A general integral

Dissertationes Mathematicae

Volumn , Issue 483, 2012, Pages 3-49

  Estrada, R ^a   Vindas, J ^b  

^a LOUISIANA STATE UNIVERSITY   (United States)

^b GHENT UNIVERSITY   (Belgium)

Author keywords

Distributional integration; Distributions; General integral; Non absolute integrals; ojasiewicz point values

Indexed keywords

EID: 84858817162     PISSN: 00123862     EISSN: 17306310     Source Type: Journal    
DOI: 10.4064/dm483-0-1     Document Type: Article

References (63)

1. P. Antosik, A. Kamiński and S. Sorek, On the distributional compositions of functions and distributions, Integral Transforms Spec. Funct. 20 (2009), 247-255.
2. P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier, Amsterdam, and PWN, Warszawa, 1973.
3. R. G. Bartle, A Modern Theory of Integration, Amer. Math. Soc., Providence, RI, 2001.
4. B. Bäumer, G. Lumer and F. Neubrander, Convolution kernels and generalized functions, in: Generalized Functions, Operator Theory, and Dynamical Systems (Brussels, 1997), Chapman & Hall/CRC Res. Notes Math. 399, Chapman & Hall/CRC, Boca Raton, FL, 1999, 68-78.
5. E. J. Beltrami and M. R. Wohlers, Distributions and the Boundary Values of Analytic Functions, Academic Press, New York, 1966.
6. H. Bremermann, Distributions, Complex Variables and Fourier Transforms, Addison- Wesley, Reading, MA, 1965.
7. J. Campos Ferreira, Introduction to the Theory of Distributions, Longman, Harlow, 1997.
8. J. Campos Ferreira, On some general notions of superior limit, inferior limit and value of a distribution at a point, Portugal. Math. 58 (2001), 139-158.
9. A. Denjoy, Sur l'intégration des coefficients différentiels d'ordre supérieur, Fund. Math. 25 (1935), 273-326.
10. J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1969.
11. Yu. N. Drozhzhinov and B. I. Zav'yalov, Tauberian theorems for generalized functions with values in Banach spaces, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 4, 47-118 (in Russian).English transl.: Izv. Math. 66 (2002), 701-769.
12. R. Estrada, Characterization of the Fourier series of a distribution having a value at a point, Proc. Amer. Math. Soc. 124 (1996), 1205-1212.
13. R. Estrada, The Cesàro behaviour of distributions, Proc. Roy. Soc. London Ser. A 454 (1998), 2425-2443.
14. R. Estrada, The nonexistence of regularization operators, J. Math. Anal. Appl. 286 (2003), 1-10.
15. R. Estrada, A distributional version of the Ferenc Lukács theorem, Sarajevo J. Math. 1 (13) (2005), 75-92.
16. R. Estrada, One-sided cluster sets of boundary values of analytic functions, Complex Var. Elliptic Equations 51 (2006), 661-673.
17. R. Estrada, Distributions that are functions, in: Linear and Non-Linear Theory of Generalized Functions and its Applications, Banach Center Publ. 88, Inst. Math., Polish Acad. Sci., Warszawa, 2010, 91-110.
18. R. Estrada and S. A. Fulling, Distributional asymptotic expansions of spectral functions and of the associated Green kernels, Electron. J. Differential Equations 1999, no. 07, 37 pp.
19. R. Estrada and S. A. Fulling, How singular functions define distributions, J. Phys. A 35 (2002), 3079-3089.
20. Corrigendum, ibid. 38 (2005), 7785.
21. R. Estrada and S. A. Fulling, Functions and distributions in spaces with thick points, Int. J. Appl. Math. Statist. 10 (2007), no. S07, 25-37.
22. R. Estrada and R. P. Kanwal, A distributional theory for asymptotic expansions, Proc. Roy. Soc. London Ser. A 428 (1990), 399-430.
23. R. Estrada and S. A. Fulling, A Distributional Approach to Asymptotics. Theory and Applications, 2nd ed., Birkhäuser, Boston, 2002.
24. R. Estrada and J. Vindas, Determination of jumps of distributions by differentiated means, Acta Math. Hungar. 124 (2009), 215-241.
25. R. Estrada and J. Vindas, On Romanovski's lemma, Real Anal. Exchange 35 (2010), 431-444.
26. G. W. Ford and R. F. O'Connell, Derivative of the hyperbolic tangent, Nature 380 (1996), 113.
27. R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Amer. Math. Soc., Providence, RI, 1994.
28. H. Hake, Über de la Vallée Poussins Ober- und Unterfunktionen einfacher Integrale und die Integraldefinition von Perron, Math. Ann. 83 (1921), 119-142.
