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The Wiener algebra of absolutely convergent Fourier integrals: an overview

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Abstract

In this survey, results on the representation of a function as an absolutely convergent Fourier integral are collected, classified and discussed. Certain applications are also given.

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References

  1. Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Fizmatgiz, Moscow (1961, in Russian). English transl.: Oliver & Boyd, Edinburgh and London (1965)

  2. Askey, R.: Radial characteristic functions. Tech. Report No. 1262, Math. Research Center, University of Wisconsin-Madison (1973)

  3. Balan R., Krishtal I.: An almost periodic noncommutative Wiener’s lemma. J. Math. Anal. Appl. 370, 339–349 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bary, N.K.: A Treatise on Trigonometric Series. Fizmatgiz, Moscow (1961, in Russian). English transl.: Pergamon Press, MacMillan, New York (1964)

  5. Beckenbach, E.F., Bellman, R.: Inequalities. Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 30. Springer, Berlin (1961)

  6. Belinsky, E.S., Trigub, R.M.: Summability on a Lebesgue set and a Banach algebra. Theory Func. Approx., Saratov 1982 Winter School Proc. Part II, pp. 29–34 (1983, in Russian)

  7. Belinsky E.S., Dvejrin M.Z., Malamud M.M.: Multipliers in L 1 and estimates for systems of differential operators. Russ. J. Math. Phys. 12, 6–16 (2005)

    MathSciNet  MATH  Google Scholar 

  8. Belinsky E.S., Liflyand E., Trigub R.M.: The Banach algebra A* and its properties. J. Fourier Anal. Appl. 3, 103–129 (1997)

    Article  MathSciNet  Google Scholar 

  9. Berens H., Görlich E.: Über einen Darstellungssatz für Funktionen als Fourierintegrale und Anwendungen in der Fourieranalysis. Tôhoku Math. J. 18(2), 429–453 (1966)

    Article  MATH  Google Scholar 

  10. Berens H., Xu Y.: l-1 summability of multiple Fourier integrals and positivity. Math. Proc. Cambridge Philos. Soc. 122, 149–172 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bernstein, S.N.: On majorants of finite or quasi-finite growth. Dokl. Akad. Nauk SSSR. 65, 117–120 (1949, in Russian)

    Google Scholar 

  12. Bernstein, S.N.: Complete Works, vol. II. Constructive Function Theory [1931–1953]. Izdat. Akad. Nauk SSSR Moscow (1954, in Russian)

  13. Berry A.C.: Necessary and sufficient conditions in the theory of Fourier transforms. Ann. Math. 32, 830–838 (1931)

    Article  Google Scholar 

  14. Besov, O.V.: Hörmander’s theorem on Fourier multipliers. Trudy Mat. Inst. Steklov 173, 164–180 (1986, in Russian). English transl.: Proc. Steklov Inst. Math. 4, 4–14 (1987)

  15. Besov, O.V., Il’in, V.P., Nikol’skii, S.M.: Integral Representations of Functions and Embedding of Functions, 2nd edn. Nauka, Fizmatlit, Moscow (1996, in Russian)

  16. Besov, O.V., Lizorkin, P.I.: Singular integral operators and sequences of convolutions in L p -spaces. Mat. Sb. (N.S.) 73(115), 65–88 (1967, Russian). English transl.: Math. USSR-Sbornik 2(1), 57–76 (1967)

  17. Beurling, A.: Sur les integrales de Fourier absolument convergentes et leur application à une transformation fonctionelle. Proc. IX Congrès de Math. Scand. Helsingfors, pp. 345–366 (1938)

  18. Beurling A.: On the spectral synthesis of bounded functions. Acta Math. 81, 225–238 (1949)

    Article  MATH  Google Scholar 

  19. Beurling A.: Construction and analysis of some convolution algebras. Ann. Inst. Fourier 14, 1–32 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bochkarev, S.V.: On a problem of Zygmund. Izv. AN SSSR 37, 630–638 (1973, in Russian). English transl. in Math. USSR-Izvestiya 7, 629–637 (1973)

