Skip to main content
SpringerLink
Log in

Embedding theorems for functions of one variable

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Literature cited

  1. V. A. Andrienko, “Embedding of certain classes of functions,” Izv. Akad. Nauk SSSR, Ser. Matem.,31, No. 6, 1311–1326 (1967).

    Google Scholar 

  2. V. A. Andrienko, “Necessary conditions for the embedding of function classes H wp ,” Matem. Sborn.,78, No. 2, 280–300 (1969).

    Google Scholar 

  3. V. V. Arestov, “Best uniform approximation of differentiation operators,” Matem. Zametki,5, No. 3, 273–284 (1969).

    Google Scholar 

  4. N. K. Bari [Bary], Trigonometric Series [in Russian], Fizmatgiz, Moscow (1961) [English translation: Pergamon, New York (1964)].

    Google Scholar 

  5. N. K. Bari, “Best approximation of two conjugate functions by trigonometric polynomials,” Izv. Akad. Nauk SSSR, Ser. Matem.,19, No. 5, 285–302 (1955).

    Google Scholar 

  6. N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two conjugate functions,” Trudy Moskov. Matem. Obshch.,5, 483–522 (1956).

    Google Scholar 

  7. E. F. Beckenbach and R. Bellman, Inequalities [Russian translation], Moscow (1965) [English ed.: Springer, Berlin (1961)].

  8. O. V. Besov, “On a certain family of function spaces; embedding and continuation theorems,” Dokl. Akad. Nauk SSSR,126, No. 6, 1163–1165 (1959).

    Google Scholar 

  9. O. V. Besov, “Certain conditions for membership in Lp of the derivatives of periodic functions,” Nauch. Dokl. Vyssh. Shkoly, Fiz.-Matem. Nauk, No. 1, 13–17 (1959).

    Google Scholar 

  10. O. V. Besov, “Investigation of a family of function spaces in connection with embedding and continuation theorems,” Trudy Matem. Inst. Akad. Nauk SSSR,60, 42–81 (1961).

    Google Scholar 

  11. O. V. Besov, “Continuation of functions, preserving the properties of the second-order integral modulus of continuity,” Matem. Sborn.,58, No. 2, 673–684 (1962).

    Google Scholar 

  12. O. V. Besov, On Embedding and Continuation Theory for Classes of Differentiable Functions (Doctoral Dissertation) [in Russian], Matem. Inst. Akad. Nauk SSSR (1966).

  13. O. V. Besov and V. P. Il'in, “An embedding theorem for a limiting exponent,” Matem, Zametki,6, No. 2, 129–138 (1969).

    Google Scholar 

  14. O. V. Besov, V. P. Il'in, L. D. Kudryavtsev, P. I. Lizorkin, and S. M. Nikol'skii, “Embedding theory for classes of differentiable functions of many variables,” in: Partial Differential Equations [in Russian], Nauka, Moscow (1970), pp. 38–63.

    Google Scholar 

  15. Yu. A. Brudnyi, “Existence criteria for derivatives in Lp,” Matem. Sborn.,73, No. 1, 42–62 (1967).

    Google Scholar 

  16. Yu. A. Brudnyi, “Spaces defined by variation of local approximations,” Dokl. Akad. Nauk SSSR,190, No. 4, 753–755 (1970).

    Google Scholar 

  17. Yu. A. Brudnyi and I. E. Gopengauz, “Generalization of a theorem of Hardy and Littlewood,” Matem. Sborn.,52, No. 3, 891–894 (1960).

    Google Scholar 

  18. V. I. Burenkov, “Imbedding and continuation for classes of differentiable functions of several variables defined in the whole space,” in: Progress in Mathematics, Vol. 2, Plenum Press, New York (1968), pp. 73–161.

    Google Scholar 

  19. L. R. Volevich and B. P. Paneyakh, “A new approach to embedding theorems,” Izv. Akad. Nauk Azerb. SSR, Ser. Fiz.-Tekh. i Matem. Nauk, No. 1, 5–13 (1966).

    Google Scholar 

  20. L. R. Volevich and V. M. Kagan, “Pseudodiffrential hypoelliptic operators in the theory of function spaces,” Trudy Moskov. Matem. Obshch.,20, 241–275 (1969).

    Google Scholar 

  21. L. R. Volevich and A. G. Mekhtiev, “Multipliers of Fourier integrals in weighted classes and embedding theorems,” Izv. VUZov, Matematika, No. 10, 27–38 (1967).

    Google Scholar 

  22. L. R. Volevich and B. P. Paneyakh, “Some spaces of generalized functions and embedding theorems,” Usp. Matem. Nauk,20, No. 1, 3–74 (1965).

    Google Scholar 

  23. V. N. Gabushin, “Inequalities for the norms of a function and its derivatives in Lp metrics,” Matem. Zametki,1, No. 3, 291–298 (1967).

    Google Scholar 

  24. V. N. Gabushin, “Exact constants in inequalities between the norms of the derivatives of functions,” Matem. Zametki,4, No. 2, 221–232 (1968).