29. E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier's Series, Vol. 1, Dover, New York, 1958.
30. R. P. Kanwal, Generalized Functions: Theory and Technique, 2nd ed., Birkhäuser, Boston, 1998.
31. S. Łojasiewicz, Sur la valeur et la limite d'une distribution en un point, Studia Math. 16 (1957), 1-36.
32. S. Łojasiewicz, Sur la fixation des variables dans une distribution, ibid. 17 (1958), 1-64.
33. P. Mikusiński and K. Ostaszewski, Embedding Henstock integrable functions into the space of Schwartz distributions, Real Anal. Exchange 14 (1988-89), 24-29.
34. O. P. Misra and J. L. Lavoine, Transform Analysis of Generalized Functions, North- Holland, Amsterdam, 1986.
35. I. P. Natanson, Theory of Functions of a Real Variable, Frederick Ungar, New York, 1955.
36. J. Peetre, On the value of a distribution at a point, Portugal. Math. 27 (1968), 149-159.
37. S. Pilipovic, B. Stankovic and A. Takaci, Asymptotic Behaviour and Stieltjes Transformation of Distributions, Teubner-Texte Math. 116, Teubner, Leipzig, 1990.
38. S. Pilipovic and J. Vindas, Multidimensional Tauberian theorems for wavelet and nonwavelet transforms, arXiv:1012.5090v2 [math.FA].
39. P. Romanovski, Essai d'une exposition de l'intégrale de Denjoy sans nombres transfinis, Fund. Math. 19 (1932), 38-44.
40. S. Saks, Theory of the Integral, 2nd ed., Dover, New York, 1964.
41. L. Schwartz, Théorie des Distributions, Hermann, Paris, 1966.
42. J. Sebastião e Silva, Integrals and orders of growth of distributions, in: Theory of Distributions (Lisbon, 1964), Inst. Gulbenkian Ci., Lisbon, 1964, 327-390.
43. R. Sikorski, Integrals of distributions, Studia Math. 20 (1961), 119-139.
44. R. S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, World Sci., Singapore, 2003.
45. E. Talvila, The distributional Denjoy integral, Real Anal. Exchange 33 (2008), 51-82.
46. J. Vindas, Structural theorems for quasiasymptotics of distributions at infinity, Publ. Inst. Math. (Beograd) (N.S.) 84 (98) (2008), 159-174.
47. J. Vindas, Local behavior of distributions and applications, Dissertation, Louisiana State Univ., Baton Rouge, 2009.
48. J. Vindas, The structure of quasiasymptotics of Søchwartz distributions, in: Linear and Non-Linear Theory of Generalized Functions and its Applications, Banach Center Publ. 88, Inst. Math., Polish Acad. Sci. Warszawa, 2010, 297-314.
49. J. Vindas and R. Estrada, Distributionally regulated functions, Studia Math. 181 (2007), 211-236.
50. J. Vindas and R. Estrada, Distributional point values and convergence of Fourier series and integrals, J. Fourier Anal. Appl. 13 (2007), 551-576.
51. J. Vindas and R. Estrada, On the jump behavior of distributions and logarithmic averages, J. Math. Anal. Appl. 347 (2008), 597-606.
52. J. Vindas and R. Estrada, Measures and the distributionaø-transform, Integral Transforms Spec. Funct. 20 (2009), 325-332.
53. J. Vindas and R. Estrada, On the support of tempered distributions, Proc. Edinburgh Math. Soc. 53 (2010), 255-270.
54. J. Vindas and R. Estrada, On the order of summability of the Fourier inversion formula, Anal. Theory Appl. 26 (2010), 13-42.
55. J. Vindas and S. Pilipovic, Structural theorems for quasiasymptotics of distributions at the origin, Math. Nachr. 282 (2009), 1584-1599.
56. V. S. Vladimirov, Methods of the Theory of Generalized Functions, Taylor & Francis, London, 2002.
57. V. S. Vladimirov, Yu. N. Drozhzhinov and B. I. Zavialov, Tauberian Theorems for Generalized Functions, Kluwer, Dordrecht, 1988.
58. G. Walter, Pointwise convergence of distribution expansions, Studia Math. 26 (1966), 143-154.
59. G. Walter, Abel summability for a distribution sampling theorem, in: Generalized Functions, Convergence Structures, and their Applications (Dubrovnik, 1987), Plenum Press, New York, 1988, 349-357.
60. G. Walter, Pointwise convergence of wavelet expansions, J. Approx. Theory 80 (1995), 108-118.
61. G. G. Walter and X. Shen, Wavelets and Other Orthogonal Systems, 2nd ed., Stud. Adv. Math., Chapman & Hall/CRC, Boca Raton, FL, 2001.
62. A. H. Zemanian, Generalized Integral Transformations, Interscience, New York, 1968.
63. Z. Zielezny, Über die Mengen der regulären und singulären Punkte einer Distribution, Studia Math. 19 (1960), 27-52.