  21. Bochner S.: Summation of multiple Fourier series by spherical means. Trans. Am. Math. Soc. 40, 175–207 (1936)

    Article  MathSciNet  Google Scholar 

  22. Bochner S.: Lectures on Fourier Integrals. Princeton University Press, Princeton (1959)

    MATH  Google Scholar 

  23. Boas R.P.: Absolute convergence and integrability of trigonometric series. J. Ration. Mech. Anal. 5, 621–632 (1956)

    MathSciNet  MATH  Google Scholar 

  24. Borwein D.: Linear functionals connected with strong Cesáro summability. J. Lond. Math. Soc. 40, 628–634 (1965)

    Article  MATH  Google Scholar 

  25. Brenner Ph., Thomée V., Wahlbin L.B.: Besov Spaces and Applications to Difference Methods for Initial Value Problems. Lecture Notes in Mathematics, vol. 434. Springer, Berlin (1975)

    Google Scholar 

  26. Bugrov, Ya.S.: Summability of Fourier transforms and absolute convergence of multiple Fourier series. Trudy Mat. Inst. Steklov. 187, 22–30 (1989, in Russian). English transl.: Proc. Steklov Inst. Math. Studies in the theory of differentiable functions of several variables and its applications 13(3), 25–34 (1990)

  27. Buhmann M.D.: A new class of radial basis functions with compact support. Math. Comp. 70(23), 307–318 (2000)

    Article  MathSciNet  Google Scholar 

  28. Buhmann, M.D.: Radial Basis Functions: Theory and Implementation. Cambridge University Press (2004)

  29. Butzer, P.L., Nessel, R.J.: Fourier analysis and approximation. One-Dimensional Theory, vol. 1. Pure and Applied Mathematics, vol. 40. Academic Press, New York (1971)

  30. Butzer P.L., Nessel R.J., Trebels W.: On summation processes of Fourier expansions in Banach spaces, II. Saturation theorems. Tohoku Math. J. 24(2), 551–569 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  31. Carleman, T.: L’intégrale de Fourier et Questions qui s’y rattachent. Almquist and Wiksells Boktyckere (1944)

  32. Carleson, L.: Appendix to the paper of J.-P. Kahane and Y. Katznelson. Stud. Pure Math. Mem. P. Turan, Budapest, pp. 411–413 (1983)

  33. Chelidze V.G.: On the absolute convergence of double Fourier series. Doklady AN SSSR 54(2), 117–120 (1946)

    MathSciNet  MATH  Google Scholar 

  34. Cossar J.: A theorem on Cesàro summability. J. Lond. Math. Soc. 16, 56–68 (1941)

    Article  MathSciNet  Google Scholar 

  35. Dappa H., Trebels W.: On L 1-criteria for quasiradial Fourier multipliers with applications to some anisotropic function spaces. Anal. Math. 9, 275–289 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  36. Dappa H., Trebels W.: On hypersingular integrals and anisotropic Bessel potential spaces. Trans. Am. Math. Soc. 286, 419–429 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  37. Duoandikoetxea, J.: Fourier Analysis. Grad. Studies Math., vol. 29. AMS, Providence (2001)

  38. Edwards R.E.: Criteria for Fourier transforms. J. Australian Math. Soc. 7, 239–246 (1967)

    Article  MATH  Google Scholar 

  39. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G.: Tables of Integral Transforms, vols. 1, 2. McGraw-Hill, New York (1954)

    Google Scholar 

  40. Fefferman Ch.: Inequalities for strongly singular convolution operators. Acta Math. 124, 9–36 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  41. Feichtinger H.G.: A characterization of Wiener’s algebra on locally compact groups. Arch. Math. (Basel) 29, 136–140 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  42. Flett T.M.: Temperatures, Bessel potentials and Lipschitz spaces. Proc. Lond. Math. Soc. 22, 385–451 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  43. Fomin, G.A.: A class of trigonometric series. Mat. Zametki 23, 213–222 (1978, in Russian). English transl.: Math. Notes 23, 117–123 (1978)

    Google Scholar 

  44. Fridli S.: Hardy spaces generated by an integrability condition. J. Approx. Theory 113, 91–109 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  45. Fridli S.: An inverse Sidon type inequality. Stud. Math. 105, 283–308 (1993)

