Probability and interpolation
Authors:
G. G. Lorentz and R. A. Lorentz
Journal:
Trans. Amer. Math. Soc. 268 (1981), 477486
MSC:
Primary 41A05; Secondary 05B20, 15A52, 60C05
MathSciNet review:
632539
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Abstract 
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Abstract: An matrix with ones and zeros, which satisfies the Pólya condition, may be regular and singular for Birkhoff interpolation. We prove that for random distributed ones, is singular with probability that converges to one if , . Previously, this was known only if . For constant and , the probability is asymptotically at least .
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D. Riemenschneider, Probabilistic approach to Schoenberg’s
problem in Birkhoff interpolation, Acta Math. Acad. Sci. Hungar.
33 (1979), no. 12, 127–135. Special issue
dedicated to George Alexits on the occasion of his 80th birthday. MR 515126
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Georg
Gunther Lorentz and K.
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39–48. MR
0420967 (54 #8976)
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 K. Atkinson and A. Sharma, A partial characterization of poised HermiteBirkhoff interpolation problems, SIAM J. Numer. Anal. 6 (1969), 230235. MR 0264828 (41:9419)
 [2]
 G. D. Birkhoff, General mean value and remainder theorems with applications to mechanical differentiation and integration, Trans. Amer. Math. Soc. 7 (1906), 107136. MR 1500736
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 W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 1330. MR 0144363 (26:1908)
 [4]
 S. Karlin and J. M. Karon, Poised and nonpoised HermiteBirkhoff interpolations, Indiana Univ. Math. J. 21 (1972), 11311170. MR 0315328 (47:3877)
 [5]
 J. H. B. Kemperman, Moment problems for sampling without replacement, Indag. Math. 35 (1973), 149188. MR 0345259 (49:9997a)
 [6]
 G. G. Lorentz, Birkhoff interpolation and the problem of free matrices, J. Approx. Theory 6 (1972), 283290. MR 0340889 (49:5639)
 [7]
 , "The Birkhoff interpolation problem: New methods and results" in Linear operators and approximation, II edited by P. L. Butzer and B. Sz.Nagy, ISMN No. 25, Birkhäuser Verlag, Basel, 1974, pp. 481501. MR 0393939 (52:14746)
 [8]
 , Coalescence of matrices, regularity and singularity of Birkhoff interpolation problems, J. Approx. Theory 20 (1977), 178190. MR 0454452 (56:12703)
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 G. G. Lorentz and S. D. Riemenschneider, "Recent progress in Birkhoff interpolation" in Approximation theory and functional analysis edited by J. B. Prolla, NorthHolland, Amsterdam, 1979, pp. 187236. MR 553421 (81a:41006)
 [10]
 , Probabilistic approach to Schoenberg's problem in Birkhoff interpolation theory, Acta Math. Acad. Sci. Hungar. 33 (1979), 127135. MR 515126 (80f:41002)
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 G. G. Lorentz and K. L. Zeller, Birkhoff interpolation, SIAM J. Numer. Anal. 8 (1971), 4348. MR 0295529 (45:4595)
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 R. A. Lorentz, "Interpolation and probability" in Approximation theory, III edited by E. W. Cheney, Academic Press, New York, 1980, pp. 595600. MR 602775 (82f:41007)
 [13]
 I. J. Schoenberg, On HermiteBirkhoff interpolation, J. Math. Anal. Appl. 16 (1966), 538543. MR 0203307 (34:3160)
 [14]
 R. J. Serfling, Probability inequalities for the sum in sampling without replacement, Ann. Statist. 2 (1974), 3948. MR 0420967 (54:8976)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198106325390
PII:
S 00029947(1981)06325390
Keywords:
Birkhoff interpolation,
Pólya matrix,
regularity and singularity,
coalescence of rows,
probability of singularity,
hypergeometric distribution
Article copyright:
© Copyright 1981 American Mathematical Society
