On the divergence of linear summation methods of Fourier series

[0, 1],fL(0,2),

Detonation Wave Propagation in Rotational Gas Flows

This paper studies the propagation of detonation and shock waves in vortex gas flows, in which the initial pressure, density, and velocity are generally functions of the coordinate — the distance from the symmetry axis. Rotational axisymmetric flow having a transverse velocity component in addition to a nonuniform longitudinal velocity is considered. The possibility of propagation of Chapman–Jouguet detonation waves in rotating flows is analyzed. A necessary conditions for the existence of a Chapman–Jouguet wave is obtained.     

The order of growth of quadratic partial sums of a double Fourier series

Translated from Matematicheskie Zametki, Vol. 51, No. 6, pp. 69–79, June, 1992.     

2-Adic wavelet bases

Within the theory of multiresolution analysis, a method of constructing 2-adic wavelet systems that form Riesz bases in L ²(ℚ₂) is developed. A realization of this method for some infinite family of multiresolution analyses leading to nonorthogonal Riesz bases is presented.     

-adic refinable functions and MRA-based wavelets

The main goal of this paper is the development of the MRA theory in . We described a wide class of p-adic refinement equations generating p-adic multiresolution analyses. A method for the construction of p-adic orthogonal wavelet bases within the framework of the MRA theory is suggested. A realization of this method is illustrated by an example which gives a new 3-adic wavelet basis. Another realization leads to the p-adic Haar bases which were known before.     

p-Adic Haar Multiresolution Analysis and Pseudo-Differential Operators

The notion of p-adic multiresolution analysis (MRA) is introduced. We discuss a “natural” refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of p characteristic functions of mutually disjoint discs of radius p ⁻¹. This refinement equation generates a MRA. The case p=2 is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ℒ²(ℚ₂) generated by the same Haar MRA. All of these new bases are described. We also constructed infinity many different multidimensional 2-adic Haar orthonormal wavelet bases for ℒ²(ℚ₂ ⁿ ) by means of the tensor product of one-dimensional MRAs. We also study connections between wavelet analysis and spectral analysis of pseudo-differential operators. A criterion for multidimensional p-adic wavelets to be eigenfunctions for a pseudo-differential operator (in the Lizorkin space) is derived. We proved also that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary prerequisites for intensive using our wavelet bases in applications. Our results related to the pseudo-differential operators develop the investigations started in Albeverio et al. (J. Fourier Anal. Appl. 12(4):393–425, 2006).      

Comparison of the >L 1-Norms of Total and Truncated Exponential Sums

The paper is concerned with a conjecture stated by S. V. Bochkarev in the seventies. He assumed that there exists a stability for the L ¹-norm of trigonometric polynomials when adding new harmonics. In particular, the validity of this conjecture implies the well-known Littlewood inequality. The disproof of a statement close to Bochkarev's conjecture is given. For this, the following method is used: the L ¹-norm of a sum of one-dimensional harmonics is replaced by the Lebesgue constant of a polyhedron of sufficiently high dimension.     

Approximation by frame-like wavelet systems

A wide class of MRA-based wavelet systems which are not frames in L₂(Rd), generally speaking, is studied. Frame-type expansions over a pair of dual wavelet systems (with the series converging in different senses) and their approximation order are investigated.