    MathSciNet  MATH  Google Scholar 

  46. Gabisoniya O.D.: On absolute convergence of double Fourier series and integrals. Soobshch. AN GSSR 42, 3–9 (1966) (in Russian)

    Google Scholar 

  47. Geĭsberg S.P.: Fractional derivatives of functions bounded on the axis. Izv. Vysš. Učebn. Zaved. Matematika 11(78), 51–69 (1968) (in Russian)

    Google Scholar 

  48. Gel’fand I.M., Raikov A.A., Shilov G.E.: Commutative normed rings. Uspekhi Matem. Nauk. I, 48–146 (1946) (Russian)

    Google Scholar 

  49. Gel’fand I.M., Raikov, A.A., Shilov, G.E.: Commutative Normed Rings. Moscow, GIFML (1960, in Russian). English transl.: AMS, Chelsea, Bronx (1964)

  50. Giang D.V., Móricz F.: Lebesgue integrability of double Fourier transforms. Acta Sci. Math. (Szeged) 58, 299–328 (1993)

    MathSciNet  MATH  Google Scholar 

  51. Gneiting T.: Criteria of Pólya type for radial positive definite functions. Proc. Am. Math. Soc. 129, 2309–2318 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  52. Gohberg, I., Fel’dman, I.A.: Convolution type equations and projection methods for their solution. Nauka, Moscow (1971, in Russian). English transl.: Amer. Math. Soc. Transl. of Math. Monographs, Providence, R.I., vol. 41 (1974)

  53. Gohberg, I., Krein, M.: The fundamentals on defect numbers, root numbers, and indices of linear operators. Uspekhi Matem. Nauk 12, 44–118 (1957, in Russian). English transl.: Amer. Math. Soc. Transl. 13, 185–264 (1960)

  54. Gol’dman, M.L.: An isomorphism of generalized H ölder classes. Diff. Uravnenija 7, 1449–1458, 1541 (1971, in Russian). English transl.: Differ. Equ. 7, 1100–1107 (1973)

    Google Scholar 

  55. Gol’dman, M.L.: Generalized kernels of fractional order. Diff. Uravnenija 7, 2199–2210 (1971, in Russian). English transl.: Differ. Equ. 7, 1655–1664 (1974)

    Google Scholar 

  56. Goldman, M.L.: Estimates for Multiple Fourier Transforms of Radially Symmetric Monotone Functions. Sibirsk. Mat. Ž. 18, 549–569 (1977, in Russian). English transl.: Sib. Math. J. 18, 391–406 (1978)

  57. Golovkin, K.K., Solonnikov, V.A.: Estimates of convolution operators. Zap. Nauch. Semin. LOMI. 7, 6–86 (1968, in Russian). English transl.: Semin. Math., Steklov Math. Inst., Leningrad 7, 1–36 (1968)

  58. Golovkin, K.K.: Uniform equivalence of parametric norms in ergodic and approximation theories. Izvestiya AN SSSR 35(4), 900–921 (1971, in Russian). English transl.: Math. USSR-Izvestiya 5(4), 915–934 (1972)

  59. Golubov, B.I.: Multiple series and Fourier integrals. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow. Mathematical analysis, vol. 19, 3–54, 232 (1982, in Russian). English transl.: J. Soviet Math. 24, 639–673 (1984)

  60. Gradshtein, I.S., Ryzhik, I.M.: Tables of Integrals, Sums, Series and Products, 5th edn. Academic Press (1994)

  61. Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, Upper Saddle River (2004)

  62. Grishin, A.F., Skoryk, M.V.: Some properties of Fourier integrals. ArXiv: http://arxiv.org/abs/1108.2890v (2011, in Russian)

  63. Hardy G.H., Littlewood J.E., Pólya G.: Inequalities. Cambridge University Press, Cambridge (1934)

    Google Scholar 

  64. Herz, C.S.: Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms. J. Math. Mech. 18, 283–323

  65. Izumi S.I., Tsuchikura T.: Absolute convergence of trigonometric expansions. Tôhoku Math. J. 7, 243–251 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  66. Kahane J.-P.: Séries de Fourier absolument convergentes. Springer, Berlin (1970)

