Multiple fourier series and integrals

One gives a brief survey of the investigations on the theory of multiple Fourier series and integrals, reviewed in Referativnyi Zhurnal “Matematika” in the period 1953–1980. Principal attention is given to the following questions: localization principles, uniform convergence and summability, convergence and summability at a point, in the Lp metric, and almost everywhere, absolute convergence, uniqueness theorems, conjugate Fourier series and integrals, equiconvergence and equisummability of Fourier series and integrals, properties of the kernel and of the Lebesgue constant of summation methods of Fourier series, Fourier coefficients and Fourier transforms.

[1]  P. Sjölin.  On the convergence almost everywhere of double Fourier series , 1976 .

[2]  P. Sjölin.  Convergence almost everywhere of certain singular integrals and multiple Fourier series , 1971 .

[3]  Albert Wilansky.  On the convergence of double series , 1947 .

[4]  L. Hörmander,et al.  Oscillatory integrals and multipliers onFLp , 1973 .

[5]  N. C. Krutitskaya.  Localization in bounded summation by Cesaro, Riesz, and Abel methods for multiple Fourier series , 1972 .

[6]  C. Fefferman.  On the convergence of multiple Fourier series , 1971 .

[7]  The conjugate Fourier-Stieltjes integral in the plane , 1959 .

[8]  Über absolute summierbarkeit von n-dimensionalen Fourierreihen und Fourierintegralen , 1969 .

[9]  V. A. Yudin.  The approximation of functions of many variables by their Féjér sums , 1973 .

[10]  V. Shapiro.  Cantor-type uniqueness of multiple trigonometric integrals. , 1955 .

[11]  V. A. Yudin,et al.  The multidimensional Jackson theorem , 1976 .

[12]  Elias M. Stein,et al.  Localization and summability of multiple fourier series , 1958 .

[13]  S. Igari.  On the localization property of multiple Fourier series , 1968 .

[14]  P. A. Tomas.  A restriction theorem for the Fourier transform , 1975 .

[15]  Charles Fefferman,et al.  A note on spherical summation multipliers , 1973 .

[16]  A. Zygmund.  A Cantor-Lebesgue theorem for double trigonometric series , 1972 .

[17]  W. Trebels.  SOME NECESSARY CONDITIONS FOR RADIAL FOURIER MULTIPLIERS , 1976 .

[18]  E M Nikišin,et al.  A RESONANCE THEOREM AND SERIES IN EIGENFUNCTIONS OF THE LAPLACIAN , 1972 .

[19]  L. Hörmander.  Fourier integral operators. I , 1995 .

[20]  On a multiplier transform , 1972 .

[21]  Discrete imbedding theorems and Lebesgue constants , 1977 .

[22]  Convergence of double fourier series for functions of bounded generalized variation. II , 1974 .

[23]  G. Sunouchi.  Some criteria for the absolute summability of a Fourier series , 1967 .

[24]  I. L. Bloshanskii.  Uniform convergence of expansions into a multiple trigonometric Fourier series or a Fourier integral , 1975 .

[25]  J. J. Gergen.  Convergence criteria for double Fourier series , 1933 .

[26]  Generalized Laplacians and multiple trigonometric series , 1973 .

[27]  Torbjörn Hedberg.  On the uniqueness of summable trigonometric series and integrals , 1971 .

[28]  Lebesgue summability of double trigonometric series , 1977 .

[29]  Application of the Fourier transform to summability of fourier series , 1977 .

[30]  L. Jeśmanowicz.  Application of the Nörlund summability to the theory of localization for single and double trigonometric series (I) , 1959 .

[31]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[32]  R. Nessel.  Contributions to the Theory of Saturation for Singular Integrals in Several Variables, III Radial Kernels , 1967 .

[33]  E. M. Nikishin.  Resonance theorems and functional series , 1971 .

[34]  V. Shapiro.  Sets of uniqueness on the 2-torus , 1972 .

[35]  Antoni Zygmund,et al.  On the summability of double Fourier series , 1939 .

[36]  Summation of multiple Fourier series by spherical means , 1936 .

[37]  S. Igari.  FUNCTIONS OF L^p-MULTIPLIERS , 1969 .

[38]  E. C. Titchmarsh.  Introduction to the Theory of Fourier Integrals , 1938 .

[39]  A. Zygmund.  On Fourier coefficients and transforms of functions of two variables , 1974 .

[40]  A. Zygmund.  On the boundary values of functions of several complex variables, I , 1949 .

[41]  A lower bound for Lebesgue constants , 1979 .

[42]  H. Helson,et al.  Prediction theory and Fourier Series in several variables , 1958 .

[43]  Roger Cooke.  A Cantor-Lebesgue theorem in two dimensions , 1971 .

[44]  Convergence and divergence of series conjugate to a convergent multiple Fourier series , 1975 .

[45]  B. Golubov.  ON CONVERGENCE OF RIESZ SPHERICAL MEANS OF MULTIPLE FOURIER SERIES , 1975 .

[46]  R. V. Churchill,et al.  Lectures on Fourier Integrals , 1959 .