    Google Scholar 

  25. V. N. Gabushin, “Estimate of the extrema of a function and its derivatives,” in: Extremum Searching [in Russian], Tomsk Univ. (1969), pp. 243–244.

  26. G. Gaimnazarov, “Generalization of an embedding theorems of P. L. Ul'yanov,” Dokl. Akad. Nauk Tadzh. SSR,13, No. 1, 7–10 (1970).

    Google Scholar 

  27. I. G. Galyautdinov and V. R. Fridlender, “Generalizations and applications of the Kolmogorov and Gornyi-Cartan inequalities,” Izv. VUZov, Matematika, No. 3, 20–26 (1967).

    Google Scholar 

  28. V. É. Geit, “Structural properties of two conjugate functions,” Matem. Zap. Ural'skii Univ.,6, No. 2, 24–31 (1968).

    Google Scholar 

  29. V. É. Geit, 1) “Embedding theorems for certain classes of continuous periodic functions”; 2) “On the structural and constructive properties of a function and its conjugate in L” (abstracts of articles in press), Izv. VUZov, Matematika, No. 10, 20 (1970).

    Google Scholar 

  30. Ya. L. Geronimus, “Properties of analytic functions continuous in a closed disk or closed sector,” Matem. Sborn.,38, No. 3, 319–330 (1956).

    Google Scholar 

  31. Ya. L. Geronimus, “Certain properties of functions in class Lp,” Izv. VUZov, Matemaika, No. 1, 24–32 (1958).

    Google Scholar 

  32. Ya. L. Geronimus, “On certain embedding theorems,” Izv. VUZov, Matemaika, No. 6, 53–62 (1965).

    Google Scholar 

  33. B. L. Golinskii, “Certain local properties of functions in class Lp,” Izv. VUZov, Matematika, No. 3, 43–52 (1959).

    Google Scholar 

  34. K. K. Golovkin, “On the impossibility of certain inequalities between function norms,” Trudy Matem. Inst. Akad. Nauk SSSR,70, 5–25 (1964).

    Google Scholar 

  35. B. I. Golubov, “On continuous functions of bounded p-variation,” Matem. Zametki,1 No. 3, 305–312 (1967).

    Google Scholar 

  36. B. I. Golubov, “The Fourier integral and continuity of functions of bounded p-variation,” Izv. VUZov, Matematika, No. 11, 83–92 (1968).

    Google Scholar 

  37. B. I. Golubov, “On the p-variation of a function,” Matem. Zametki,5, No. 2, 195–204 (1969).

    Google Scholar 

  38. B. I. Golubov, “On functions of bounded p-variation,” Izv. Akad. Nauk SSSR, Ser. Matem.,32, No. 4, 837–858 (1968).

    Google Scholar 

  39. B. I. Golubov, “Analogs of the Wiener and Lozinskii theorems for functions of several variables,” Trudy Tbilissk. Matem. Inst.,38, 31–43 (1970).

    Google Scholar 

  40. Yu. I. Gribanov and P. K. Belobrov, “A class of Banach function spaces,” Izv. VUZov, Matematika, No. 4, 44–55 (1963).

    Google Scholar 

  41. A. I. Guseinov, “A class of nonlinear singular integral equations,” Izv. Akad. Nauk SSSR, Ser. Matem.,12, No. 2, 193–212 (1948).

    Google Scholar 

  42. A. I. Guseinov and Kh. Sh. Mukhtarov, “A method for the investigation of nonlinear singular equations,” Dokl. Akad. Nauk SSSR,168, No. 5, 982–985 (1966).

    Google Scholar 

  43. A. S. Dzhafarov, “Sequences of differentiable functions,” Uch. Zap. Azerb. Gos. Pedagog. Inst., Ser. 11, 71–81 (1968).

  44. V. K. Dzyadyk, “Continuation of functions satisfying the Lipschitz condition in Lp metric,” Matem. Sborn.,40, No. 2, 239–242 (1956).

    Google Scholar 

  45. V. A. Dubovik and B. I. Korenblyum, “Inequalities of the Hadamard-Kolmogorov type in the presence of relationships,” Matem. Zametki,5, No. 1, 13–20 (1969).

    Google Scholar 

  46. V. V. Zhuk, “On certain relations between the moduli of continuity and functionals specified on sets of periodic functions,” Izv. VUZov, Matematika, No. 5, 24–33 (1970).

    Google Scholar 

  47. A. Zygmund, Trigonometric Series [Russian translation], Vols. I and II, Mir, Moscow (1965) m[2nd English ed.: Cambridge Univ. Press, New York (1959).

    Google Scholar 

  48. V. P. Il'in, “On a theorem of Hardy and Littlewood,” Trudy Matem. Inst. Akad. Nauk SSSR,53, 128–144 (1959).

    Google Scholar 

  49. V. P. Il'in, “Properties of certain classes of differentiate functions of several variables, specified in an n-dimensional domain,” Trudy Matem. Inst. Akad. Nauk SSSR,66, 227–363 (1962).