    MATH  Google Scholar 

  67. Karapetyants, A.: On a generalization of the Beurling inequality. Unpublished manuscript, Rostov State University (1992, in Russian)

  68. Koldobsky, A.: Schoenberg’s problem on positive definite functions. Algebra i Analiz 3, 78–85 (1991, in Russian). English transl.: St.-Petersburg Math. J. 3, 563–570 (1991)

  69. Koldobsky, A.: Fourier Analysis in Convex Geometry. (Math. Surveys and Monographs.) AMS (2005)

  70. Kolmogorov A.N.: Une série de Fourier-Lebesgue divergente partout. C.R. Acad. Sci. Paris 183, 1327–1328 (1926)

    Google Scholar 

  71. Kolomoitsev, Yu., Liflyand, E.: Sufficient conditions for absolute convergence of multiple Fourier integrals. ArXiv: http://arxiv.org/abs/1108.5470v (2010)

  72. Kostetskaya, G.S., Samko, S.G.: A sufficient condition for a function to belong to the Wiener ring \({\mathcal{R}\it ({R}^n )}\) of Fourier integrals of absolutely integrable functions I. Deposited in VINITI, no. 3483-86 (1986, in Russian)

  73. Kostetskaya, G.S., Samko, S.G.: A sufficient condition for a function to belong to the Wiener ring \({\mathcal{R}\it ({R}^n )}\) of Fourier integrals of absolutely integrable functions, II. Deposited in VINITI, no. 5801-86 (1986, in Russian)

  74. Kostetskaya, G.S., Samko, S.G.: Inversion of potential type operators with a difference characteristic. Deposited in VINITI, no. 5800-86 (1986, in Russian)

  75. Kostetskaya, G.S., Samko, S.G.: A criterion for absolute integrability of Fourier integrals. Izv. Vuzov. Matematika. 3, 72–75 (1988, in Russian). English transl.: Soviet Math. (Izv. VUZ) 32(3), 103–108 (1988)

  76. Krein M.G.: On the problem of extension of Hermite-positive continuous functions. Doklady AN SSSR 26(1), 17–20 (1940) (in Russian)

    MathSciNet  Google Scholar 

  77. Krein M.G.: On the representation of functions by Fourier-Stieltjes integrals. Uchenye Zapiski Kuibyshev Gos. Pedagog. Inst. 7, 123–148 (1943) (in Russian)

    Google Scholar 

  78. Kuznetsova, O.I.: On the integrability of a class of N-dimensional trigonometric series. Ukr. Mat. Zh. 52, 837–840 (2000, in Russian). English transl.: Ukr. Math. J. 52, 960–963 (2000)

  79. Larsson, L., Maligranda, L., Pecaric, J., Persson, L.E.: Multiplicative Inequalities of Carlson Type and Interpolation. World Scientific, Singapore (2006)

  80. Lebedev, V.V., Olevskii, A.M.: L p-Fourier multipliers with bounded powers. Izv. RAN. Ser. Mat. 70(3), 129–166 (2006, in Russian). English transl.: Izvestiya: Mathematics 70(3), 549–585 (2006)

    Google Scholar 

  81. Liflyand, E.R.: Some questions of absolute convergence of multiple Fourier integrals. Theory of functions and mappings. Naukova Dumka, Kiev, pp. 110–132, 179 (1979, in Russian)

  82. Liflyand E.: Fourier transform of functions from certain classes. Anal. Math. 19, 151–168 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  83. Liflyand E.: Fourier Transforms of Radial Functions. Integr. Transf. Special Funct. 4, 279–300 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  84. Liflyand E.: On quasi-monotone functions and sequences. Comput. Methods Funct. Theory 1, 345–352 (2002)

    MathSciNet  Google Scholar 

  85. Liflyand, E.: Lebesgue constants of multidimensional Fourier series. Online J. Anal. Combin. 1, Art. 5, 112 pp (2006)

    Google Scholar 

  86. Liflyand E.: Necessary conditions for integrability of the Fourier transform. Georgian Math. J. 16, 553–559 (2009)

    MathSciNet  MATH  Google Scholar 

  87. Liflyand, E.: On absolute convergence of Fourier integrals. Real Anal. Exchange (accepted)

  88. Liflyand E., Ournycheva E.: Two spaces conditions for integrability of the Fourier transform. Analysis 19, 331–366 (2001)

