Some criteria for the decomposability of finite groups

In this paper we prove the following fundamental results.Theorem 1: A finite unsolvable group, every involution of which is contained in a proper isolated subgroup, is decomposable.Theorem 2: Suppose the finite unsolvable group G contains a strongly isolated subgroup M of odd order with isolated normalizer N(M) of even order. If ¦N(M) (M) ¦ > 2, the group G is isomorphic with one of the groups: 1) PSL(2, q), q odd; 2) PGL (2, q), qodd.Translated from Matematicheski Zametki, Vol. 12, No. 6, pp. 717–725, December, 1972.     

Marcinkiewicz multiplier theorem and the Sunouchi operator for Ciesielski–Fourier series

Some classical results due to Marcinkiewicz, Littlewood and Paley are proved for the Ciesielski–Fourier series. The Marcinkiewicz multiplier theorem is obtained for L_p spaces and extended to Hardy spaces. The boundedness of the Sunouchi operator on L_p and Hardy spaces is also investigated.     

Fundamental solution and Fourier series in eigenfunctions of degenerate elliptic operator

We study the degenerate elliptic differential operator of the second order in the divergence form. The operator is assumed to be symmetric. The weight function which is describing the degeneration of the coefficients (or singularity) assumed to be in the Muckenhoupt class. We prove the uniform estimates for the fundamental solution of this operator and obtain the conditions which guarantee the absolute and uniform convergence of Fourier series in eigenfunctions. These results might be applied to the ground of Fourier method.     

Marcinkiewicz r-classes and Fourier series expansions of operator ergodic Stieltjes convolutions

We study the Fourier series expansions in the strong operator topology for operator-valued Stieltjes convolutions of Marcinkiewicz r-classes against spectral decompositions of modulus-mean-bounded operators. The vector-valued harmonic analysis resulting can be viewed as an extension of traditional Calderón–Coifman–G. Weiss transference without being constrained by the latter?s requirement of power-boundedness.     

Absolute convergence of Fourier series in eigenfunctions of an elliptic operator

In this article we investigate absolute convergence of Fourier series in eigenfunctions of an m-th order elliptic operator on functions in the Besov class B _2, ^N/2 . We show that in terms of Besov classes the theorem of Peetre on absolute convergence of series in eigenfunctions in the class B _2,1 ^N/2 is best possible. We construct a function in B _2, ^N/2 whose Fourier series is absolutely divergent at any preassigned point.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 435–448, March, 1976.In conclusion, the author thanks Sh. A. Alimov for guiding this work.     

Fourier series for H-function

The object of this paper is to establish two Fourier series expansions for the H-functions.     

Integrability properties of the maximal operator on partial sums of Fourier series

In the present work we extend a result ofP. Sjölin on the integrability properties of the maximal operator on partial sums of Fourier Series,S ^*. We actually show that the result is applicable to more general operators and could be considered as an abstract extrapolation principle.     

The Cauchy operator for basic hypergeometric series

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's transformation formula and Sears' transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T(bD_q). Using this operator, we obtain extensions of the Askey–Wilson integral, the Askey–Roy integral, Sears' two-term summation formula, as well as the q-analogs of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers–Szegö polynomials, or the continuous big q-Hermite polynomials.     

Numerical pseudodifferential operator and Fourier regularization

The concept of numerical pseudodifferential operator, which is an extension of numerical differentiation, is suggested. Numerical pseudodifferential operator just is calculating the value of the pseudodifferential operator with unbounded symbol. Many ill-posed problems can lead to numerical pseudodifferential operators. Fourier regularization is a very simple and effective method for recovering the stability of numerical pseudodifferential operators. A systematically theoretical analysis and some concrete examples are provided.     

On Bohr's theorem for multiple Fourier series

The problem of Pringsheim uniform convergence of multiple Fourier series in the trigonometric system is considered. A multidimensional analog of Bohr's theorem on the uniform convergence of the Fourier series of a continuous function after a homeomorphic chance of variable is proved.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 913–924, December, 1998.     

Fourier series for zeta function via Sinc

In this paper we derive some Fourier series and Fourier polynomial approximations to a function F which has the same zeros as the zeta function, ζ(z) on the strip {z∈C:0<Rz<1}. These approximations depend on an arbritrary positive parameter h, and which for arbitrary ε∈(0,1/2), converge uniformly to ζ(z) on the rectangle {z∈C:ε<Rz<1-ε,-π/h<Iz<π/h}.     

The Littlewood-Paley theory for multiple Fourier series

We study the Littlewood-Paley theory for multiple Fourier series with arbitrary period lattice. It is shown that the constants in the Littlewood-Paley inequality can be chosen to be independent of the mutual arrangement of the period lattice and the set of dyadic parallelepipeds. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 155–169.