Über die Verteilung der Fekete-Punkte, Math. Ann. 168 (1967), 111-127. Zbl0145.31701
- [16] Ch. Pommerenke, Über die Verteilung der Fekete-Punkte II, ibid. 179 (1969), 212-218.
- [17] P. Sjögren, Estimates of mass distributions from their potentials and energies, Ark. Mat. 10 (1972), 59-77. Zbl0232.31009
- [18] V. Totik, Distribution of simple zeros of polynomials, Acta Math. 170 (1993), 1-28. Zbl0888.41003
- [19] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, New York, 1950. Zbl0087.28401
- [20] J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc. Colloq. Publ. 20, 5th ed., Providence, 1969.
- [21] H. Widom, Extremal polynomials associated with a system of curves in the complex plane, Adv. in Math. 3 (1969), 127-232. Zbl0183.07503

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