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Boas-Type Formulas and Sampling in Banach Spaces with Applications to Analysis on Manifolds

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New Perspectives on Approximation and Sampling Theory

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

My teachers were Vladimir Abramovich Rokhlin (my Master’s thesis advisor) and Selim Grigorievich Krein (my Ph.D. thesis advisor). I first met Paul Butzer when I was about 50 years old, but I also consider him as my teacher since his work had an enormous influence on my carrier and ultimately on my life.

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References

  1. Berens, H., Butzer, P.L., Pawelke, S.: Limitierungsverfahren von Reihen mehr dimensionaler Kugelfunktionen und deren Saturationsverhalten (German). Publ. Res. Inst. Math. Sci. Ser. A 4, 201–268 (1968/1969)

    Google Scholar 

  2. Boas, R.: The derivative of a trigonometric integral. J. Lond. Math. Soc. 1–12, 164 (1937)

    Article  MathSciNet  Google Scholar 

  3. Boas, R.: Entire Functions. Academic, New York (1954)

    MATH  Google Scholar 

  4. Butzer, P.L., Berens, H.: Semi-Groups of Operators and Approximation. Springer, Berlin (1967)

    Book  MATH  Google Scholar 

  5. Butzer, P.L., Scherer, K.: Approximation theorems for sequences of commutative operators in Banach spaces. In: Constructive Theory of Functions (Proc. Internat. Conf., Varna, 1970) (Russian), pp. 137–146. Izdat. Bolgar. Akad. Nauk, Sofia (1972)

    Google Scholar 

  6. Butzer, P.L., Scherer, K.: Jackson and Bernstein-type inequalities for families of commutative operators in Banach spaces. J. Approx. Theory 5, 308–342 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  7. Butzer, P.L., Splettstößer, W., Stens, R.: The sampling theorem and linear prediction in signal analysis. Jahresber. Deut. Math. Ver. 90, 1–70 (1988)

    MATH  Google Scholar 

  8. Butzer, P.L., Schmeisser, G., Stens, R.: Shannons sampling theorem for bandlimited signals and their Hilbert transform, boas-type formulae for higher order derivatives-the aliasing error involved by their extensions from bandlimited to non-bandlimited signals. Entropy 14, 2192–2226 (2012). doi:10.3390/e14112192

    Article  MATH  MathSciNet  Google Scholar 

  9. Folland, G.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  10. Geller, D., Pesenson, I.: Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21(2), 334–337 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Helgason, S.: Groups and Geometric Analysis. Academic, New York (1984)

    MATH  Google Scholar 

  12. Krein, S.G.: Lineikhye differentsialnye uravneniya v Banakhovom prostranstve (Russian) [Linear Differential Equations in a Banach Space], 464 pp. Izdat. “Nauka”, Moscow (1967) [English translation: Linear differential equations in Banach space. Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, vol. 29, v+390 pp. American Mathematical Society, Providence, (1971)]

    Google Scholar 

  13. Krein, S.G., Petunin, Ju.I.: Scales of Banach spaces (Russian). Uspehi Mat. Nauk 21(2), 89–168 (1966)

    MathSciNet  Google Scholar 

  14. Krein, S., Pesenson, I.: Interpolation Spaces and Approximation on Lie Groups. The Voronezh State University, Voronezh (1990)

    Google Scholar 

  15. Krein, S.G., Petunin, Ju.I., Semenov, E.M.: Interpolyatsiya lineinykh operatorov (Russian) [Interpolation of Linear Operators], 400 pp. “Nauka”, Moscow (1978) [English translation: Krein, S., Petunin, Y., Semenov, E.: Interpolation of linear operators. Translations of Mathematical Monographs, vol. 54. AMS, Providence (1982)]

    Google Scholar 

  16. Nelson, E.: Analytic vectors. Ann. Math. (2) 70, 572–615 (1959)

    Google Scholar 

  17. Pesenson, I.: Moduli of continuity on Lie groups (Russian). In: Collection of Articles on Applications of Functional Analysis (Russian), pp. 115–121. Voronezh. Tehnolog. Inst., Voronezh (1975)

    Google Scholar 

  18. Pesenson, I.: Interpolation of noncommuting operators (Russian). Uspehi Mat. Nauk 33 3(201), 183–184 (1978)

    Google Scholar 

  19. Pesenson, I.: Interpolation spaces on Lie groups (Russian). Dokl. Akad. Nauk SSSR 246(6), 1298–1303 (1979)

    MathSciNet  Google Scholar 

  20. Pesenson, I.: Nikolskii- Besov spaces connected with representations of Lie groups (Russian). Dokl. Akad. Nauk SSSR 273(1), 45–49 (1983)