    Google Scholar 

  50. A. P. Calderon, “Intermediate spaces and interpolation, the complex method,” Studia Math.,23, 113–190 (1964).

    Google Scholar 

  51. V. S. Klimov, “Embedding theorems for symmetric spaces,” Matem. Sborn.,79, No. 2, 171–178 (1969).

    Google Scholar 

  52. V. S. Klimov, “Embedding theorems for symmetric spaces,” Matem. Sborn.,82, No. 3, 371–386 (1970).

    Google Scholar 

  53. V. M. Kokilashvili, “Estimation of best approximations and moduli of continuity in various Lebesgue spaces of periodic functions with a transformed Fourier series,” Soobshch. Akad. Nauk Gruz. SSR,35, No. 1, 3–8 (1964).

    Google Scholar 

  54. V. M. Kokilashvili, “On a function spaces and the Fourier coefficients,” Soobshch. Akad. Nauk Gruz. SSR,35, No. 3, 523–530 (1964).

    Google Scholar 

  55. V. M. Kokilashvili, “A converse theorem in the constructive theory of functions in Lp space (1<p<∞),” Trudy Tbilissk. Matem. Inst.,29, 183–189 (1964).

    Google Scholar 

  56. V. M. Kokilashvili, “A converse theorem in the constructive theory of functions in Orlicz space,” Soobshch. Akad. Nauk Gruz. SSR,37, No. 2, 263–270 (1965).

    Google Scholar 

  57. V. M. Kokilashvili, “Best approximations by trigonometric polynomials in Orlicz spaces and the Lebesgue-Fourier coefficients,” Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Phys.,13, No. 5, 357–362 (1965).

    Google Scholar 

  58. V. M. Kokilashvili, “Structural and constructive characteristics of a class of periodic functions,” Soobshch. Akad. Nauk Gruz. SSR,43, No. 1, 3–8 (1966).

    Google Scholar 

  59. A. N. Kolmogorov, “Inequalities between the upper bounds of successive derivatives of an arbitrary function on an infinite interval,” Uch. Zap. MGU, Matematika,30, No. 3, 3–16 (1939).

    Google Scholar 

  60. A. A. Konyushkov, “On the Lipschitz classes,” Izv. Akad. Nauk SSSR, Ser. Matem.,21, No. 3, 423–448 (1957).

    Google Scholar 

  61. A. A. Konyushkov, “Best approximations by trigonometric polynomials and the Fourier coefficients,” Matem. Sborn.,44, No. 1, 53–84 (1958).

    Google Scholar 

  62. A. A. Konyushkov, “On certain classes of functions (I, II),” Izv. Akad. Nauk SSSR, Ser. Matem.,22, No. 6, 841–870 (1958).

    Google Scholar 

  63. A. A. Konyushkov, “Higher-order finite differences of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Matem.,24, No. 4, 549–566 (1960).

    Google Scholar 

  64. M. A. Krasnosel'skii and Ya. B. Rutitskii, Convex Functions and Orlicz Spaces [in Russian], Fizmatgiz, Moscow (1958).

    Google Scholar 

  65. S. G. Krein, “Interpolation theorems in operator theory and embedding theorems,” Proc. Fourth All-Union Mathematics Congress [in Russian], Vol. 2, Nauka, Leningrad (1964), pp. 504–510.

    Google Scholar 

  66. S. G. Krein and Yu. N. Petunin, “Scale of Banach spaces,” Usp. Matem. Nauk,21, No. 2, 89–168 (1966).

    Google Scholar 

  67. S. G. Krein, Yu. N. Petunin, and E. M. Semenov, “Embedding theorems and interpolation of linear operators,” in: Embedding Theorems and Their Applications [in Russian], Nauka, Moscow (1970), pp. 127–131.

    Google Scholar 

  68. P. I. Lizorkin, “Embedding theorems for functions in H rp spaces,” Dokl. Akad. Nauk SSSR,143, No. 5, 1042–1045 (1962).

    Google Scholar 

  69. P. I. Lizorkin, “Generalized Liouville differentiation and the function spaces H rp (En); embedding theorems,” Matem. Sborn.,60, No. 3, 325–353 (1963).

    Google Scholar 

  70. S. M. Lozinskii, “On a theorem of N. Wiener (I, II),” Dokl. Akad. Nauk SSSR, No. 8, 562–565 (1945);53, No. 8, 691–694 (1946).

    Google Scholar 

  71. Yu. I. Lyubich, “On inequalities between the degrees of a linear operator,” Izv. Akad. Nauk SSSR, Ser. Matem.,24, No. 6, 825–864 (1960).

    Google Scholar 

  72. A. M. Lyustig, “Certain problems in the classification of infinitely differentiable functions,” Uch. Zap. Elabuzhsk. Pedagog. Inst.,1, 63–72 (1956).

    Google Scholar 

  73. É. Madzhenes, “Interpolation spaces and partial differential equations,” Usp. Matem. Nauk,21, No. 2, 169–218 (1966).