[47]  Capacity and absolute abel summability of multiple Fourier series , 1974 .

[48]  I. Hirschman.  On multiplier transformations , 1959 .

[49]  On fourier transforms of radial functions , 1965 .

[50]  WEYL MULTIPLIERS FOR MULTIPLE FOURIER SERIES , 1972 .

[51]  V. Shapiro.  Circular summability $C$ of double trigonometric series , 1954 .

[52]  Fon-che Liu.  On the localization of rectangular partial sums for multiple Fourier series , 1972 .

[53]  M. Morse,et al.  The Fréchet Variation and the Convergence of Multiple Fourier Series. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[54]  W. H. Young.  On Multiple Fourier Series , 1913 .

[55]  H. D. Miller.  Generalization of a theorem of Marcinkiewicz. , 1967 .

[56]  E. Stein.  A Maximal Function With Applications to Fourier Series , 1958 .

[57]  S. B. Stechkin.  On the Cantor-Lebesgue theorem for double trigonometric series , 1972 .

[58]  J. Ash,et al.  Convergence, uniqueness, and summability of multiple trigonometric series , 1972 .

[59]  R. Cooke,et al.  The Cantor-Lebesgue Theorem , 1979 .

[60]  L. Zhizhiashvili.  SOME PROBLEMS IN THE THEORY OF SIMPLE AND MULTIPLE TRIGONOMETRIC AND ORTHOGONAL SERIES , 1973 .

[61]  B. Golubov.  ON THE CONVERGENCE OF RIESZ SPHERICAL MEANS OF MULTIPLE FOURIER SERIES AND INTEGRALS OF FUNCTIONS OF BOUNDED GENERALIZED VARIATION , 1972 .

[62]  J. Ash,et al.  Convergence, summability, and uniqueness of multiple trigonometric series , 1971 .

[63]  M. Timan.  BEST PARTIAL APPROXIMATIONS OF FUNCTIONS, ABSOLUTE CONVERGENCE OF FOURIER SERIES, AND NUCLEARITY OF HILBERT-SCHMIDT OPERATORS , 1968 .

[64]  Y. Sagher.  Some remarks an interpolation of operators and Fourier coefficients , 1972 .

[65]  Julian Musielak.  On absolute convergence of multiple Fourier series , 1958 .

[66]  S. Minakshisundaram,et al.  On absolute convergence of multiple Fourier series , 1947 .

[67]  N. S. Nikol'skaya.  The approximation in the Lp metric of differentiable functions of several variables by Fourier sums , 1974 .

[68]  C. Fefferman.  On the divergence of multiple Fourier series , 1971 .

[69]  J. Marcinkiewicz,et al.  On the Convergence of Fourier Series , 1935 .

[70]  Ernest William Hobson.  The Theory of Functions of a Real Variable and the Theory of Fourier's Series , 1907 .

[71]  P. Sjölin,et al.  Oscillatory integrals and multiplier problem for the disc , 1972 .

[72]  V. Shapiro.  Summability and uniqueness of double trigonometric integrals , 1954 .

[73]  S. Alimov,et al.  CONVERGENCE PROBLEMS OF MULTIPLE TRIGONOMETRIC SERIES AND SPECTRAL DECOMPOSITIONS. I , 1976 .

[74]  T. Teichmann,et al.  Harmonic Analysis and the Theory of Probability , 1957, The Mathematical Gazette.

[75]  V. Yudin.  Behavior of Lebesgue constants , 1975 .

[76]  Summability Rr for double series , 1977 .

[77]  S. Bochner,et al.  Lectures on Fourier integrals : with an author's supplement on monotonic functions, Stieltjes integrals, and harmonic analysis , 1959 .

[78]  A generalization of a theorem of M. Riesz to the case of functions of several variables , 1968 .

[79]  Convergence of the Hausdorff Means of Double Fourier Series* , 1968, Canadian Mathematical Bulletin.

[80]  A. Zygmund.  On Singular Integrals , 1956 .

[81]  Stephen Wainger,et al.  Special Trigonometric Series in K Dimensions , 1965 .

[82]  B. Golubov.  ON THE SUMMABILITY OF FOURIER INTEGRALS BY RIESZ SPHERICAL MEANS , 1977 .

[83]  Hiroshi C. Watanabe.  Summability of double Fourier series , 1965 .

[84]  S. Alimov,et al.  PROBLEMS OF CONVERGENCE OF MULTIPLE TRIGONOMETRIC SERIES AND SPECTRAL DECOMPOSITIONS. II , 1977 .

[85]  Yu. A. Brychkov,et al.  Asymptotic behavior of Fourier transforms , 1970 .

[86]  K. Kanno.  On the summation of multiple Fourier series , 1959 .

[87]  On Certain Exponential Sums Arising In Multiple Fourier Series , 1961 .

[88]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[89]  W. Trebels.  On the approximation behavior of the riesz means in Lp(Rn) , 1976 .

[90]  On Fourier-Stieltjes Transforms , 1955, Canadian Journal of Mathematics.

[91]  Josephine Mitchell.  On the spherical summability of multiple orthogonal series , 1951 .

