Abstract
What is interpolation? “Perhaps it would be interesting to dig to the roots of the theory and to indicate its historical origin. Newton, who wanted to draw conclusions from the observed location of comets at equidistant times as to their location at arbitrary times arrived at the problem of determining a ‘geometric’ curve passing through arbitrarily many given points. He solved this problem by the interpolation polynomial bearing his name” (Pál Tuán {128, p. 23}.)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Freud, Géza, Orthogonale Polynome, Akadémiai Kiadó (Budapest), Birkhäuser (Basel, 1969). Orthogonal Polynomials, Pergamon Press (London-Toronto-New York, 1971).
Szegö, Gábor, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. XXIII., 1939, revised 1959, 3rd edition 1967, 4th edition 1975.
R. Askey, Mean convergence of orthogonal series and Lagrange interpolation, Acta Math. Acad. Sci. Hungar., 23 (1972), 71–85.
N. S. Baiguzov, Some estimates for the dérivátes of algebraic polynomials and an application to numerical differentiation, Mat. Zametki, 5 (1969), 183–194 (in Russian).
J. Balázs and P. Tuán, Notes on interpolation, II, Acta Math. Acad. Sci. Hungar., 8 (1957), 201–215.
J. Balázs and P. Tuán, Notes on interpolation, III, Acta Math. Acad. Sci. Hungar., 9 (1958), 195–214.
J. Balázs, On the convergence of Hermite-Fejér interpolation process, Acta Math. Acad. Sci. Hungar., 9 (1958), 259–267.
J. Balázs, Weighted (0,2) interpolation on the ultrasherical nodes, MTA III. Osztálya Közl, 11(3) (1961), 305–338.
D. L. Berman, On some linear operators, Dokl. Akad. Nauk SSSR., 73 (1950), 249–252 (Russian).
L. Brutman and A. Pinkus, On the Erdös conjecture concerning minimal norm interpolation on the unit circle, SIAM J. Numer. Anal., 17 (1980), 373–375.
L. Brutman and A. Pinkus, On the Erdös conjecture concerning minimal norm interpolation on the unit circle, SIAM J. Numer. Anal., 17 (1980), 373–375.
L. Brutman, Lebesgue functions for polynomial interpolation. A survey, Ann. of Numer. Math., 4 (1997), 111–127.
A. S. Cavaretta, Jr., A. Sharma and R. S. Varga, Hermite Birkhoff interpolation in the n-th roots of unity, Trans. Amer. Math. Soc., 259 (1980), 621–628.
A. M. Chak, A. Sharma and J. Szabados, On a problem of P. Tuán, Studia Sci. Math. Hungar., 15 (1980), 441–455.
S. Damelin, The Lebesgue function and Lebesgue constant of Lagrange interpolation for Erdös weights, J. Approx. Theory, to appear.
C. de Boor and A. Pinkus, Proof of the conjectures of Bernstein and Erdös concerning the optimal nodes for polynomial interpolation, J. Approx. Theory, 24 (1978), 289–303.
B. della Vecchia, Direct and converse results by rational operators, Constr. Approx., 12 (1996), 271–285.
B. della Vecchia, G. Mastroianni and P. Vértesi, One-sided convergence of Lagrange interpolation based on generalized Jacobi weights, to appear.
E. Egerváry and P. Tuán, Notes on interpolation, V., Acta Math. Acad. Sci. Hungar., 9 (1958), 259–267.
S. A. N. Eneduanya, On the convergence of interpolation polynomials, Anal Math., 11 (1985), 13–22.
P. Erdös and E. Feldheim, Sur le mode de convergence pour l’interpolation de Lagrange, C. R. Acad. Sci. Paris, 203 (1936), 913–915.
P. Erdös and G. Grünwald, Über die aritmetischen Mittelwerte der Lagrangeschen Interpolationspolynome, Studia Math., 7 (1938), 82–95.
P. Erdös and G. Grünwald, Über einen Faber’sehen Satz, Ann. Math., 39 (1938), 257–261.
P. Erdös and G. Halász, On the arithmetic means of Lagrange interpolation, in: Approximation Theory (J. Szabados, K. Tandori, eds.), Colloq. Math. Soc. J. Bolyai, 58 (1991), pp. 263–274.
P. Erdös and J. Szabados, On the integral of the Lebesgue function of interpolation, Acta Math. Acad. Sci. Hungar., 32 (1978), 191–195.
P. Erdös and P. Tuán, An extremal problem in the theory of interpolation, Acta Math. Acad. Sci. Hungar., 12 (1961), 221–234.