    Google Scholar 

  74. B. S. Mityagin and A. S. Shvarts, “Functors in categories of Banach spaces,” Usp. Matem. Nauk,19, No. 2, 65–130 (1964).

    Google Scholar 

  75. Kh. Sh. Mukhtarov, “On certain inequalities and their applications to the study of nonlinear singular equations,” Dokl. Akad. Nauk Azerb. SSR,21, No. 4, 3–8 (1965).

    Google Scholar 

  76. Kh. Sh. Mukhtarov, “On certain multiplicative inequalities and their applications to linear singular integral operators,” Dokl. Akad. Nauk Azerb. SSR,182, No. 4, 764–767 (1968).

    Google Scholar 

  77. Kh. Sh. Mukhturov, “Inequalities of the embedding-theorem type,” Matem. Zametki,8, No. 2, 159–167 (1970).

    Google Scholar 

  78. S. M. Nikol'skii, “Fourier series of a function with a given modulus of continuity,” Dokl. Akad. Nauk SSSR,52, No. 3, 191–194 (1946).

    Google Scholar 

  79. S. M. Nikol'skii, “Inequalities for entire functions of finite power and their applications in the theory of differential functions of several variables,” Trudy Matem. Inst. Akad. Nauk SSSR,38, 244–278 (1951).

    Google Scholar 

  80. S. M. Nikol'skii, “Properties of certain classes of functions of several variables on differentiable manifolds,” Matem. Sborn.,33, No. 2, 261–326 (1953).

    Google Scholar 

  81. S. M. Nikol'skii, “An embedding theorem for functions with partial derivatives analyzable in various metrics,” Izv. Akad. Nauk SSSR, Ser. Matem.,22, No. 3, 321–336 (1958).

    Google Scholar 

  82. S. M. Nikol'skii, “On embedding, continuation, and approximation theorems for differentiable functions of several variables,” Usp. Matem. Nauk,16, No. 5, 63–114 (1961).

    Google Scholar 

  83. S. M. Nikol'skii, Approximation of Functions of Many Variables and Embedding Theorems [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  84. L. K. Pandzhikidze, “On an embedding theorem,” Soobshch. Akad. Nauk Gruz. SSR,56, No. 1, 21–24 (1969).

    Google Scholar 

  85. M. K. Potapov, “On the Fourier coefficients of periodic functions belonging to the H-classes of S. M. Nikol'skii,” Dokl. Akad. Nauk SSSR,141, No. 3, 564–567 (1961).

    Google Scholar 

  86. M. K. Potapov, “On the Fourier coefficients,” in: Modern Problems of Constructive Function Theory [in Russian], AN Azerb. SSR, Baku (1965), pp. 475–483.

    Google Scholar 

  87. M. K. Potapov, “On the equivalence of conditions for the convergence of Fourier series,” Matem. Sborn.,68, No. 1, 111–127 (1965).

    Google Scholar 

  88. M. K. Potapov, “Interrelationship of certain classes of functions,” Matem. Zametki,2, No. 4, 361–372 (1967).

    Google Scholar 

  89. M. K. Potapov, “Membership in Lp of a transformed Fourier series,” Vest. Moskov. Univ., Matem., Mekh., No. 5, 3–14 (1968).

    Google Scholar 

  90. M. K. Potapov, “On the embedding and congruence of certain classes of functions,” Izv. Akad. Nauk SSSR, Ser. Matem.,33, No. 4, 840–861 (1969).

    Google Scholar 

  91. L. B. Rabinovich, “Certain embedding theorems in Orlicz spaces,” Izv. VUZov, Matematika, No. 9, 78–85 (1968).

    Google Scholar 

  92. D. A. Raikov, “On differences and derivatives,” Matem. Sborn.,7, No. 2, 379–384 (1940).

    Google Scholar 

  93. Ya. B. Rutitskii, “New criteria of the continuity and complete continuity of integral operators in Orlicz spaces,” Izv. VUZov, Matematika, No. 5, 87–100 (1962).

    Google Scholar 

  94. Ya. B. Rutitskii, “On certain classes of measurable functions,” Usp. Matem. Nauk,20, No. 4, 205–208 (1965).

    Google Scholar 

  95. B. V. Ryazanov, “On functions of class Vγ,” Vest. Moskov. Univ., Matem., Mekh., No. 6, 36–39 (1968).

    Google Scholar 

  96. D. V. Salekhov, “On the representabilify of functions belongingto Orlicz classes by singular integrals,” Ukrainsk. Matem. Zh.,13, No. 4, 34–50 (1961).

    Google Scholar 

  97. D. V. Salekhov, “Properties of functions belonging to an Orlicz space,” Matem. Zametki,3, No. 2, 145–156 (1968).

    Google Scholar 

  98. D. V. Salekhov, “A characteristic property of Lorentz space in the class of symmetric spaces,” Matem. Zametki,7, No. 5, 627–633 (1970).

    Google Scholar 

  99. E. M. Semenov, Scales of Banach Spaces Connecting Spaces L1 and L∞ (dissertation) [in Russian], Voronezh (1964).

  100. E. M. Semenov, “Embedding theorems for Banach spaces of measurable functions,” Dokl. Akad. Nauk SSSR,156, No. 6, 1292–1295 (1964).

