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Introduction and Preliminaries

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Approximation by Max-Product Type Operators

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In this chapter we introduce the reader into the topic of the book and present some preliminaries useful for the next chapters.

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References

  1. Abel, U., Butzer, P.L.: Complete asymptotic expansion for generalized Favard operators. Constr. Approx. 35, 73–88 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals. J. Math. Anal. Appl. 316, 269–306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Sampl. Theory Signal Image Process. 6 (1), 19–52 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Prediction by samples from the past with error estimates covering discontinuous signals. IEEE Trans. Inf. Theory 56 (1), 614–633 (2010)

    Article  MathSciNet  Google Scholar 

  5. Baskakov, V.A.: An example of a sequence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk SSSR 113, 249–251 (1957, in Russian)

    Google Scholar 

  6. Bede, B., Coroianu, L., Gal, S.G.: Approximation and shape preserving properties of the truncated Baskakov operator of max-product kind. Rev. Union Matematica Argent. 52 (1), 89–107 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Bede, B., Gal, S.G.: Approximation by nonlinear Bernstein and Favard-Szász-Mirakjan operators of max-product kind. J. Concr. Appl. Math. 8 (2), 193–207 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Bernstein, S.N.: Démonstration du théorém de Weierstrass fondeé sur le calcul des probabilités. Commun. Soc. Math. Kharkov 13, 1–2 (1912/1913)

    Google Scholar 

  9. Bernstein, S.: Quelques remarques sur l’interpolation. Math. Ann. 79 (1–2), 1–12 (1918)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bezuglaya, L., Katsnelson, V.: The sampling theorem for functions with limited multi-band spectrum. Z. Anal. Anwend. 12 (3), 511–534 (1993)

    MathSciNet  MATH  Google Scholar 

  11. Bleimann, G., Butzer, P.L., Hahn, L.: A Bernstein-type operator approximating continuous functions on the semi-axis. Indag. Math. 42, 255–262 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bojanic, R.: A note on the precision of interpolation by Hermite-Fejér polynomials. In: Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969), pp. 69–76. Akadémiai Kiadó, Budapest (1972)

    Google Scholar 

  13. Borel, E.: Sur l’interpolation. C. R. Acad. Sci. Paris 124, 673–676 (1897)

    Google Scholar 

  14. Burinska, Z., Runovski, K., Schmeisser, H.-J.: On the approximation by generalized sampling series in L p -metrics. Sampl. Theory Signal Image Process. 5 (1), 59–87 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Butzer, P.L.: A survey of the Whittaker-Shannon sampling theorem and some of its extensions. J. Math. Res. Expos. 3, 185–212 (1983)

    MathSciNet  MATH  Google Scholar 

  16. Butzer, P.L., Engels, W., Ries, S., Stens, R.L.: The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines. SIAM J. Appl. Math. 46 (2), 299–323 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation, vol. I. Academic, New York/London (1971)

    Book  MATH  Google Scholar 

  18. Butzer, P.L., Ries, S., Stens, R.L.: Approximation of continuous and discountinuous functions by generalized sampling series. J. Approx. Theory 50, 25–39 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  19. Butzer, P.L., Splettstöβer, W., Stens, R.L.: The sampling theorems and linear prediction in signal analysis. Jahresber. Dtsch. Math-Vereinigung 90, 1–70 (1988)

    Google Scholar 

  20. Butzer, P.L., Stens, R.L.: The Poisson summation formula, Whittaker’s cardinal series and approximate integration. In: Proceedings of the Second Edmonton Conference on Approximation Theory, Canadian Mathematical Society, vol. 3, pp. 19–36 (1983)