P. Erdös and P. Tuán, On interpolation, I.Quadrature and mean convergence in the Lagrange interpolation, Ann.of Math., 38 (1937), 142–155.
P. Erdös and P. Tuán, On interpolation, III., Ann. of Math., 41 (1940), 510–553.
P. Erdös and P. Tuán, On the role of the Lebesgue function the theory of Lagrange interpolation, Acta Math. Acad. Sci. Hungar., 6 (1955), 47–66.
P. Erdös and P. Vértesi, Correction of some misprints in our paper: “On the almost everywhere divergence of Lagrange interpolation polynomials for arbitrary systems of nodes” [Acta Math. Acad. Sci. Hungar., 36, no. 1–2 (1980), 71–89, Acta Math. Acad. Sci. Hungar. 38 (1981), 263.
P. Erdös and P. Vértesi, On the almost everywhere divergence of Lagrange interpolation of polynomials for arbitrary systems of nodes, Acta Math. Acad. Sci. Hungar., 36 (1980), 71–89.
P. Erdös and P. Vértesi, On the almost everywhere divergence of Lagrange interpolation, in: Approximation and Function Spaces (Gdańsk, 1979), North-Holland (Amsterdam-New York, 1981), pp. 270–278.
P. Erdös and P. Vértesi, On the Lebesgue function of interpolation, in: Functional Analysis and Approximation (P. L. Butzer, B. Sz.-Nagy, E. Görlich, eds.), ISNM, 60, Birkhäuser (1981), pp. 299–309.
P. Erdös, A. Kroó and J. Szabados, On convergent interpolatory polynomials, J. Approx. Theory, 58 (1989), 232–241.
P. Erdös, J. Szabados and P. Vértesi, On the integral of Lebesgue function of interpolation. II, Acta Math. Hungar., 68 (1995), 1–6.
P. Erdös, J. Szabados, A. K. Varma and P. Vértesi, On an interpolation theoretical extremal problem, Studia Sci. Math. Hungar., 29 (1994), 55–60.
P. Erdös, On some convergence properties in the interpolation polynomials, Ann.of Math., 44 (1943), 330–337.
P. Erdös, Problems and results on the theory of interpolation, I, Acta Math. Acad. Sci. Hungar., 9 (1958), 381–388.
P. Erdös, Problems and results on the theory of interpolation, I, Acta Math. Acad. Sci. Hungar., 9 (1958), 381–388.
P. Erdös, Problems and results on the theory of interpolation, II, Acta Math. Acad. Sci. Hungar., 12 (1961), 235–244.
P. Erdös, Some theorems and remarks on interpolation, Acta Sci. Math. (Szeged), 12 (1950), 11–17.
G. Faber, Über die interpolatorische Darstellung steiger Funktionen, Jahresber. der Deutschen Math. Verein., 23 (1914), 190–210.
L. Fejér, Über Interpolation, Göttinger Nachrichten (1916), 66–91.
L. Fejér, Interpolation und konforme Abbildung, Gött. Nachr. (1918), 319–331.
L. Fejér, Interpolation-röl (elsö közlemeny), Mat. és Term. Értesító, 34 (1916), 209–229.
L. Fejér, Lagrangesche interpolation und die zugehörigen konjugierten Punkte, Math. Ann., 106 (1932), 1–55.
L. Fejér, Lebeguesche Konstanten und divergente Fourierreihen, J. Reine Angew, Math., 138 (1910), 22–53.
L. Fejér, On the characterization of some remarkable systems point of interpolation by means of conjugate points, American Math. Monthly, 41 (1934), 1–14.
L. Fejér, Sur les fonctions bornées et integrables, C. R. Acad. Sci. Paris, 131 (1900), 984–987.
E. Feldheim, Sur le mode de convergence dans l’interpolation de Lagrange, Dokl. Nauk. USSR, 14 (1937), 327–331.
E. Feldheim, Théorie de la convergence des procédés d’interpolation et de quadrature mécanique, Mémorial des Sciences Mathématiques, 95 (1939), 1–9, Paris, Gauthier-Villars.
G. Freud, Bemarkung über die Konvergenz eines interpolations verfahren von P. Tuán, Acta Math. Acad. Sci. Hungar., 9 (1958), 337–341.
D. Gaier, Lectures on Complex Approximation, Birkhäuser (1987), pp. 196+IX.
H. H. Gonska and H.-B. Knoop, On Hermite-Fejér Interpolation; A Biography (1914–1987), Studia Sci. Math. Hungar., 25 (1990), 147–198.
G. Grünwald, Über Divergenzerscheinungen der Lagrangeschen Interpolationspoly-nome stetiger Funktionen, Ann. Math., 37 (1936), 908–918.
