Automorphic L-Functions in the Weight Aspect

Let S_k(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let S_k(Γ)⁺ be the set of all Hecke eigenforms from this space with the first Fourier coefficient a_f(1) = 1. For f ∈ S_k(Γ)⁺, consider the Hecke L-function L(s, f). Let

On generalized local time for the process of brownian motion

We prove that the functionals of a d-dimensional Brownian process are Hida distributions, i.e., generalized Wiener functionals. Here, δ_Γ(·) is a generalization of the δ-function constructed on a bounded closed smooth surface Γ⊂R ^d , k≥1 and acting on finite continuous functions φ(·) in R ^d according to the rule where ι(·) is a surface measure on Γ.     

Representation of a solution of the dirichlet problem in a planar cusp domain as a logarithmic single-layer potential

The problem of representing the solution of the Dirichlet problem for the Laplace equation as a single-layer potential V _ϱ with unknown density ϱ is known to lead to the equation V _ϱ = f for density ϱ, where f is the Dirichlet boundary data. Let Γ be the boundary of a bounded planar domain with an outward or inward peak and T(Γ) be the space of the traces on Γ of functions with finite Dirichlet integral over R ². It is shown that the operator $ L_2 \left( \Gamma \right) \ominus 1 \mathrel\backepsilon \varrho \to V\left. \varrho \right|\Gamma \in T\left( \Gamma \right) $ L_2 \left( \Gamma \right) \ominus 1 \mathrel\backepsilon \varrho \to V\left. \varrho \right|\Gamma \in T\left( \Gamma \right) is continuous, and the operator $ \varrho \to V\varrho - \overline {V\varrho } $ \varrho \to V\varrho - \overline {V\varrho } (where $ \bar u $ \bar u denotes u averaged over Γ) can be uniquely extended to the isomorphism      

Integral estimates of the solutions to the helmholtz equation in unbounded domains

The following boundary value problem is studied:

Protuberance effect in the generalized functional Strassen-Révész law

The set of increments of the Wiener process

Asymptotics and weighted estimates of meixner polynomials orthogonal on the gird {0, δ, 2δ, ...}

Suppose that 0<δ≤1,N=1/δ, and α, ga≥0, is an integer. For the classical Meixner polynomials orthonormal on the gird {0, δ, 2δ, ...} with weight ρ(x)=(1-e ^−δ)^αг(Nx+α+ 1)/г(Nx+1), the following asymptotic formula is obtained: . The remainderv _n,N ^α (z) forn≤λN satisfies the estimate

On non-closure of range of values of elliptic operator for a plane angle

Summary Let a plane angle of opening α∈(π, 2π). LetP _D andP _N the Dirichlet and Neumann problems associated to the Poisson equation in . ForP _D andP _N it is proved non existence of solution in L _p ( ) whenp=2/(1±π/α). In other words, the ranges of elliptic operators naturally associated toP _D andP _N are not-closed in L _p ( ) forp=2/(1±π/α).

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Sunto Sia } un angolo piano di apertura α∈(π, 2π). SianoP _D eP _N i problemi di Dirichlet e di Neumann associati all'equazione di Poisson in . PerP _D eP _N si prova non esistenza di soluzioni in L _p ( ) quandop=2/(1±π/α). Vale a dire i ranges degli operatori ellittici naturalmente associati aP _D eP _N sono non-chiusi in π^--AgBrα _K L _p ( ) perp=2/(1±π/α).

     

Certain inverse problems for parabolic equations

In the paper, we study the inverse problem of finding the solution u and the coefficient q from the following data:

Isometric embeddings of coinvariant subspaces of the shift operator

Let θ be an inner function. The main aim of this paper is to describe all positive measures on the unit circle such that for all continuous functions f∈H²⊖θH². Bibliography: 8 titles. Translated fromZapiski Nauchnykh Serninarov POMI, Vol. 232, 1996, pp. 5‐15. Translated by V. V. Kapustin.     

On a uniqueness theorem for functions with a sparse spectrum

We present an example of a set { } satisfying the following two conditions: (1) there exists a nonzero positive singular measure on the unit circle { } with spectrum in Λ; (2) if the spectrum of f∈L¹ { } is contained in Λ and f vanishes on a set of positive measure, then f=0. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 7–14. Translated by A. B. Aleksandrov.     