    MathSciNet  MATH  Google Scholar 

  21. Chanas, S.: On the interval approximation of a fuzzy number. Fuzzy Sets Syst. 122, 353–356 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cheney, E.W., Sharma, A.: Bernstein power series. Can. J. Math. 16 (2), 241–252 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cobzas, S., Muntean, I.: Condensation of singularities and divergence results in approximation theory. J. Approx. Theory 31 (2), 138–153 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Coroianu, L., Gal, S.G.: Approximation by nonlinear Hermite-Fejér interpolation operators of max-product kind on Chebyshev knots. Rev. d’Anal. Numér. Théor. Approx. (Cluj) 39 (1), 21–31 (2010)

    Google Scholar 

  25. Coroianu, L., Gal, S.G.: Approximation by nonlinear Lagrange interpolation operators of max-product kind on Chebyshev nodes. J. Comput. Anal. Appl. 13 (2), 211–224 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Coroianu, L., Gal, S.G.: Approximation by max-product Lagrange interpolation operators. Stud. Univ. Babes-Bolyai Math. 56 (2), 315–325 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Coroianu, L., Gal, S.G.: Approximation by nonlinear generalized sampling operators of max-product kind. Sampl. Theory Signal Image Process. 9 (1–3), 59–75 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Coroianu, L., Gal, S.G.: Approximation by max-product sampling operators based on sinc-type kernels. Sampl. Theory Signal Image Process. 10 (3), 211–230 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Coroianu, L., Gal, S.G., Opris, B.D., Trifa, S.: Feller’s scheme in approximation by nonlinear possibilistic integral operators (2016, submitted)

    Google Scholar 

  30. Costarelli, D., Vinti, G.: Order of approximation for sampling Kantorovich type operators. J. Integr. Equ. Appl. 26 (3), 345–368 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. De Cooman, G.: Possibility theory. I. The measure-and integral-theoretic groundwork. Int. J. Gen. Syst. 25 (4), 291–323 (1997)

    MATH  Google Scholar 

  32. Delgado, M., Vila, M.A., Voxman, W.: On a canonical representation of a fuzzy number. Fuzzy Sets Syst. 93, 125–135 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. DeVore, R.A.: The Approximation of Continuous Functions by Positive Linear Operators. Lecture Notes in Mathematics, vol. 293. Springer, Heidelberg/Berlin/New York (1972)

    Google Scholar 

  34. Diamond, P., Kloeden P.: Metric spaces of fuzzy sets. Theory and Applications. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  35. Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  36. Dubois, D., Prade, H.: The mean value of a fuzzy number. Fuzzy Sets Syst. 24, 279–300 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  37. Dubois, D., Prade, H.: Possibility Theory. Plenum, New York (1988)

    Book  MATH  Google Scholar 

  38. Erdös, P., Vértesi, P.: On the almost everywhere divergence of Lagrange interpolatory polynomials for arbitrary system of nodes. Acta Math. Acad. Sci. Hung. 36 (1–2), 71–89 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  39. Favard, J.: Sur les multiplicateurs d’interpolation. J. Math. Pures Appl. 23 (9), 219–247 (1944)

    MathSciNet  MATH  Google Scholar 

  40. Fejér, L.: Über Interpolation. Göttingen Nachrichten 66–91 (1916)

    Google Scholar 

  41. Finta, Z.: Direct local and global approximation theorems for some linear positive operators. Anal. Theory Appl. 20 (4), 307–322 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Gal, S.G.: Shape-Preserving Approximation by Real and Complex Polynomials. Birkhäuser, Boston/Basel/Berlin (2008)

    Book  MATH  Google Scholar 

  43. Gal, S.G.: A possibilistic approach of the max-product Bernstein operators. Results Math. 65 (3–4), 453–462 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Gal, S.G., Szabados, J.: On the preservation of global smoothness by some interpolation operators. Stud. Sci. Math. Hung. 35, 397–414 (1999)

    MathSciNet  MATH  Google Scholar 

  45. Grünwald, G.: Über Divergenzerscheinungen der Lagrangeschen Interpolationspolynome stetiger Functionen. Ann. Math. 37, 908–918 (1936)