    Google Scholar 

  101. E. M. Semenov, “Interpolation of linear operators in symmetric spaces,” Dokl. Akad. Nauk SSSR,164, No. 4, 746–749 (1965).

    Google Scholar 

  102. E. M. Semenov, “Interpolation of linear operators and estimates of the Fourier coefficients,” Dokl. Akad. Nauk SSSR,176, No. 6, 1251–1254 (1967).

    Google Scholar 

  103. E. M. Semenov, “A new interpolation theorem,” Funktsional. Anal. i Ego Prilozhen.,2, No. 2, 68–80 (1968).

    Google Scholar 

  104. L. N. Slobodetskii, “Generalized Sobolev spaces and their application to boundary-value problems for partial differential equations,” Uch. Zap. Leningrad. Pedagog. Inst.,197, 54–112 (1958).

    Google Scholar 

  105. S. L. Sobolev, “On a theory of functional analysis,” Matem. Sborn.,4, No. 3, 471–497 (1938).

    Google Scholar 

  106. S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics [in Russian], Leningrad (1950); Sibirsk. Otdel. AN SSSR, Novosibirsk (1965).

  107. S. B. Stechkin, “On the order of best approximations of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Matem.,15, No. 3, 219–242 (1951).

    Google Scholar 

  108. S. B. Stechkin, “On the Kolmogorov-Seliverstov theorem,” Izv. Akad. Nauk SSSR, Ser. Matem.,17, No. 6, 499–512 (1953).

    Google Scholar 

  109. S. B. Stechkin, “On the best approximation of conjugate functions by trigonometric polynomials,” Izv. Akad. Nauk SSSR, Ser. Matem.,20, No. 2, 197–206 (1956).

    Google Scholar 

  110. S. B. Stechkin, “Inequalities between the norms of the derivatives of an arbitrary function,” Acta Sci. Math.,26, Nos. 3–4, 225–230 (1965).

    Google Scholar 

  111. S. B. Stechkin, “Inequalities between the upper bounds of the derivatives of an arbitrary function on a half-line,” Matem. Zametki,1, No. 6, 665–674 (1967).

    Google Scholar 

  112. Yu. N. Subbotin, “Relationship between finite differences and the corresponding derivatives,” Usp. Matem. Nauk,20, No. 5, 303 (1965); Trudy Matem. Inst. Akad. Nauk SSSR,78, 24–42 (1965).

    Google Scholar 

  113. Yu. N. Subbotin and L. V. Taikov, “Best approximation of the differentiation operator in L2 space,” Matem. Zametki,3, No. 2, 157–164 (1968).

    Google Scholar 

  114. N. T. Temirgaliev, “Certain embedding theorems for classes H ωpm of functions of several variables,” Izv. Akad. Nauk Kaz, SSR, Ser. Fiz.-Matem., No. 5, 90–92 (1970).

    Google Scholar 

  115. A. P. Terekhin, “Approximation of functions of bounded p-variation,” Izv. VUZov, Matematika, No. 2, 171–187 (1965).

    Google Scholar 

  116. A. P. Terekhin, “Integral smoothness properties of periodic functions of bounded p-variation,” Matem. Zametki,2, No. 3, 288–300 (1967).

    Google Scholar 

  117. A. F. Timan, Studies in the Theory of Approximation of Functions (author's abstract of doctoral dissertation [in Russian], Dnepropetrovsk (1951).

  118. A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Fizmatgiz, Moscow (1960).

    Google Scholar 

  119. A. F. Timan and M. F. Timan, “Generalized modulus of continuity and best approximations in the mean,” Dokl. Akad. Nauk SSSR,71, No. 1, 17–20 (1950).

    Google Scholar 

  120. M. F. Timan, “Converse theorems in the constructive theory of functions in Lp spaces (1<p<∞),” Matem. Sborn.,46, No. 1, 125–132 (1958).

    Google Scholar 

  121. M. F. Timan, “Best approximation and modulus of continuity of functions specified on the entire real axis,” Izv. VUZov, Matematika, No. 6, 108–120 (1961).

    Google Scholar 

  122. M. F. Timan, “On certain embedding theorems for function classes Lp,” Dokl. Akad. Nauk SSSR,193, No. 6, 1251–1254 (1970).

    Google Scholar 

  123. P. L. Ul'yanov, “Generalization of a theorem of Marcinkiewicz,” Izv. Akad. Nauk SSSR, Ser. Matem.,17, No. 6, 513–524 (1953).

    Google Scholar 

  124. P. L. Ul'yanov, “On certain equivalent conditions for the convergence of series and integrals,” Usp. Matem. Nauk,8, No. 6, 133–141 (1953).

