On Lagrange interpolation at disturbed roots of unity
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- by Charles K. Chui, Xie Chang Shen and Le Fan Zhong PDF
- Trans. Amer. Math. Soc. 336 (1993), 817-830 Request permission
Abstract:
Let ${z_{nk}} = {e^{i{t_{nk}}}}$, $0 \leq {t_{n0}} < \cdots < {t_{nn}} < 2\pi$, $f$ a function in the disc algebra $A$, and ${\mu _n} = \max \{ |{t_{nk}} - 2k\pi /(n + 1)|:0 \leq k \leq n\}$. Denote by ${L_n}(f;\; \cdot )$ the polynomial of degree $n$ that agrees with $f$ at $\{ {z_{nk}}:k = 0, \ldots ,n\}$ . In this paper, we prove that for every $p$, $0 < p < \infty$, there exists a ${\delta _p} > 0$, such that $||{L_n}(f;\cdot ) - f|{|_p} = O(\omega (f;\frac {1} {n}))$ whenever ${\mu _n} \leq {\delta _p}/(n + 1)$. It must be emphasized that ${\delta _p}$ necessarily depends on $p$, in the sense that there exists a family $\{ {z_{nk}}:k = 0, \ldots ,n\}$ with ${\mu _n} = {\delta _2}/(n + 1)$ and such that $||{L_n}(f;\cdot ) - f|{|_2} = O(\omega (f;\frac {1} {n}))$ for all $f \in A$, but $\sup \{ ||{L_n}(f;\cdot )|{|_p}:f \in A,||f|{|_\infty } = 1\}$ diverges for sufficiently large values of $p$. In establishing our estimates, we also derive a Marcinkiewicz-Zygmund type inequality for $\{ {z_{nk}}\}$.References
- S. Ja. AlâČper and G. I. Kalinogorskaja, The convergence of Lagrange interpolation polynomials in the complex domain, Izv. VysĆĄ. UÄebn. Zaved. Matematika 1969 (1969), no. 11 (90), 13â23 (Russian). MR 0259138
- Charles K. Chui and Xie Chang Shen, On Hermite-FejĂ©r interpolation in a Jordan domain, Trans. Amer. Math. Soc. 323 (1991), no. 1, 93â109. MR 1018573, DOI 10.1090/S0002-9947-1991-1018573-6
- J. G. Clunie and J. C. Mason, Norms of analytic interpolation projections on general domains, J. Approx. Theory 41 (1984), no. 2, 149â158. MR 747398, DOI 10.1016/0021-9045(84)90108-4
- J. H. Curtiss, Convergence of complex Lagrange interpolation polynomials on the locus of the interpolation points, Duke Math. J. 32 (1965), 187â204. MR 210902
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655 L. FejĂ©r, Ăber Interpolation, Göttlinger Nachr. (1916), 66-91.
- Dieter Gaier, Vorlesungen ĂŒber Approximation im Komplexen, BirkhĂ€user Verlag, Basel-Boston, Mass., 1980 (German). MR 604011
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- Paul Koosis, Introduction to $H_{p}$ spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolffâs proof of the corona theorem. MR 565451
- S. M. Lozinski, Ăber Interpolation, Rec. Math. [Mat. Sbornik] N.S. 8(50) (1940), 57â68 (German, with Russian summary). MR 0003294
- Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207â226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
- E. B. Saff and J. L. Walsh, On the convergence of rational functions which interpolate in the roots of unity, Pacific J. Math. 45 (1973), 639â641. MR 409831
- A. Sharma and P. VĂ©rtesi, Mean convergence and interpolation in roots of unity, SIAM J. Math. Anal. 14 (1983), no. 4, 800â806. MR 704493, DOI 10.1137/0514061 X. C. Shen, The convergence of $(0,1, \ldots ,q)$ Hermite-FejĂ©r interpolating polynomials on the roots of unity, Chinese Ann. Math. Ser. (to appear).
- Xie Chang Shen and Le Fan Zhong, The mean order of approximation by Lagrange interpolation polynomials in the complex plane, Kexue Tongbao (Chinese) 33 (1988), no. 11, 810â814 (Chinese). MR 964197
- P. VĂ©rtesi, On the almost everywhere divergence of Lagrange interpolation (complex and trigonometric cases), Acta Math. Acad. Sci. Hungar. 39 (1982), no. 4, 367â377. MR 653848, DOI 10.1007/BF01896703
- J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR 0218587
- J. L. Walsh and A. Sharma, Least squares and interpolation in roots of unity, Pacific J. Math. 14 (1964), 727â730. MR 162076
- Yuan Xu, The generalized Marcinkiewicz-Zygmund inequality for trigonometric polynomials, J. Math. Anal. Appl. 161 (1991), no. 2, 447â456. MR 1132120, DOI 10.1016/0022-247X(91)90344-Y
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 336 (1993), 817-830
- MSC: Primary 30E05; Secondary 41A05, 41A10
- DOI: https://doi.org/10.1090/S0002-9947-1993-1087054-8
- MathSciNet review: 1087054