[92]  P. Butzer,et al.  Contributions to the Theory of Saturation for Singular Integrals in Several Variables. IV , 1966 .

[93]  Singular integrals and spherical convergence , 1972 .

[94]  The convergence of the double Fourier series of continuous functions , 1973 .

[95]  B. Golubov.  On the Gibbs phenomenon for Riesz spherical means of multiple Fourier series and Fourier integrals , 1975 .

[96]  Spherical convergence, bounded variation, and singular integrals on the N-torus , 1971 .

[97]  Singular integrals and cardinal series , 1972 .

[98]  L. Carleson.  On convergence and growth of partial sums of Fourier series , 1966 .

[99]  S. Igari.  On the (Lp, Lp) multipliers , 1974 .

[100]  P. Sharma,et al.  On the strong summability of double fourier series , 1958 .

[101]  N. Temirgaliev.  A connection between inclusion theorems and the uniform convergence of multiple Fourier series , 1972 .

[102]  N. Bary,et al.  Treatise of Trigonometric Series , 1966 .

[103]  E. Stein,et al.  On the limits of sequences of operators , 1961 .

[104]  G. Welland,et al.  Order summability of multiple Fourier series , 1975 .

[105]  L. V. Žižiašvili.  ON TRIGONOMETRIC FOURIER SERIES , 1976 .

[106]  V. Shapiro.  An extension of results in the uniqueness theory of double trigonometric series , 1953 .

[107]  B. Golubov.  The Abel-Poisson summation method for multiple Fourier series , 1980 .

[108]  On Gibbs' phenomenon for Riesz spherical means of multiple Fourier integrals and Fourier series , 1978 .

[109]  G. Welland.  Norm Convergence of Riesz-Bochner Means For Radial Functions , 1975, Canadian Journal of Mathematics.

[110]  M. Nikolskii,et al.  Approximation of Functions of Several Variables and Embedding Theorems , 1971 .

[111]  A. Zygmund,et al.  On the existence of certain singular integrals , 1952 .

[112]  Charles Fefferman,et al.  The Multiplier Problem for the Ball , 1971 .

[113]  V. Panferov.  Cantor-Lebesgue theorem for double trignometric series , 1973 .

[114]  T. Walsh.  On the summation of conjugate Fourier integrals of functions of several variables , 1972 .

[115]  V. Shapiro.  Fourier series in several variables , 1964 .

[116]  A. Zygmund,et al.  Singular integrals and periodic functions , 1954 .

[117]  T. J. White.  On the restricted Cesro summability of double Fourier series , 1961 .

[118]  On sufficient conditions for the convergence of double series over rectangles , 1974 .

[119]  L. Gogoladze.  ON $ (H,k)$-SUMMABILITY OF MULTIPLE TRIGONOMETRIC FOURIER SERIES , 1977 .

[120]  Contributions to the Theory of Saturation for Singular Integrals in Several Variables, II Applications , 1967 .

[121]  C. Herz.  ON THE MEAN INVERSION OF FOURIER AND HANKEL TRANSFORMS. , 1954, Proceedings of the National Academy of Sciences of the United States of America.

[122]  P. Sjölin.  Fourier multipliers and estimates of the Fourier transform of measures carried by smooth curves in R , 1974 .

[123]  V. Shapiro.  Localization of Conjugate Multiple Trigonometric Series , 1955 .

[124]  M. L. Gol’dman.  Estimates for multiple Fourier transforms of radially symmetric monotone functions , 1977 .

[125]  J. Horváth.  Sur Les Fonctions Conjuguées À Plusieurs Variables , 1953 .

[126]  E. C. Titchmarsh.  On Conjugate Functions , 1929 .

[127]  C. Goffman,et al.  Some Aspects of Fourier Series , 1970 .

[128]  Localization on spheres , 1957 .

[129]  V. Shapiro.  Logarithmic Capacity of Sets and Double Trigonometric Series , 1954, Canadian Journal of Mathematics.

[130]  Joseph G. Marcinkiewicz.  Sur une méthode remarquable de sommation des séries doubles de Fourier , 1939 .

[131]  The Abel summability of conjugate multiple Fourier-Stieltjes integrals , 1971 .

[132]  Charles Fefferman,et al.  Inequalities for strongly singular convolution operators , 1970 .

[133]  S. Bochner.  Lectures on Fourier Integrals. (AM-42) , 1959 .

[134]  V. A. Il'in.  Conditions for the localization of partial sums of S. M. Nikol'skii's class of multiple trigonometric Fourier series , 1970 .

[135]  R. P. Gosselin.  On the divergence of Fourier series , 1958 .

[136]  A. Zygmund.  On the Summability of Multiple Fourier Series , 1947 .

[137]  H. Fédérer.  On Spherical Summation of the Fourier Transform of a Distribution Whose Partial Derivatives are Representable by Integration , 1970 .

[138]  C. Herz.  Fourier Transforms Related to Convex Sets , 1962 .

[139]  INTEGRALS AND n-DIMENSIONAL CONJUGATE FUNCTIONS , 1974 .

[140]  C. L. Kaul.  Fourier series of a generalised function of two variables , 1972 .