    Google Scholar 

  125. P. L. Ul'yanov, “On the continuation of functions,” Dokl. Akad. Nauk SSSR,105, No. 5, 913–915 (1955).

    Google Scholar 

  126. P. L. Ul'yanov, “On series with respect to the Haar system,” Matem. Sborn.,63, No. 3, 356–391 (1964).

    Google Scholar 

  127. P. L. Ul'yanov, “On the absolute and uniform convergence of Fourier series,” Matem. Sborn.,72, No. 2, 193–225 (1967).

    Google Scholar 

  128. P. L. Ul'yanov, “On the embedding of certain classes of functions,” Matem. Zametki,1 No. 4, 405–414 (1967).

    Google Scholar 

  129. P. L. Ul'yanov, “Embedding of certain classes of functions of one variable,” Dokl. Akad. Nauk SSSR,178, No. 6, 1259–1262 (1968).

    Google Scholar 

  130. P. L. Ul'yanov, “Embedding of certain function classes H ωp ,” Izv. Akad. Nauk SSSR, Ser. Matem.,32, No. 3, 649–686 (1968).

    Google Scholar 

  131. P. L. Ul'yanov, “Embedding theorems and best approximations,” Dokl. Akad. Nauk SSSR,184, No. 5, 1044–1047 (1969).

    Google Scholar 

  132. P. L. Ul'yanov, “Embedding theorems and relations between best approximations (moduli of continuity) in various metrics,” Matem. Sborn.,81, No. 1, 104–131 (1970).

    Google Scholar 

  133. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities [Russian translation], Moscow (1948) [2nd English ed.: Cambridge Univ. Press (Eng.) (1952)].

  134. L. Hörmander, Linear Partial Differential Operators [Russian translation], Mir, Moscow (1965) [English ed.: Springer, Berlin (1963)].

    Google Scholar 

  135. I. V. Shragin, “On the theory of Orlicz spaces,” Dokl. Akad. Nauk SSSR,179, No. 5, 1064–1066 (1968).

    Google Scholar 

  136. T. Ando, “On products of Orlicz spaces,” Math. Ann.,140, No. 3, 174–186 (1960).

    Google Scholar 

  137. A. P. Andrews, Differential Inequalities in Hilbert Space (Doctoral Dissertation), Univ. California, Los Angeles (1968); Dissert. Abstr.,B29, No. 4, 1424–1425 (1968).

    Google Scholar 

  138. P. J. Aranda and E. P. Cattaneo, “Sur les espaces de Lorentz avec poids,” Compt. Rend.,264, No. 3, A109-A112 (1967).

    Google Scholar 

  139. N. Aronszajn and K. T. Smith, “Theory of Bessel potentials (I),” Ann. Inst. Fourier,11, 385–475 (1961).

    Google Scholar 

  140. G. C. Barozzi, “Su una generalizazione degli spazi L(q, λ) di Morrey,” Ann. Scuola Norm. Super. Pisa, Sci. Fis. Matem.,19, No. 4, 609–626 (1965).

    Google Scholar 

  141. A. P. Calderon, Singular Integrals, Notes on a Course Taught at the Massachusetts Institute of Technology (1959).

  142. Z. Ciesielski and J. Musielak, “On absolute convergence of Haar series” Colloq. Math.,7, No. 1, 61–65 (1959).

    Google Scholar 

  143. G. Dankert, “Über Produkte von Orlicz-Räumen,” Arch. Math.,19, No. 6, 635–645 (1969).

    Google Scholar 

  144. G. Dankert and H. König, “Über die Höldersche Ungleichung in Orlicz-Räumen,” Arch. Math.,18, No. 1, 61–75 (1967).

    Google Scholar 

  145. J. Diedonné, “Sur les espaces de Köthe,” J. Analyse Math.,1, 81–115 (1951).

    Google Scholar 

  146. H. W. Ellis and I. Halperin, “Function spaces determined by a levelling length function,” Canad. J. Math.,5, No. 4, 576–592 (1953).

    Google Scholar 

  147. R. B. Fraser, “Banach spaces of functions satisfying a modulus of continuity condition,” Studia Math.,32, No. 3, 277–283 (1969).

    Google Scholar 

  148. G. Georganopoulos, “Sur les fonctions satisfaisantes a la condition de Hölder-Lipschitz d'ordre h,” Bull. Sci. Math.,86, No. 3, 89–100 (1962).

    Google Scholar 

  149. C. Goulaouic, “Interpolation entre des espaces localement convexes définis a l'aide de semi-groupes; cas des espaces de Gevrey,” Ann. Inst. Fourier,19, No. 2, 269–278 (1969–1970).

    Google Scholar 

  150. I. Halperin, “Function spaces,” Canad. J. Math.,5, No. 3, 273–288 (1953).

    Google Scholar 

  151. G. H. Hardy and J. E. Littlewood, “A convergence criterion for Fourier series,” Math. Z.,28, No. 4, 612–634 (1928).

    Google Scholar 

  152. G. H. Hardy, “Some properties of fractional integrals (I, III),” Math. Z.,27, No. 4, 565–606 (1928);34, No. 3, 403–439 (1932).

    Google Scholar 

  153. J. A. Johnson, “Banach spaces of Lipschitz functions and vector-valued Lipschitz functions,” Bull. Amer. Math. Soc.,75, No. 6, 1334–1338 (1969).

