Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

On unconditionally convergent basis expansions in the space of continuous functions

[0,1], - H .

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This paper was written during the author's scholarship at the State University of Odessa in the USSR.     

Produktsätze für Verfahren zur Limitierung von Doppelfolgen

{p _mn } - ₀₀>0, (1, 1) (1.1) (1.2). {s _mn } J _p - ( bJ _p -lims _mn =), (1.3) 0<x,y<1 p _s (, )/p(x, y) x, y 1-. {r _mn } - , (1.5) 0<, <1. N _rp - , (1.6). , bJ _p -lims _mn = bJ _q -lim(N _rps) _mn =. J _p - . , .     

On the convergence in a restricted sense of multiple series

Z ^d — k=(k ₁, ...,k _d) k _j,d1.d- (8), . . a _k s _m= a _k s, >0 N, min (m ₁,...,m _d)N, ¦s _m–s¦. , , >0 N, min (m ₁,...,m _d)N min (n ₁,...,n _d)N, ¦s _m–s _n. . , (8) , >0 N, max (b ₁,...,b _d) N, m — Z ^d , m1, ¦s(b, m)¦ where      

Summability of multiple Fourier series. Growth of Lebesgue constants

. , . . , H ₂ . , .     

Limiting behavior of certain mixtures of distributions

U — [0, 1] Y — . X=[1–U ^1/v /Y], U Y.     

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.     

On cubature formulas on the basis of lattices

Q (.. , L). Q . P(S_r(2)) — 2 (S _r(2) (r — ). , M(P(S _r(m=sup{t(·)t(·)₁:t P(S _r(2)),t 0}. , /4+(1)M(P(S _r(2)))/r ²15/17+(1)(r+). (Q), Q L.     

On the Cesàro summability of orthogonal sequences

(C, ). , . 0<<1. 1) - ( _k ), _k =a _k , (C, ), . 2) , , (C, ) ; _k = =¦a _k ¦.     

On some problems of interpolation by ℒ-splines

_n (D) — ,s — _n (D), _v (v=1, 2, ...,s/2) — . _m={0x ₀<x ₁<...<x _2m–1<2,x _2m =x ₀+2} , x _j +1–x _j <(4s max _v )^–1,j=0, 1, ..., 2m –1, ( ) 2- - _n,m 2m , _m . , L _q - (1q) W ( _n )={f ₂ :f ^(n–1)AC ₂ , _n (D)f 1} 2- - (s _n f), _m . , - - _n,m .

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The author expresses his gratitude to Yu. N. Subbotin for a useful discussion on the results of this paper.     

Orthonormal systems satisfying an inequality of S. M. Nikol'skii

N- (p, q) (1 pN-, L _p - L _q -. , , , L L _q - , , .     

Groups acting simply transitively on the vertices of a building of typeà 2, II: The casesq=2 andq=3

If (P, L) is a projective plane and is a triangle presentation compatible with a point-line correspondence :P L, then gives rise to a group and a thick building of typeà ₂ on the vertices of which acts simply transitively. We find all triangle presentations (up to natural equivalence) compatible with some point-line correspondence :P L, when (P, L) is the projective plane of orderq=2 orq=3. For some, but not all, of these , is isomorphic to the building associated withG=PGL(3,K) whereK is a local field with discrete valuation and residual field of orderq. We identify the for which this is the case, and in these cases, find embeddings of intoG. We also describe the arithmetic nature of these groups.     

A relation between Chung's and Strassen's laws of the iterated logarithm

Summary Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)^–1/2–f(x)¦, 0x1 suitably normalized as T.This extends Chung's result valid for f(x)0, stating that lim inf.[ sup ¦(2T loglogT)^–1/2 W(xT)¦(loglog T)^–1]=/4 a.s. T 0x1     

A generalized Laplace transform of generalized functions

- . . . , .     

Monotonicity properties of zeros of generalized Airy functions

Summary We show, among other things, that the positive zeros of a solution ofy +x y=0,y(0)=0 decrease to 1 as increases, 0.

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Sommario Si dimostra, tra l'altro, che gli zeri positivi d'una soiuzione diy +x y=0,y(0)=0 decrescono al limite 1, quando cresce, 0.

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To the memory of Milo Háik

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This research was supported by grants from the Natural Sciences and Engineering Research Council (Canada) and Consiglio Nazionale delle Ricerche (Italy). Some of the work was done while the second-named author was visiting the Department of Mathematics, University of Torino.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

On the central limit theorem for geometrically ergodic Markov chains

Let X₀,X₁,... be a geometrically ergodic Markov chain with state space and stationary distribution . It is known that if h: R satisfies (|h|²⁺)< for some >0, then the normalized sums of the X_is obey a central limit theorem. Here we show, by means of a counterexample, that the condition (|h|²⁺)< cannot be weakened to only assuming a finite second moment, i.e., (h²)<.Reasearch supported by the Swedish Research Council.     

On the interrelation of the generalized Besov-Nikol'skii and Weyl-Nikol'skii classes of functions

, , , . .

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Dedicated to Professors K. Tandori and L. Leindler on the occasion of their anniversaries

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This work was completed in support of the Russian Foundation of Fundamental Research (Project # 96-01-00094) and of the International Scientific Foundation (Grant # NCI-300).     

Morera type theorems on the unit disc

, . . : , , , , .     

On the newly generalized absolute Riesz summability of orthogonal series

[3] , [1], ¦C,,(t)¦ _k - . , , ¦R, , (t)¦ _k - ¦R, , (t)¦ _k - . (t)=1,k=1 [7], [9].

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This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant#T 016 393.