    Article  MATH  Google Scholar 

  46. Grzegorzewski, P.: Metrics and orders in space of fuzzy numbers. Fuzzy Sets Syst. 97, 83–94 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  47. Guo, S., Liu, L., Wang, Z.: Pointwise approximation by Meyer–König and Zeller operators. Numer. Funct. Anal. Optim. 29 (7–8), 770–778 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  48. Heilpern, S.: The expected value of a fuzzy number. Fuzzy Sets Syst. 47, 81–86 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  49. Hermann, T., Vértesi, P.: On the method of Somorjai. Acta Math. Hung. 54 (3–4), 253–262 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  50. Jerry, A.J.: The Shannon sampling-its various extensions and applications: a tutorial review. Proc. IEEE 65, 1565–1596 (1977)

    Article  Google Scholar 

  51. Khan, R.A.: A note on a Bernstein-type operator of Bleimann, Butzer and Hahn. J. Approx. Theory 53 (3), 295–303 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  52. Khan, R.A.: Some properties of a Bernstein-type operator of Bleimann, Butzer, and Hahn. In: Progress in Approximation Theory, pp. 497–504. Academic, Boston (1991)

    Google Scholar 

  53. Kivinukk, A., Tamberg, G.: Interpolating generalized Shannon sampling operators, their norms and approximation properties. Sampl. Theory Signal Image Process. 8 (1), 77–95 (2009)

    MathSciNet  MATH  Google Scholar 

  54. Kivinukk A., Tamberg, G.: On approximation properties of sampling operators by dilated kernels. In: 8th International Conference on Sampling Theory and Applications, SAMPTA’09, Marseille, May 18–22 (2009). Poster sessions, electronic access at www.latp.univ-mrs.fr/SAMPTA09/FinalSubmissions/187.pdf

  55. Knoop, H.B., Zhou, X.-L.: The lower estimate for linear positive operators, (II). Results Math. 25, 315–330 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  56. Lupas, A.: Some properties of the linear positive operators, I. Mathematica (Cluj) 9 (32), 77–83 (1967)

    MathSciNet  MATH  Google Scholar 

  57. Lupas, A.: Some properties of the linear positive operators, II. Mathematica (Cluj) 9 (32), 295–298 (1967)

    MathSciNet  MATH  Google Scholar 

  58. Marcinkiewicz, J.: Sur la divergence des polynômes d’interpolation. Acta Sci. Math. (Szeged) 8, 131–135 (1937)

    Google Scholar 

  59. Mastroianni, G., Szabados, J.: Jackson order of approximation by Lagrange interpolation. Suppl. Rend. Circ. Mat. Palermo II 33, 375–386 (1993)

    MathSciNet  MATH  Google Scholar 

  60. Meyer-König, W., Zeller, K.: Bernsteinsche Potenzreihen. Stud. Math. 19, 89–94 (1960)

    Google Scholar 

  61. Mirakjan, G.M.: Approximation des fonctions continues au moyen de polynômes de la forme \(e^{-nx}\sum _{k=0}^{n}C_{k,n}x^{k}\). Dokl. Akad. Nauk SSSR 31, 201–205 (1941, in French)

    Google Scholar 

  62. Moldovan, E.: Observations sur certains procédés d’interpolation généralisés (Romanian, Russian and French summaries). Acad. Republicii Pop. Romane Bul. Stiint. Sect. Stiint. Mat. Fiz. 6, 477–482 (1954)

    MathSciNet  MATH  Google Scholar 

  63. Muntean, I.: The Lagrange interpolation operators are densely divergent. Stud. Univ. Babes-Bolyai (Cluj) Math. 21, 28–30 (1976)

    Google Scholar 

  64. Păltănea, R.: The preservation of the property of quasiconvexity of higher order by Bernstein polynomials. Rev. d’Anal. Numér. Théor. Approx. 25 (1–2), 195–201 (1996)