G. Grünwald, Über Divergenzerscheinungen der Lagrangeschen Interpolationspoly-nome, Acta Sci. Math. (Szeged), 7 (1935), 207–221.
G. Grünwald, On the theory of interpolation, Acta Math., 75 (1942), 219–245.
G. Halász, On projections into the space of trigonometric polynomials, Acta Sci. Math. (Szeged), 57 (1993), 353–366.
G. Halász, The “coarse and fine theory of interpolation” of Erdös and Tuán in a boarder view, Constr. Approx., 8 (1992), 169–185.
D. Jackson, A formula of trigonometric interpolation, Rendiconti der circolo matematico di Palermo, 37 (1914), 371–375.
I. Joó, On Pál interpolation, Ann. Univ. Sci. Budapest. Sect. Math., 37 (1994), 247–262.
I. Joö and V. E. S. Szabó, A generalization of Pál interpolation process, Acta Sci. Math. (Szeged), 60 (1995), 429–438.
A. Kalmár, Az interpolációról, Math, és Phys. Lapok, 33 (1926), 120–149 (in Hungarian).
T. A. Kilgore, A characterization of the Lagrange interpolating projection with minimal Tchebycheff norm, J. Approx. Theory, 24 (1978), 273–288.
O. Kis and J. Szabados, On the convergence of Lagrange interpolation, Acta Math. Acad. Sci. Hungar., 16 (1965), 389–430 (in Russian).
O. Kis, Convergence of interpolation in some function spaces, MTA Mat. Kut. Int. Közl., 7 (1962), 95–111 (Russian).
O. Kis, On certain interpolatory processes, II, Acta Math. Acad. Sci. Hungar., 26 (1973), 171–190 (in Russian).
O. Kis, On the convergence of the trigonometrical and harmonical interpolation, Acta Math. Acad. Sci. Hungar., 7 (1956), 173–200 (in Russian).
O. Kis, On trigonometric (0,2) interpolation, Acta Math. Acad. Sci. Hungar., 11 (1960), 255–276 (in Russian).
O. Kis, Remarks on interpolation, Acta Math. Acad. Sci. Hungar., 11 (1960), 49–64 (in Russian).
O. Kis, Remarks on the error of interpolation, Acta Math. Acad. Sci. Hungar., 20 (1969), 339–346 (in Russian).
P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan (Delhi, 1960).
M. Lénárd, On (0; 1) Pál-type interpolation with boundary conditions, Puhl. Math. Debrecen, 55 (3–4) (1999), 465–478.
M. Lénárd, Birkhoff quadrature formulae based on the zeros of Jacobi polynomials, Mathematical and Computer Modelling, 38 (2003), 917–927.
A. L. Levin and D.S. Lubinsky, Christoffel functions, orthogonal polynomials, and Nevai’s conjecture for Freud weights, Constr. Approx., 8 (1992), 463–535.
A. L. Levin, D. S. Lubinsky and T. Z. Mthembu, Christoffel functions and or-thogonal polynomials for Erdös weights on (−∞,∞), Rendiconti di Matematica di Roma, 14 (1994), 199–289.
G. G. Lorentz, K. Jetter and S. D. Riemenschneider, Birkhoff interpolation, in: Encyclopedia of Mathematics and its Applications, 19, Addison-Wesley (London-Amsterdam-Sydney-Tokyo, 1983).
D.S. Lubinsky and G. Mastroianni, Mean convergence of extended Lagrange interpolation with Freud weights, to appear.
D.S. Lubinsky, An update on orthogonal polynomials and weighted approximation on the real line, Acta Appl. Math., 33 (1993), 121–164.
D. S. Lubinsky, On converse Marcinkiewicz-Zygmund inequalities in L p p > 1 constr., Approx., 15 (1999), 577–610.
S. M. Lozinskii, The spaces \( \tilde C_\omega ^* \) and \( \tilde C_\omega ^* \) and the convergence of interpolation processes in them, Dokl. Akad. Nauk SSSR (N.S.), 59 (1948), 1389–1392 (in Russian).
T. M. Mills and P. Vértesi, An extension of the Grünwald-Marcinkiewicz interpolation theorem, Bull. Austral Math. Soc., 63 (201), 299-320.
J. Marcinkiewicz, Sur la divergence des polynômes d’interpolation, Acta Sci. Math. (Szeged), 8 (1937), 131–135.
J. Marczinkiewicz, On interpolation, I., Studia Math., 6 (1936), 1–17 (in French).
G. Mastroianni and P. Nevai, Mean convergence of derivatives of Lagrange interpolation, J. Comput. Appl. Math., 34(3) (1991), 385–396.