    Google Scholar 

  89. Liflyand E., Trebels W.: On asymptotics for a class of radial Fourier transforms. ZAA 17, 103–114 (1998)

    MathSciNet  MATH  Google Scholar 

  90. Liflyand, E., Trigub, R.: Known and new results on absolute integrability of Fourier integrals. Preprint CRM 859, 29 pp (2009)

    Google Scholar 

  91. Liflyand, E., Trigub, R.: On the representation of a function as an absolutely convergent Fourier integral. Trudy Mat. Inst. Steklov 269, 153–166 (2010, in Russian). English transl.: Proc. Steklov Inst. Math. 269, 146–159 (2010)

  92. Liflyand E., Trigub R.: Conditions for the absolute convergence of Fourier integrals. J. Approx. Theory 163, 438–459 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  93. Lizorkin, P.I.: On the theory of Fourier multipliers. Studies in the theory of differentiable functions of several variables and its applications, 11. Trudy Mat. Inst. Steklov. 173, 149–163 (1986, in Russian). English transl.: Proc. Steklov Inst. Math. 4, 161–176 (1987)

  94. Lizorkin, P.I.: Limit cases of theorems on \({\mathcal{F} L_p}\) -multipliers. Trudy Mat. Inst. Steklov 173, 164–180 (1986, in Russian). English transl.: Proc. Steklov Inst. Math. 4, 177–194 (1987)

  95. Löfström J.: Some theorems on interpolation spaces with applications to approximations in L p . Math. Ann. 172, 176–196 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  96. Löfström J.: Besov spaces in theory of approximation. Ann. Mat. Pura Appl. 85(4), 93–184 (1970)

    MathSciNet  MATH  Google Scholar 

  97. Long F.-L.: Sommes partielles de Fourier des fonctions born‘ees. C.R. Acad. Sc. Paris Ser. A 288, 1009–1011 (1979)

    MATH  Google Scholar 

  98. Lucacs, E.: Characteristic Functions, 2nd edn. Charles Griffin & Co. Ltd, London (1970)

  99. Madych W.R.: On Littlewood-Paley functions. Stud. Math. 50, 43–63 (1974)

    MathSciNet  MATH  Google Scholar 

  100. Madych W.R.: Absolute summability of Fourier transforms on R n. Indiana Univ. Math. J. 25, 467–479 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  101. Mateu, J., Porcu, E.: Positive Definite Functions: From Schoenberg to Space-Time Challenges. Universitat Jaume I (2008)

  102. Miyachi A.: On some Fourier multipliers for \({H^p(\mathbb R^n)}\) . J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 157–179 (1980)

    MathSciNet  MATH  Google Scholar 

  103. Miyachi A.: On some singular Fourier multipliers. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28, 267–315 (1981)

    MathSciNet  MATH  Google Scholar 

  104. Mukherjee R.N.: A note on the criteria for absolute integrability of Fourier transforms. Aligarh Bull. Math. 9(10), 57–59 (1980)

    Google Scholar 

  105. Müller, C.: Spherical Harmonics. Lect. Notes in Math., vol. 17. Springer, Berlin (1966)

  106. Musielak J.: On the absolute convergence of multiple Fourier series. Ann. Polon. Math. 5, 107–120 (1958)

    MathSciNet  MATH  Google Scholar 

  107. Nessel R.J., Trebels W.: Multipliers with respect to spectral measures in Banach spaces and approximation. II. One-dimensional Fourier multipliers. J. Approx. Theory 14, 23–29 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  108. Nikol’skii, S.M.: The Approximation of Functions of Several Variables and the Imbedding Theorems, 2nd edn. Nauka, Moscow (1977, in Russian). English transl. of 1st edn.: Wiley, New York (1978)

  109. Peetre J.: Applications de la théorie des espaces d’interpolation dans l’analyse harmonique. Ricerche Mat. 15, 3–36 (1966)