K-Theory for C*-Algebras of One-Relator Groups

We compute the K-theory groups of the reduced C*-algebra C _r ^* () of a one-relator group . We prove that every such group is K-amenable in the sense of Cuntz. For a torsion-free one-relator group =<X|r> such that r is not a product of commutators, we give a direct proof of the fact that the Baum–Connes analytical assembly map

Problems similar to the additive divisor problem

For multiplicative functions ƒ(n), let the following conditions be satisfied: ƒ(n)≥0 ƒ(p ^r)≤A ^r,A>0, and for anyε>0 there exist constants ,α>0 such that and Σ _p≤x ƒ(p) lnp≥αx. For such functions, the following relation is proved:

Approximation of functions in\widetilde{S_1^\Gamma L} S 1 r H by entire functionsS 1 r H by entire functions

This paper is a continuation of [1], we study the approximation of the function classes , S ₁ ^Λ H in L_q metric (1≤q<∞) by entire functions whose spectrals lie in step hyperbolic crosses and obtain the asymptotic estimates of these quantities. In Memory of Professor M. T. Cheng Supported by Beijing Natural Science Foundation.     

Estimates of the independence number of a hypergraph and the Ryser conjecture

For the Legendre-Sobolev orthonormal polynomials depending on the parametersM≥0,N≥0, weighted and uniform estimates on the orthogonality interval are obtained. It is shown that, unlike the Legendre orthonormal polynomials, the polynomials forM>0,N>0 decay at the rate ofn ^−3/2 at the points 1 and −1. The values of are calculated. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 871–880, December, 1997. Translated by N. K. Kulman     

The behavior of Riesz means of the coefficients of a symmetric square L-function

Let f(z) be a holomorphic Hecke eigencuspform of even weight k with respect to SL(2, Z) and let L(s, sym ² f) = ∑ _n=1 ^∞ c_nn^−s, Re s > 1, be the symmetric square L-function associated with f. Represent the Riesz mean (ρ ≥ 0)

Estimates of maximal distances between spaces whose norms are invariant under a group of operators

We consider the class A_г of n-dimensional normed spaces with unit balls of the form: , where B _n ¹ n is the unit ball of ℓ _n ¹ , Γ is a finite group of orthogonal operators acting in ℝⁿ, and U is a “random” orthogonal transformation. It is proved that this class contains spaces with a large Banach-Mazur distance between them. If the cardinality of Γ is of order n^c, it is shown that, in the power scale, the diameter of A_г in the modified Banach-Mazur distance behaves as the classical diameter and is of order n. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 33–42.     

Strict inequalities for the derivatives of functions satisfying certain boundary conditions

For functions satisfying the boundary conditions

Estimates of the rate of decay of solutions to the impedance mixed problem for the wave equation in a region with noncompact boundary

The paper is devoted to the study of the behavior of the following mixed problem for large values of time:

Parallelepipeds circumscribed about a convex body in 3-space

It is proved that each convex body K ⊂ ℝ³ of volume V(K) is contained in a parallelepiped of volume . Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 79–87.     

Kernels of Toeplitz operators,smooth functions,and Bernstein type inequalities

Let ϕ be a unimodular function on the unit circle and let K_p(ϕ) denote the kernel of the Toeplitz operator T_ϕ in the Hardy space H^p, p≥1; . Suppose K_p(ϕ)≠{0}. The problem is to find out how the smoothness of the symbol ϕ influences the boundary smoothness of functions in K_p(ϕ). One of the main results is as follows. Theorem 1 Let 1<p, q<+∞, 1<r≤+∞, q⁻¹=p⁻¹+r⁻¹. Suppose |ϕ|≡1 on and ϕ∈W _r ¹ (i.e., ). Then K_p(ϕ)⊂W _q ¹ . Moreover, for any f∈K_p(ϕ) we have ‖f′‖_q≤c(p, r)‖ϕ′‖_r ‖f‖. Bibliography: 19 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 5–21. Translated by K. M. D'yakonov.