    MathSciNet  MATH  Google Scholar 

  65. Plana, G.: Sur une nouvelle expression analytique des nombers Bernoulliens. Acad. Torino 25, 403–418 (1820)

    Google Scholar 

  66. Popoviciu, T.: About the Best Polynomial Approximation of Continuous Functions. Mathematical Monography, fasc. III. Sect. Mat. Univ., Cluj (1937, in Romanian)

    Google Scholar 

  67. Popoviciu, T.: On the proof of Weierstrass’ theorem with the interpolation polynomials. In: Acad. Republicii Pop. Romane “Lucrarile sesiunii generale stiintifice din 2–12 Iunie 1950”, vol. 1-4, 1664-1667 (1950) (in Romanian)

    Google Scholar 

  68. Prasad, J.: On the degree of approximation of the Hermite and Hermite-Fejér interpolation. Int. J. Math. Math. Sci. 15 (1), 47–56 (1992)

    Article  MATH  Google Scholar 

  69. Sklyarov, V.P.: On the best uniform sinc-approximation on a finite interval. East J. Approx. 14 (2), 183–192 (2008)

    MathSciNet  MATH  Google Scholar 

  70. Stens, R.L.: Approximation of functions by Whittaker’s cardinal series. In: Walter, W. (ed.) Proceedings of the Fourth International Conference on General Inequalities, Oberwolfach, Germany, May 1983. International Series of Numerical Mathematics, vol. 71, pp. 137–149. Birkhäuser Verlag, Basel (1984)

    Google Scholar 

  71. Szabados, J.: On the convergence and saturation problem of the Jackson polynomials. Acta Math. Acad. Sci. Hung. 24 (3–4), 399–406 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  72. Szabados, J., Vértesi, P.: Interpolation of Functions. World Scientific, Singapore/New Jersey/London/Hong Kong (1990)

    Book  MATH  Google Scholar 

  73. Szász, O.: Generalization of S.N. Bernstein’s polynomials to the infinite interval. J. Res. Natl. Bur. Stand. 45, 239–245 (1950)

    Google Scholar 

  74. Theis, M.: Über eine Interpolationsformel von de la Vallée-Poussin. Math. Z. 3, 93–113 (1919)

    Article  MathSciNet  MATH  Google Scholar 

  75. Totik, V.: Approximation by Bernstein polynomials. Am. J. Math. 116, 995–1018 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  76. Trynin, A.Y.: A criterion for the uniform convergence of sinc-approximation on a segment. Russ. Math. (Iz. VUZ) 52 (6), 58–69 (2008)

    Google Scholar 

  77. Vértesi, P.: On the convergence of Hermite-Fejér interpolation. Acta Math. Acad. Sci. Hung. 22, 151–158 (1971)

    Article  MATH  Google Scholar 

  78. Whittaker, E.T.: On the functions which are represented by expansions of the interpolation theory. Proc. R. Soc. Edinb. 35, 181–194 (1915)

    Article  MATH  Google Scholar 

  79. Vinti, G.: Approximation in Orlicz spaces for linear integral operators and applications. Suppl. Rend. Circ. Mat. Palermo II 76, 103–127 (2005)

    MathSciNet  MATH  Google Scholar 

  80. Vinti, G., Zampogni, L.: Approximation results for a general class of Kantorovich type operators. Adv. Nonlinear Stud. 14 (4), 991–1012 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  81. Xie, H.S., Jiang, D.: On the convergence rate of Hermite-Fejér interpolation at zeros of Jacobi polynomials. Soochow J. Math. 23 (3), 293–304 (1997)

    MathSciNet  MATH  Google Scholar 

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Bede, B., Coroianu, L., Gal, S.G. (2016). Introduction and Preliminaries. In: Approximation by Max-Product Type Operators. Springer, Cham. https://doi.org/10.1007/978-3-319-34189-7_1

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