    MathSciNet  MATH  Google Scholar 

  110. Peetre, J.: New thoughts on Besov spaces. Duke Univ. Math. Series, No. 1. Math. Dept. Duke Univ. Durham, NC (1976)

  111. Pitt H.R.: A note on the representation of functions by absolutely convergent Fourier integrals. Proc. Cambridge Phil. Soc. 44, 8–12 (1940)

    Article  MathSciNet  Google Scholar 

  112. Reiter H., Stegeman J.D.: Classical Harmonic Analysis and Locally Compact Groups. Clarendon Press, Oxford (2000)

    MATH  Google Scholar 

  113. Ryan R.: Fourier transforms of certain classes of integrable functions. Trans. Am. Math. Soc. 105, 102–111 (1962)

    Article  MATH  Google Scholar 

  114. Salem, R.: Essais sur les séries trigonometriques. Actual. Sci. et Industr., vol. 862. Paris (1940)

  115. Samko, S.G.: The spaces \({L_{p, r}^{\alpha}({R}^n)}\) and hypersingular integrals. Studia Math. 61, 193–230 (1977, in Russian)

    Google Scholar 

  116. Samko, S.G.: Hypersingular integrals and their applications. Rostov-on-Don, Izdat. Rostov Univ. (1984, in Russian)

  117. Samko, S.G.: Hypersingular integrals and differences of fractional order. Trudy Mat. Inst. Steklov. 192, 164–182 (1990, in Russian). English transl.: Proc. Steklov Inst. Math. 192, 175–194 (1992)

  118. Samko, S.G.: On absolute integrability of Fouriers transforms in terms of fractional derivatives. Unpublished manuscript, Rostov State University (1992, in Russian)

  119. Samko, S.G.: Hypersingular Integrals and their Applications. Analytical Methods and Special Functions, vol. 5. Taylor & Francis, London-New-York (2002)

  120. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, London-New-York (1993)

  121. Samko, S.G., Umarkhadzhiev, S.M.: Applications of hypersingular integrals to multidimensional integral equations of the first kind. Trudy Mat. Inst. Steklov. 172, 299–312 (1985, in Russian). English transl.: Proc. Steklov Inst. Math. 3, 325–339 (1987)

  122. Samko, S.G., Kostetskaya, G.S.: Absolute integrability of Fourier integrals. Vestnik RUDN (Russian Peoples Friendship Univ.), Math. 1, 138–168 (1994)

  123. Schaback R.: Multivariate interpolation by polynomials and radial basis functions. Constr. Approx. 21(3), 293–317 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  124. Schoenberg I.J.: A remark on the preceding note by Bochner. Bull. Am. Math. Soc. 40, 277–278 (1934)

    Article  MathSciNet  Google Scholar 

  125. Schoenberg I.J.: Metric spaces and completely monotone functions. Ann. Math. 39, 811–841 (1938)

    Article  MathSciNet  Google Scholar 

  126. Seeger A.: Necessary conditions for quasiradial Fourier multipliers. Tohoku Math. J. 39(2), 249–257 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  127. Shapiro, H.S.: Topics in approximation theory. With appendices by Jan Boman and Torbjörn Hedberg. Lecture Notes in Math., vol. 187. Springer, Berlin (1971)

  128. Stein, E.M.: Singular integrals, harmonic functions, and differentiability properties of functions of several variables. Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pp. 316–335. Amer. Math. Soc., Providence (1967)

  129. Stein E.M.: Note on the class L log L. Stud. Math. XXXII, 305–310 (1969)

    Google Scholar 

  130. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1970)

  131. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Space. Princeton University Press (1971)

  132. Stein E.M.: Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  133. Stepanov, V.D.: Integral representation of the resolvent of a self-adjoint elliptic operator. Diff. Uravnenija. 12, 129–136 (1976, in Russian). English transl.: Differ. Equ. 12, 89–94 (1977)

  134. Stepanov, V.D.: Absolute summability of the Fsourier transform of functions of several variables. Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, Trudy Sem. S.L. Soboleva, No. 1, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat. Novosibirsk, 116–121 (1978, in Russian)

  135. Sz-Nagy B.: Sur une classe générale de procédés de sommation pour les séries de Fourier. Hungarica Acta Math. 1, 14–52 (1948)