    MathSciNet  Google Scholar 

  21. Pesenson, I.: On the abstract theory of Nikolskii-Besov spaces (Russian). Izv. Vyssh. Uchebn. Zaved. Mat. 1988(6), 59–68 (1988) [Translation in Soviet Math. (Iz. VUZ) 32(6), 80–92 (1988)]

    Google Scholar 

  22. Pesenson, I.: The best approximation in a representation space of a lie group. Dokl. Acad. Nauk USSR 302(5), 1055–1059 (1988) [English Translation in Soviet Math. Dokl. 38(2), 384–388 (1989)]

    Google Scholar 

  23. Pesenson, I.: Approximations in the representation space of a Lie group (Russian). Izv. Vyssh. Uchebn. Zaved. Mat. 1990(7), 43–50 (1990) [Translation in Soviet Math. (Iz. VUZ) 34(7), 49–57 (1990)]

    Google Scholar 

  24. Pesenson, I.: The Bernstein inequality in the space of representation of Lie group. Dokl. Acad. Nauk USSR 313, 86–90 (1990) [English translation in Soviet Math. Dokl. 42 (1991)]

    Google Scholar 

  25. Pesenson, I.: A sampling theorem on homogeneous manifolds. Trans. Am. Math. Soc. 352(9), 4257–4269 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Pesenson, I.: Sampling of band limited vectors. J. Fourier Anal. Appl. 7(1), 93–100 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pesenson, I.: Bernstein-Nikolskii inequalities and Riesz interpolation formula on compact homogeneous manifolds. J. Approx. Theory 150(2), 175–198 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Pesenson, I.: Bernstein-Nikolskii and Plancherel-Polya inequalities in Lp-norms on non-compact symmetric spaces. Math. Nachr. 282(2), 253–269 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Pesenson, I.: Paley-Wiener approximations and multiscale approximations in Sobolev and Besov spaces on manifolds. J. Geom. Anal. 19(2), 390–419 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  30. Pesenson, I., Geller, D.: Cubature formulas and discrete fourier transform on compact manifolds. In: Farkas, H.M., Gunning, R.C., Knopp, M.I., Taylor, B.A. (eds.) From Fourier Analysis and Number Theory to Radon Transforms and Geometry: In Memory of Leon Ehrenpreis. Developments in Mathematics, vol. 28. Springer, New York (2013)

    Chapter  Google Scholar 

  31. Pesenson, I., Pesenson, M.: Approximation of Besov vectors by Paley-Wiener vectors in Hilbert spaces. In: Neamtu, M., Schumaker, L. (eds.) Approximation Theory XIII: San Antonio 2010. Springer Proceedings in Mathematics, pp. 249–263. Springer, New York (2012)

    Google Scholar 

  32. Pesenson, I., Zayed, A.: PaleyWiener subspace of vectors in a Hilbert space with applications to integral transforms. J. Math. Anal. Appl. 353, 566–582 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  33. Riesz, M.: Eine trigonometrische Interpolationsformel und einige Ungleichungen fur Polynome. Jahresber. Deut. Math. Ver. 23, 354–368 (1914)

    MATH  Google Scholar 

  34. Riesz, M. Les fonctions conjuguees et les series de Fourier. C. R. Acad. Sci. 178, 1464–1467 (1924)

    MATH  Google Scholar 

  35. Schmeisser, G.: Numerical differentiation inspired by a formula of R. P. Boas. J. Approx. Theory 160, 202–222 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zelobenko, D.: Compact Lie groups and their representations. Translations of Mathematical Monographs, vol. 40, viii+448 pp. American Mathematical Society, Providence (1973)

    Google Scholar 

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Acknowledgements

The author was supported in part by the National Geospatial-Intelligence Agency University Research Initiative (NURI), grant HM1582-08-1-0019.

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Authors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA

    Isaac Z. Pesenson

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  1. Isaac Z. Pesenson
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Correspondence to Isaac Z. Pesenson .

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Editors and Affiliations

  1. Department of Mathematical Sciences, DePaul University, Chicago, Illinois, USA

    Ahmed I. Zayed

  2. Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany

    Gerhard Schmeisser

Additional information

Dedicated to 85th Birthday of my teacher Paul Butzer

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Pesenson, I.Z. (2014). Boas-Type Formulas and Sampling in Banach Spaces with Applications to Analysis on Manifolds. In: Zayed, A., Schmeisser, G. (eds) New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-08801-3_3

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