Sur la répartition des valeurs de la fonction d'Euler

Let (x) denote the number of those integers n with (n) x, where denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that (x)-Ax R(x), where A=(2)(3)/(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.     

Über das Verzweigungsverhalten eines nichtlinearen Eigenwertproblems in der Nähe eines unendlich vielfachen Eigenwertes der Linearisierung

We consider a nonlinear operator G=A+R in Hilbert space, where A is linear, bounded and self-adjoint, and R is completely continous with R(x)=0(x) for x0. We establish various conditions under which an eigenvalue ₀ of A ofinfinite multiplicity is a bifurcation point of the nonlinear eigenvalue problem G(x)= x. To this end, we first prove the existence of a certain type of approximation procedure the construction of bifurcation solutions in the case of infinite multiplicity. The existence theorems for the problem G(x)= x are then obtained by applying this procedure to various situations.     

Interpolation and best polynomial approximation in the domain ofp-adic integers

, , . , . Lip

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The authors are indebted to Professor R. Bojanic for his valuable remarks and suggestions, especially for the simplification of the proof of Theorem 4.     

A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

Orthogonal systems invariant under an automorphism

, , - ( ) . , - , ( ) .     

Some expansion formulae for Fox'sH-function of two variables involving associated Legendre functions

H- .     

OnU ϕ-sets of trigonometric integrals

, (t) >0 E(–, +),E<, , ¦f(t)¦ (t) xE, f(t)=0 (–, +).     

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.     

Asymptotics of Eigenvalues of a Plate with Small Clamped Zone

Different types of asymptotic expansions are constructed and justified for eigenvalues of the Dirichlet problem for the biharmonic operator in a plane domain with a small hole of the diameter (the Kirchhoff-Love plate clamped at ). Depending on properties of an eigenfunction of the limiting problem, the expansions happen either to be series in powers of , or to contain terms, holomorphic in | ln |^–1     

Rate of approximation by rectangular partial sums of double orthogonal series

( ) . .

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Dedicated to Professor K. Tandori on his seventieth birthday

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This research was supported in part by Grant # K41 100 of the Joint Fund of the Government of Ukraine and the International Science Foundation.     

Linearization of operator functions on arbitrary open sets

In this paper it is shown that after a suitable extension an operator function A, which is holomorphic on an open set in , is equivalent on to a linear pencil S-V. If is bounded, then V turns out to be right invertible, and in that case a further extension is equivalent on to a linear pencil T-I.     

On differentiability of SRB states for partially hyperbolic systems

Consider a one parameter family of diffeomorphisms f such that f ₀ is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let be any u-Gibbs state for f . We prove (Theorem 1) that for any C function A the map (A) is differentiable at =0. This implies (Corollary 2.2) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to . We apply this result (Corollary 3.3) to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique SRB measure with good statistical properties.     

Best mean-square approximation of functions of m variables

Let be the best mean-square approximation of a functionf(x) L₂(R_m) (m=1, 2, ...) by integral functions of the exponential spherical type (in the sense of thel _q metric, 0>0, where(f,/; l _p)L₂(R_m) is the spherical (in the sense of the metricl _p, 0f(x) L₂(R_m). For the quantity two-sided estimates are obtained which are uniform in the parameters m, q, and p. Similar results are also obtained in the case of q=p=2 for classes of functions W _f2 (R_m) (=1,2,...).Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 913–924, December, 1973.The author would like to express his deep gratitude to N. I. Chernykh under whose guidance this work has been carried out.     

The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

On a Darboux problem for a third-order hyperbolic equation with multiple characteristics

A Darboux type problem for a model hyperbolic equation of the third order with multiple characteristics is considered in the case of two independent variables. The Banach space, 0, is introduced where the problem under consideration is investigated. The real number ₀ is found such that for > ₀ the problem is solved uniquely and for < ₀ it is normally solvable in Hausdorff's sense. In the class of uniqueness an estimate of the solution of the problem is obtained which ensures stability of the solution.     

Converse theorems for multidimensional Kantorovich operators

L _p [0, l]. . . - .

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Supported by National Science Foundation, Zhejiang Provincial Science Foundation of China, and Alexander von Humboldt Foundation of Germany.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

On strong summability of polynomial expansions

—.

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Dedicated to Professor L. Leindler on his 50^th birthday     

Approximation of functions of several variables by trigonometric polynomials with given number of harmonics,and estimates of ε-entropy

. - . .. . . . , 340 048 . 1      

On eigenvalues of perturbed quadratic matrix polynomials

Among other results, it is shown that ifC andK are arbitrary complexn×n matrices and if det( ₀ ² I₀ C+K)=0 for some ₀0 (resp. ₀=0), then the Newton diagram of the polynomialt(, ) = det(² I+(1+)C+K expanded in (–₀) and , has at least a point on or below the linex+y=b (resp. has no expanded in (–₀) and , has at least a point on or below the of 0 as an eigenvalue of ₀ ² I+₀ C+K. These are extensions of similar results deu to H. Langer, B. Najman, and K. Veseli proved for diagonable matricesC, and shed light on the eigenvalues of the perturbed quadratic matrix polynomials. Our proofs are independent and seem to be simpler