    MathSciNet  MATH  Google Scholar 

  136. Telyakovskii, S.A.: Integrability conditions for trigonometric series and their applications to the study of linear summation methods of Fourier series. Izv. Akad. Nauk SSSR, Ser. Matem. 28, 1209–1236 (1964, in Russian)

  137. Telyakovskii, S.A.: On a sufficient condition of Sidon for the integrability of trigonometric series. Mat. Zametki 14, 317–328 (1973, in Russian). English transl.: Math. Notes 14, 742–748 (1973)

    Google Scholar 

  138. Timan, M.F.: Absolute convergence of multiple Fourier series. Dokl. Akad. Nauk SSSR 137, 1074–1077 (1961, in Russian). English translation in Soviet Math. Dokl. 2 , 430–433 (1961)

  139. Timan, M.F.: Approximation and Properties of Periodic Functions. Naukova dumka, Kiev (2009, in Russian)

  140. Titchmarsh E.C.: A note on Fourier transforms. J. Lond. Math. Soc. 2, 148–150 (1927)

    Article  MATH  Google Scholar 

  141. Titchmarsh E.C.: Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford (1937)

    Google Scholar 

  142. Trebels W.: On a Fourier L 1(E n ) multiplier criterion. Acta Sci. Math. (Szeged) 35, 21–26 (1973)

    MathSciNet  MATH  Google Scholar 

  143. Trebels, W.: Fourier multipliers on L p(R n) in connection with bounded Riesz means. In: Approximation Theory (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1973), pp. 505–510. Academic Press, New York (1973)

  144. Trebels, W.: Multipliers for (C, α)-bounded Fourier expansions in Banach spaces and approximation theory. In: Lecture Notes in Mathematics, vol. 329. Springer, Berlin (1973)

  145. Trebels W.: Some Fourier multiplier criteria and the spherical Bochner-Riesz kernel. Rev. Roumaine Math. Pures Appl. 20, 1173–1185 (1975)

    MathSciNet  MATH  Google Scholar 

  146. Trebels, W.: Estimates for moduli of continuity of functions given by their Fourier transform. In: Constructive Theory of Functions of Several Variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976). Lecture Notes in Math., vol. 571, pp. 277–288. Springer, Berlin (1977)

  147. Trigub, R.M.: Integrability and asymptotic behavior of the Fourier transform of a radial function. In: Metric Questions of the Theory of Functions and Mappings, pp. 142–163. Naukova Dumka, Kiev (1977, in Russian)

  148. Trigub, R.M.: Absolute convergence of Fourier integrals, summability of Fourier series, and polynomial approximation of functions on the torus. Izv. Akad. Nauk SSSR Ser. Mat. 44, 1378–1408 (1980, in Russian). English translation in Math. USSR Izv. 17, 567–593 (1981)

  149. Trigub, R.M.: Certain properties of the Fourier transform of a measure and their application. In: Proc. Intern. Conf. Approx. Theory (Kiev, 1983), 439–443. Nauka, Moscow (1987, in Russian)

  150. Trigub, R.M.: Positive definite radial functions and splines. Constructive Function Theory ’87 (Varna, 25-31.05.1987), 123. Publ. House Bulgar. Acad. Sci., Sofia (1987)

  151. Trigub, R.M.: A criterion for a characteristic function and a polya-type criterion for radial functions of several variables. Teor. Ver. Pril. 34, 805–810 (1989, in Russian). English transl.: Theory Probab. Appl. 34, 738–742 (1989)

  152. Trigub, R.M.: Multipliers of Fourier series and approximation of functions by polynomials in spaces C and L. Doklady Akad. Nauk SSSR, 306, 292–296 (1989, in Russian). English transl.: Soviet Math. Dokl. 39, 494–498 (1989)

  153. Trigub, R.M.: Multipliers of Fourier series. Ukr. Mat. Zh. 43, 1686–1693 (1991, in Russian). English transl.: Ukr. Math. J. 43, 1572–1578 (1991)

  154. Trigub, R.M.: Positive definite functions and splines. Theory Funct. Approx. Proc. V Saratov Winter School (1990). Part I, Saratov, pp. 68–75 (1992, in Russian)

