Advertisement

Some Problems in Fourier Analysis and Approximation Theory

  • Michael RuzhanskyEmail author
  • Sergey Tikhonov
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We give a short overview of some questions and methods of Fourier analysis, approximation theory, and optimization theory that constitute an area of current research.

Keywords

Approximation theory Fourier analysis Harmonic analysis Optimization theory 

Notes

Acknowledgements

The first author “Michael Ruzhansky” was supported in parts by the EPSRC grant EP/K039407/1 and by the Leverhulme Grant RPG-2014-02. The second author “Sergey Tikhonov” was partially supported by MTM2014-59174-P, 2014 SGR 289, and RFFI 13-01-00043. We thank the authors for the assistance in writing this survey.

References

  1. 1.
    P. Ahern, W. Rudin, Bloch functions, BMO, and boundary zeros. Indiana Univ. Math. J. 36(1), 131–148 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    G.A. Akishev, Approximation of function classes in spaces with mixed norm. Sb. Math. 197(7–8), 1121–1144 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    K. Andersen, Inequalities with weights for discrete Hilbert transforms. Can. Math. Bull. 20, 9–16 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    P. Ausher, S. Hofmann, M. Lacey, A. McIntosh, Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on R n. Ann. Math. 2, 633–654 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Axelson, S. Keith, A. McIntosh, Quadratic estimates and functional calculi of perturbated dirac operators. Invent. Math. 163(3), 455–497 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    E.E. Berdysheva, Two related extremal problems for entire functions of several variables. Math. Notes 66(3), 271–282 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H. Bohman, Approximate Fourier analysis of distribution functions. Ark. Mat. 4, 99–157 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    E. Borel, Leçons sur les séries divergentes (Gaunter-Villars, Paris, 1901)zbMATHGoogle Scholar
  9. 9.
    P.G. Casazza, M. Fickus, J. Kovacevic, M.T. Leon, J.C. Tremain, A physical interpretation of finite frames. Appl. Numer. Harmon. Anal. 2–3, 51–76 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    P. Casazza, G. Kutyniok, Finite Frames: Theory and Applications (Birkhauser, New York, 2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    R.R. Coifman, Y. Meyer, Wavelets, Calderon-Zygmund and Multilinear Operators (Cambridge University Press, Cambridge, 1997)zbMATHGoogle Scholar
  12. 12.
    P. De Nápoli, I. Drelichman, Weighted convolution inequalities for radial functions. Ann. Mat. Pura Appl. 194, 167–181 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    E. Doubtsov, Bloch-to-BMOA compositions on complex balls. Proc. Am. Math. Soc. 140(12), 4217–4225 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    W. Ehm, T. Gneiting, D. Richards, Convolution roots of radial positive definite functions with compact support. Trans. Am. Math. Soc. 356, 4655–4685 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A.S. Kechris, A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness. London Mathematical Society lecture series, vol. 128 (Cambridge University Press, Cambridge, 1987)Google Scholar
  16. 16.
    E.G. Kwon, Hyperbolic mean growth of bounded holomorphic functions in the ball. Trans. Am. Math. Soc. 355(3), 1269–1294 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    J. Leray, Hyperbolic Differential Equations (Institute for Advanced Study, Princeton, 1953)zbMATHGoogle Scholar
  18. 18.
    E. Liflyand, S. Tikhonov, Weighted Paley-Wiener theorem on the Hilbert transform. C.R. Acad. Sci. Paris, Ser. I 348, 1253–1258 (2010)Google Scholar
  19. 19.