G. Mastroianni and P. Vértesi, Mean convergence of Lagrange interpolation on arbitrary system of nodes, Acta Sci. Math. (Szeged), 57 (1993), 425–436.
G. Mastroianni and P. Vértesi, Some applications of generalized Jacobi weights, Acta Math. Acad. Sci Hungar., 77 (1997), 323–357.
D. M. Matjila, Bounds for the weighted Lebesgue function for Freud weights on a larger interval, J. Comp. Appl. Math., 65 (1995), 293–298.
T. M. Mills, Some techniques in approximation theory, Math. Sci., 5 (1980), 105–120.
L. Neckermann and P. O. Runck, Über ApproximationScigenschaften differenzierter Lagrangescher Interpolationspolynome mit Jacobischen Abszissen, Numer. Math., 12 (1968), 1959–1969.
P. Nevai and P. Vértesi, Convergence of Hermite-Fejér interpolation at zeros of generalized polynomials, Acta Sci. Math. Szeged, 53 (1989), 77–98.
P. Nevai and P. Vértesi, Mean convergence of Hermite-Fejér interpolation, Math. Anal. Appl, 105 (1985), 26–58.
P. Nevai, Necessary conditions for weighted mean convergence of Lagrange interpolation associated with exponential weights, in preparation.
P. Nevai, Géza Freud, orthogonal polynomials and Christoffel functions: A case study, J. Approx. Th., 48 (1986), 3–167.
P. Nevai, Lagrange interpolation at zeros of orthogonal polynomials, in: Approxi-mation Theory II., Academic Press (1976), pp. 163–201.
P. Nevai, Mean convergence of Lagrange interpolation. III, Trans. Amer. Math. Soc., 282 (1984), 669–698., and J. Approx. Theory, 60 (1990), 360–363.
P. Nevai, Orthogonal Polynomials, Mem. Amer. Math. Soc., 213 (Amer. Mathematical Soc. Providence RI, 1979).
P. Nevai, Remarks on interpolation, Acta Math. Acad. Sci. Hungar., 25 (1974), 129–144 (in Russian).
P. Nevai, Solution of Tuán’ s problem on divergence of Lagrange interpolation in L p with p > 2, J. Approx Theory, 43 (1985), 190–193.
L. G. Pál, A new modification of the Hermite-Fejér interpolation, Anal. Math., 1 (1975), 197–205.
J. Prasad and A. K. Varma, An analogue of a problem of P. Erdös and E. Feldheim on L p convergence of interpolating process, to appear.
Z. F. Sebestyén, Pál-type interpolation on the roots of Hermite polynomials, Pure Math. Appt., 9(3–4) (1998), 429–439.
Z. F. Sebestyén, Supplement to the Pál type (0; 0,1) lacunary interpolation Anal Math., 25(2) (1999), 147–154.
A. Sharma and A. K. Varma, Trigonometric interpolation, Duke Math J., 32 (1965), 341–357.
Y. G. Shi, Bounds and inequalities for general orthogonal polynomials on finite intervals, J. Approx. Theory, 73 (1993), 303–319.
Y. G. Shi, On critical order of Hermite-Fejér type interpolation, in: Progress in Approximation Theory, Academic Press (1991), pp. 761–766.
S. J. Smith, Generalized Hermite-Fejér interpolation polynomials, Expo. Math., 18 (2000), 389–404.
G. Somorjai, On a saturation problem, Acta Math. Acad. Sci. Hungar., 32 (1978), 377–381.
G. Somorjai, On discrete linear operators in the function space A, in: Constructive Function Theory’ 77 (Sofia, 1980) pp. 489–496.
J. Surányi and P. Tuán, Notes on interpolation, I., Acta Math. Acad. Sci. Hungar., 6 (1955), 67–80.
J. Szabados, The exact error of trigonometric interpolation for differentiable functions, Constr. Approx., 8 (1992), 203–210.
J. Szabados and P. Vértesi, A survey on mean convergence of interpolatory process, J. Comp. App. Math., 43 (1992), 3–18.
J. Szabados and P. Vértesi, Interpolation of Functions, in: World Scientific (1990), pp. 1–305+I–XII.
J. Szabados, On Hermite-Fejér interpolation for the Jacobi abscissas, Acta Math. Acad. Sci. Hungar., 23 (1972), 449–464.
J. Szabados, Optimal order of convergence of Hermite-Fejér interpolation for general system of nodes, Acad. Sci. Math. (Szeged), 57 (1993), 463–470.
J. Szabados, On the convergence of quadrature procedures in certain classes of functions, Acta Math. Acad. Sci. Hungar., 18 (1967), 97–111.