  155. Trigub, R.M.: Some Topics in Fourier Analysis and Approximation Theory. arXiv:funct-an/9612008v1 (1996)

  156. Trigub, R.M.: Multipliers in the Hardy spaces H p (D m) with \({p\in(0,1]}\) and approximation properties of summability methods for power series. Mat. Sbornik 188, 145–160 (1997, in Russian). English transl.: Sbornik Math. 188, 621–638 (1997)

  157. Trigub R.M.: Fourier multipliers and K-functionals in spaces of smooth functions. Ukr. Math. Bull. 2, 239–284 (2005)

    MathSciNet  Google Scholar 

  158. Trigub, R.M.: On comparison of linear differential operators. Matem. Zametki 82, 426–440 (2007, in Russian). English transl.: Math. Notes 82, 380–394 (2007)

    Google Scholar 

  159. Trigub R.M.: On Fourier multipliers and absolute convergence of the Fourier series of radial functions. Ukr. Math. J. 62, 1280–1293 (2010) (Russian)

    MathSciNet  MATH  Google Scholar 

  160. Trigub, R.M., Belinsky, E.S.: Fourier Analysis and Approximation of Functions. Kluwer/Springer (2004)

  161. Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge Mathematical Library, Reprint of the second (1944) edition. Cambridge University Press, Cambridge (1995)

  162. Wendland H.: Piecewise positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  163. Widder, D.V.: The Laplace Transform. Princeton University Press (1946)

  164. Wiener N.: On the representation of functions by trigonometrical integrals. Math. Zeitschr. 24, 575–616 (1926)

    Article  MathSciNet  Google Scholar 

  165. Wiener N.: Generalized harmonic analysis. Acta Math. 55, 117–258 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  166. Wiener N.: Tauberian theorems. Ann. Math. 33, 1–100 (1932)

    Article  MathSciNet  Google Scholar 

  167. Wiener N.: The Fourier Integral and Certain of its Applications. Cambridge University Press, Cambridge (1935)

    Google Scholar 

  168. Wiener N., Pitt H.R.: On absolutely convergent Fourier-Stieltjes transforms. Duke Math. J. 4, 420–436 (1938)

    Article  MathSciNet  Google Scholar 

  169. Yosida K.: Proc. Imp. Acad. Tokyo 20, 656–660 (1944)

    Google Scholar 

  170. Zastavnyi, V.P.: Positive definite functions depending on the norm. Solution of a problem of Shoenberg. Preprint N1-35, Inst. Applied Math. Mech. Acad. Sci., Ukraine, Donetsk (1991, in Russian)

  171. Zastavnyi V.P.: Positive definite functions depending on the norm. Russ. J. Math. Phys. 1, 511–522 (1993)

    MathSciNet  MATH  Google Scholar 

  172. Zastavnyi V.P.: On positive definiteness of some functions. J. Multivar. Anal. 73, 55–81 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  173. Zastavnyi, V.P.: Positive definite radial functions and splines. Doklady Ross. Akad. Nauk, 386, 446–449 (2002, in Russian). English transl.: Russ. Acad. Sci., Dokl. Math. 66, 1–4 (2002)

  174. Zastavnyi, V.P., Trigub, R.M.: Positive-definite splines of special form. Mat. Sbornik 193(12), 41–68 (2002, in Russian). English transl.: Sb. Math. 193, 1771–1800 (2002)

    Google Scholar 

  175. Zygmund A.: Trigonometric series, vols. I, II, 2nd edn. Reprinted with corrections and some additions. Cambridge University Press, London (1968)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    E. Liflyand

  2. Departamento de Matematica, Faculdade de Ciencias e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005-139, Faro, Portugal

    S. Samko

  3. Department of Mathematics, Donetsk National University, Donetsk, 83001, Ukraine

    R. Trigub

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  1. E. Liflyand
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Liflyand, E., Samko, S. & Trigub, R. The Wiener algebra of absolutely convergent Fourier integrals: an overview. Anal.Math.Phys. 2, 1–68 (2012). https://doi.org/10.1007/s13324-012-0025-6

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