    E. Liflyand, S. Tikhonov, A concept of general monotonicity and applications. Math. Nachr. 284, 1083–1098 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    E. Liflyand, W. Trebels, On asymptotics for a class of radial Fourier transforms. Z. Anal. Anwen. 17, 103–114 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    P.L. Lions, Symétrie e compacité dans les espaces de Sobolev. J. Funct. Anal. 49, 315–334 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    B.F. Logan, Extremal problems for positive-definite bandlimited functions. I. Eventually positive functions with zero integral. SIAM J. Math. Anal. 14(2), 249–252 (1983)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Y.M. Molokovich, P.P. Osipov, Basics of Relaxation Filtration Theory (Kazan University, Kazan, 1987)Google Scholar
  24. 24.
    W.M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications. Indiana Univ. Math. J. 31(6), 801–807 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    E.D. Nursultanov, N.T. Tleukhanova, On the approximate computation of integrals for functions in the spaces W p α([0, 1]n). Russ. Math. Surv. 55(6), 1165–1167 (2000)Google Scholar
  26. 26.
    M.G. Plotnikov, Quasi-measures, Hausdorff p-measures and Walsh and Haar Series. Izv. RAN: Ser. Mat. 74, 157–188 (2010); Engl. Transl. Izvestia: Math. 74, 819–848 (2010)Google Scholar
  27. 27.
    M. Plotnikov, V. Skvortsov, On various types of continuity of multiple dyadic integrals, Acta Math. Acad. Paedagog. Nyházi (N. S.) (2015, to appear)Google Scholar
  28. 28.
    A.N. Podkorytov, Linear means of spherical Fourier sums, in Operator Theory and Function Theory, ed. by M.Z. Solomyak, vol. 1 (Leningrad University, Leningrad) (1983), pp. 171–177 (Russian)Google Scholar
  29. 29.
    G. Polya, Untersuchungen über Lücken and Singularitäten von Potenzsrihen. Math. Zeits. Bd. 29, 549–640 (1929)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    M.K. Potapov, B.V. Simonov, S.Yu. Tikhonov, Mixed moduli of smoothness in L p, \(1 <p <\infty\): a survey. Surv. Approx. Theory 8, 1–57 (2013)Google Scholar
  31. 31.
    M.K. Potapov, B.V. Simonov, S.Yu. Tikhonov, Relations between the mixed moduli of smoothness and embedding theorems for Nikol’skii classes, in Proceeding of the Steklov Institute of Mathematics, vol. 269 (2010), pp. 197–207; translation from Russian: Trudy Matem. Inst. V.A. Steklova 269, 204–214 (2010)Google Scholar
  32. 32.
    W. Ramey, D. Ullrich, Bounded mean oscillation of Bloch pull-backs. Math. Ann. 291(4), 591–606 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, New York, 1993)zbMATHGoogle Scholar
  34. 34.
    J.H. Shapiro, Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics (Springer, New York, 1993)Google Scholar
  35. 35.
    B. Simonov, S. Tikhonov, Sharp Ul’yanov-type inequalities using fractional smoothness. J. Approx. Theory 162, 1654–1684 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    V.A. Skvortsov, Henstock-Kurzweil type integrals in \(\mathcal{P}\)-adic harmonic analysis. Acta Math. Acad. Paedagog. Nyházi (N. S.) 20, 207–224 (2004)MathSciNetzbMATHGoogle Scholar
  37. 37.
    W.A. Strauss, Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    S. Tikhonov, Trigonometric series with general monotone coefficients. J. Math. Anal. Appl. 326, 721–735 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    S. Tikhonov, Best approximation and moduli of smoothness: computation and equivalence theorems. J. Approx. Theory 153, 19–39 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    S. Tikhonov, Weak type inequalities for moduli of smoothness: the case of limit value parameters. J. Fourier Anal. Appl. 16(4), 590–608 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    P. Villarroya, A characterization of compactness for singular integrals. J. Math. Pure Appl. (arXiv:1211.0672) (to appear)Google Scholar
  42. 42.
    S. Yamashita, Hyperbolic Hardy class H 1. Math. Scand. 45(2), 261–266 (1979)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.ICREA, Centre de Recerca Matemàtica, and UABBellaterra (Barcelona)Spain

Personalised recommendations