    Google Scholar 

  154. V. Kokilashvili, “On approximations of periodic functions in Orlicz spaces,” Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys.,14, No. 2, 77–80 (1966).

    Google Scholar 

  155. A. Kolmogorov [Kolmogoroff], “Une série de Fourier-Lebesgue divergente presque partout,” Fund. Math., 4, 324–328 (1923).

    Google Scholar 

  156. D. Kralik, “Untersuchung der Integrale und Derivierten gebrochener Ordnung mit den Methoden der Konstruktiven Funktionentheorie,” Acta Math. Acad. Sci. Hung.,7, No. 1, 49–64 (1956).

    Google Scholar 

  157. L. Leindler, “Über vershiedene Konvergenzgarten trigonometrischer Reihen, III: Bedingungen in der Metrik von Lp,” Acta Sci. Math.,27, Nos. 3–4, 205–215 (1966).

    Google Scholar 

  158. L. Leindler, “On embedding of classes H ωp ,” Acta Sci. Math.,31, Nos. 1–2, 13–31 (1970).

    Google Scholar 

  159. R. Leśniewicz, “On Hardy-Orlicz spaces (I),” Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Phys.,14, No. 3, 145–150 (1966).

    Google Scholar 

  160. J. E. Littlewood and R. E. A. C. Paley, “Theorems on Fourier series and power series,” J. London Math. Soc.,6, No. 3, 230–233 (1931).

    Google Scholar 

  161. G. G. Lorentz, “Some new functional spaces,” Ann. Math.,51, No. 1, 37–55 (1950).

    Google Scholar 

  162. G. G. Lorentz, “On the theory of spaces A,” Pacific J. Math.,1, No. 3, 411–429 (1950).

    Google Scholar 

  163. G. G. Lorentz, “Relations between function spaces,” Proc. Amer. Math. Soc.,12, No. 1, 127–132 (1961).

    Google Scholar 

  164. G. G. Lorentz and T. Shimogaki, “Interpolation theorems for operators in function spaces,” J. Funct. Anal.,2, No. 1, 31–51 (1968).

    Google Scholar 

  165. G. G. Lorentz and D. G. Wertheim, “Representation of linear functionals on Köthe spaces,” Canad. J. Math.,5, No. 4, 568–575 (1953).

    Google Scholar 

  166. E. R. Love, “A generalization of absolute continuity,” J. London Math. Soc.,26, No. 1, 1–13 (1951).

    Google Scholar 

  167. E. R. Love and L. C. Young, “On fractional integration by parts,” Proc. London Math. Soc., Ser. 2,41, No. 1, 1–35 (1938).

    Google Scholar 

  168. W. A. J. Luxemburg, Banach Function Spaces (doctoral dissertation), Tech. Hogeschool te Delft (1955).

  169. W. A. J. Luxemburg and A. C. Zaanen, “Some remarks on Banach function spaces,” Proc. Koninkl. Netherl. Akad. Wet.,A59, No. 1, 110–119 (1956).

    Google Scholar 

  170. W. A. J. Luxemburg and A. C. Zaanen, “Notes on Banach function spaces (I, II),” Proc. Koninkl. Nederl. Akad. Wet.,A66, No. 2, 135–147; 148–153 (1963).

  171. J. Marcinkiewicz, “Sur quelques integrales du type de Dini,” Ann. Soc. Polon. Math.,17, No. 1, 42–50 (1938).

    Google Scholar 

  172. J. Marcinkiewicz, “Sur une nouvelle condition pour la convergence presque partout des séries de Fourier,” Ann. Scuola Norm. Super. Pisa, Ser. 2,8, Nos. 3–4, 239–240 (1939).

    Google Scholar 

  173. W. Matuszewska, “On generalized Orlicz spaces,” Bull. Akad. Polon. Sci., Ser. Sci. Math. Astron. Phys.,8, No. 6, 349–353 (1960).

    Google Scholar 

  174. H. P. Mulholland, “Concerning the generalization of the Young-Hausdorff theorem and the Hardy-Littlewood theorems on Fourier constants,” Proc. London Math. Soc.,35, No. 4, 257–293 (1933).

    Google Scholar 

  175. H. Nakano, Modulared Semi-ordered Linear Spaces, Tokyo (1950).

  176. R. O'Neil, “Fractional integration in Orlicz spaces (I),” Trans. Amer. Math. Soc.,115, No. 3, 300–328 (1965).

    Google Scholar 

  177. Ono Akira, “On imbedding between strong Hölder spaces and Lipschitz spaces in Lp,” Mem. Fac. Sci. Kyushu Univ.,A24, No. 1, 76–93 (1970).

    Google Scholar 

  178. W. Orlicz, “Sur les fonctions satisfaisant á une condition de Lipschitz généralisée (I),” Studia Math.,10, No. 1, 21–39 (1948).