J. Szabados, On the convergence of the derivatives of projection operators, Analysis, 7 (1987), 349–357.
J. Szabados, On the order of magnitude oof fundamental polynomials of Hermite interpolation, Acta Math. Acad. Sci. Hungar., 61 (1993), 357–368.
J. Szabados, Weighted Lagrange and Hermite-Fejér interpolation on the real line, J. oflnequal. and Appl, 1 (1997), 99–123.
J. Szabados and A. K. Varma, On a convergent Pál-type (0,2) interpolation process, Acta Math. Hungar., 66 (1995), 301–326.
V. E. S. Szabó, On Pál-type interpolation processes, Approximation and optimization, Vol. II (Cluj-Napoca, 1996), 221-226, Transilvania, Cluj-Napoca, 1997.
V. E. S. Szabó, A generalization of Pal interpolation processes. II, Acta Math. Hungar., 74 (1–2) (1997), 19–29.
V. E. S. Szabó, A generalization of Pal interpolation processes. III. The Hermite case, Acta Math. Hungar., 74 (3) (1997), 191–201.
V. E. S. Szabó, A generalization of Pál interpolation processes. IV. The Jacobi case, Acta Math. Hungar., 74 (4) (1997), 287–300.
V. E. S. Szabó, Weighted interpolation, I., to appear.
L. Szili, An interpolation process on the roots of the integrated Legendre polynomials, Anal. Math., 9 (1983), 235–245.
L. Szili, A convergence theorem for the Pal method of interpolation on the roots of Hermite polynomials, Anal. Math., 11 (1985), 75–84.
L. Szili and P. Vértesi, An Erdös type convergence process in weighted interpolation, II, Acta Math. Acad. Sci. Hungar., 98 (2003), 129–162.
A. F. Timan, Approximation Theory of Functions of a Real Variable (Moscow, 1960) (in Russian).
P. Tuán, On some open problems of approximation theory, J. Approx. Theory, 29 (1980), 23–85.
P. Vértesi, An Erdös type convergence process in weighted interpolation. I. (Freud type weights), Acta Math. Hungar., 91 (2000), 195–215.
P. Vértesi, Derivatives of projection operators, Analysis, 9 (1989), 145–156.
P. Vértesi, Hermite-Fejér type interpolations, II, Acta Math. Acad. Sci. Hungar., 33 (1979), 333–343.
P. Vértesi, Hermite-Fejér type interpolations, IV., Acta Math. Acad. Sci. Hungar., 39 (1982), 85–93.
P. Vértesi, Lebesgue function type sums of Hermite interpolations, Acta Math. Acad. Sci. Hungar., 64 (1994), 341–349.
P. Vértesi, New estimation for the Lebesgue function of Lagrange interpolation, Acta Math. Acad. Sci. Hungar., 40 (1982), 21–27.
P. Vértesi, On σ-nor mal pointsystems, in: A survey Approximation Theory (edited by M. W. Müller et. al.), Math. Res., vol. 86., Proc. IDoMAT 95, pp. 317–327.
P. Vértesi, On the Lebesgue function of weighted Lagrange interpolation, I., Constr. Approx., 15 (1999), 355–367.
P. Vértesi, On the Lebesgue function of weighted Lagrange interpolation, IL, J. Austral Math. Soc. (Series A), 65 (1998), 145–162.
P. Vértesi, One sided convergence conditions for Lagrange interpolation, Acta Sci. Math. (Szeged), 45 (1983), 419–428.
P. Vértesi, Optimal Lebesgue constant for Lagrange interpolation, SIAM J. Numer. Anal., 27 (1990), 1322–1331.
P. Vértesi, Saturation of the Shepard operators, Acta Math. Acad. Sci. Hungar., 72 (1996), 307–317.
P. Vértesi, Tuán-type problems on mean convergence, I–II., Acta Math. Acad. Sci. Hungar., 65 (1994), 115–139., and (1994), 237–242.
A. Zygmund, Trigonometric Series I, Cambridge University Press (1959).
A. Zygmund, Trigonometric Series II, Cambridge University Press (1959).
A. Zygmund, Jozef Marcinkiewicz, in: Collected works of Jozef Marcinkiewicz, pp. 4.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 János Bolyai Mathematical Society and Springer-Verlag
About this chapter
Cite this chapter
Vértesi, P. (2006). Classical (Unweighted) and Weighted Interpolation. In: Horváth, J. (eds) A Panorama of Hungarian Mathematics in the Twentieth Century I. Bolyai Society Mathematical Studies, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30721-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-30721-1_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28945-6
Online ISBN: 978-3-540-30721-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)