    Google Scholar 

  179. J. Peetre, “On the theory of Lp, λ spaces,” J. Funct. Anal.,4, No. 1, 71–87 (1969).

    Google Scholar 

  180. J. Peetre, “Sur la transformation de Fourier de fonctions á valeurs vectorialles,” Rend. Sem. Mat. Univ. Padova,42, 15–26 (1969).

    Google Scholar 

  181. L. C. Piccinini, “Inclusioni tra spazi di Morrey,” Boll. Unione Mat. Ital.,2, No. 1, 95–101 (1969).

    Google Scholar 

  182. C. T. Rajagopal, “On the Fourier coefficients of functions of L2,” Math. Student,29, Nos. 3–4, 127–132 (1961–1962).

    Google Scholar 

  183. F. Riesz, “Untersuchunge über Systeme integrierbarer Funktionen,” Math. Ann.,69, No. 4, 449–497 (1910).

    Google Scholar 

  184. R. Salem, “Sur certaines fonctions continues et les propriétés de leurs séries de Fourier,” Compt. Rend.,201, No. 17, 703–705 (1935).

    Google Scholar 

  185. R. Salem, “Essais sur les séries trigonometriques,” Actualtés sci. et Industr. (Paris), No. 862, 1–85.

  186. J. J. Schäffer, “Function spaces with translations,” Math. Ann.,137, No. 3, 209–262 (1959).

    Google Scholar 

  187. I. J. Schoenberg, “Spline interpolation and the higher derivatives,” Proc. Nat. Acad. Sci. USA,51, No. 1, 24–28 (1964).

    Google Scholar 

  188. M. S. Skaff, “Vector-valued Orlicz spaces; generalized N-functions (I, II),” Pacific J. Math.,28, No. 1, 193–206; No. 2, 413–430 (1969).

  189. J. D. Stafney, “Analytic interpolation of certain multiplier spaces,” Pacific J. Math.,32, No. 1, 241–248 (1970).

    Google Scholar 

  190. E. M. Stein, “Functions of exponential type,” Ann. Math.,65, No. 3, 582–592 (1957).

    Google Scholar 

  191. E. M. Stein and A. Zygmund, “On the fractional differentiability of functions,” Proc. London Math. Soc.,14A, 249–264 (1965).

    Google Scholar 

  192. O. Szasz, “Fourier series and mean moduli of continuity,” Trans. Amer. Math. Soc.,42, No. 3, 366–395 (1937).

    Google Scholar 

  193. M. R. Taibleson, “A note on indefinite integrals,” Proc. Amer. Math. Soc.,16, No. 5, 923–925 (1965).

    Google Scholar 

  194. E. Tarnawski, “Continuous functions in the logarithmic power classification according to Hölder's conditions,” Fund. Math.,42, No. 1, 11–37 (1955).

    Google Scholar 

  195. E. Tarnawski, “On the spaces of functions satisfying Hölder's condition,” Fund. Math.,42, No. 2, 207–214 (1955).

    Google Scholar 

  196. E. Tarnawski, “Continuous functions considered from the standpoint of Dini's conditions,” Fund. Math.,43, No. 1, 3–22 (1956).

    Google Scholar 

  197. E. Tarnawski, “On the spaces of functions satisfying Dini's condition,” Fund. Math.,43, No. 2, 141–147 (1956).

    Google Scholar 

  198. E. C. Titchmarsh, “A note on Fourier transforms,” J. London Math. Soc.,2, No. 7, 148–150 (1927).

    Google Scholar 

  199. N. S. Trudinger, “On imbeddings into Orlicz spaces and some applications,” J. Math. Mech.,17, No. 5, 473–483 (1967).

    Google Scholar 

  200. C.-J. Vallée-Poussin, Leçons surl' Approximationdes Fonctions d'une Variable Réelle, Paris (1919).

  201. M. Weiss and A. Zygmund, “A note on smooth functions,” Proc. Koninkl. Nederl. Akad. Wet.,A62, No. 1, 52–58 (1959); Indagationes Math.,21, No. 1, 52–58 (1959).

    Google Scholar 

  202. R. Welland, “Inclusion relations among Orlicz spaces,” Proc. Amer. Math. Soc.,17, No. 1, 135–139 (1966).

    Google Scholar 

  203. N. Wiener, “The quadratic variation of a function and its Fourier coefficients,” Massachusetts J. Math.,3, 72–94 (1924).

    Google Scholar 

  204. N. Wiener and L. C. Young, “The total variation of g(x+h)−g(x),” Trans. Amer. Math. Soc.,35, No. 1, 327–340 (1933).

    Google Scholar 

  205. L. C. Young, “An inequality of the Hölder type connected with Stieltjes integration,” Acta Math.,67, Nos. 1–2, 251–282 (1936).

    Google Scholar 

Download references

Authors
  1. V. A. Andrienko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Itogi Nauki, Seriya Matematika (Matematicheskii Analiz), pp. 203–262, 1970.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Andrienko, V.A. Embedding theorems for functions of one variable. J Math Sci 1, 764–804 (1973). https://doi.org/10.1007/BF01236364

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01236364

Search

Navigation