Equivalent conditions for representing analytic functions by exponential series

Let L() be an entire function of exponential type with simple zeros ₁, ₂, ...; let ¯D be the smallest closed convex set which contains all of the singularities of the function which is associated with L() in the sense of Borel. In [1] there are necessary and sufficient conditions on L() under which a function f(z) which is analytic in ¯D can be represented in D by a Dirichlet series with exponents ₁, ₂, ... We obtain new equivalent conditions on L().Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 91–104, July, 1976.     

On the equivalence of some rearrangements of the two-parameter Haar system

h ₁, L _p .

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

Exact Anomalous Dimensions of Composite Operators in the Obukhov–Kraichnan Model

We consider two stochastic equations that describe the turbulent transfer of a passive scalar field (x)(t,x) and generalize the known Obukhov–Kraichnan model to the case of a possible compressibility and large-scale anisotropy. The pair correlation function of the field (x) is characterized by an infinite collection of anomalous indices, which have previously been found exactly using the zero-mode method. In the quantum field formulation, these indices are identified with the critical dimensions of an infinite family of tensor composite operators that are quadratic in the field (x), which allows obtaining exact values for the latter (the values not restricted to the -expansion) and then using them to find the corresponding renormalization constants. The identification of the correlation function indices with the composite-operator dimensions itself is supported by a direct calculation of the critical dimensions in the one-loop approximation.     

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

H- .     

On some properties of multiple conjugate trigonometric series

We have obtained an estimate, in terms of partial and mixed moduli, of the continuity of deviation of the Cesáro (C, ) means ( = (₁,...,_n),_i , ₁ > –1, ) of the sequence of rectangular partial sums ofn-multiple (n>1) conjugate trigonometric series from then-multiple truncated conjugate function. This estimate implies the result on them -convergence (1) of (C, ) means (₁ > 0, ) provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown that them -convergence cannot be replaced by ordinary convergence.     

Structural assignment of neumann boundary feedback parabolic equations: the unbounded case in the feedback loop

Summary A parabolic equation defined on a bounded domain is considered, with input acting in the Neumann (or mixed) boundary conditions, and expressed as a specified feedback of the solution x of the form: x, wg₂ where w L₂(), gL₂() and is a continuous operator for <3/4:H ²()L ₂(). The freesystem is assumed unstable. In this case, the boundary feedback stabilization problem (in space dimension larger or equal to two) follows from an essentially more general result recently established by the authors in [L8]:under algebraic (full rank), verifiable conditions at the unstable eigenvalues, one can select boundaryvectors, so that the corresponding feedback solutions decay in the uniform operator normexponentially at t .Here, this stabilization peoblem is pushed further and made more precise, under the additional assumption that the original freesystem be self-adjoint: we show, in fact, that one can further restrict the boundaryvectors, so that the corresponding feedback solutions have the following more precise desirable structural property (the same enjoyed by free stablesystems): they can be expressed as an infinite linear combination of decaying exponentials. A semigroup approach is employed. Since structureof feedback solutions is sought, the analysis here is much more technical and vastly different from [L8],where only norm upper bound was the goal.The final version of this paper was completed while the authors were visiting the Istituto di Matematica « Pincherle » dell'Università di Bologna. Support from the Consiglio Nazionale delle Ricerche is gratefully acknowledged.     

Boundedness of singular integrals of variable rough Calderón-Zygmund kernels along surfaces

Letn2. The authors establish theL ²( ⁿ )-boundedness of singular integrals with variable rough Calderón-Zygmund kernels associated to surfaces satisfying some conditions.The research is supported in part by the NNSF and the SEDF of China.     

Der Beweis eines Satzes von G. Choodnovsky

1. If is a weakly compact cardinal then ( ⁺)( ). 2. If is measurable andU a normal ultrafilter then ( ⁺)(U ).

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Diese Arbeit ist ein Teil der Dissertation des Autors. Teilweise gefördert von der DFG.     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

Plate thermometer

Zusammenfassung Das Doppelintegral, das mit der Temperatur eines Plattenthermometers zusammenhängt, konvergiert langsam bei grossen Werten der Prandtl-Zahl . Es wird eine Methode angedeutet, um dieses Integral für alle nicht zu kleinen Werte von auszurechnen. Bei =1000 ist der ausgerechnete Wert ungefähr 20% grösser, als gewöhnlich angenommen wird.     

On the dimensions of automorphism groups of 8-dimensional ternary fields I

Let be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed subgroup of its automorphism group which is taken with the compact-open topology. It is proved that is either isomorphic to the compact exceptional Lie group G₂, or the (covering) dimension of is at most 11. This bound can be decreased to 10, if the ternary fixed fieldF of is connected.     

A generalized Laplace transform of generalized functions

- . . . , .     

On power series with positive coefficients

M. . , . , ^p () (). , , .     

On Fourier multipliers between Besov spaces with 0 <p o ≤ min{1,p 1}

, 0<0 min{1,p1}.     

Orthogonal systems invariant under an automorphism

, , - ( ) . , - , ( ) .     

Littlewood-paley operators on the generalized Lipschitz spaces

Littlewood-Paley operators defined on a new kind of generalized Lipschitz spaces ₀ ^,p are studied. It is proved that the image of a function under the action of these operators is either equal to infinity almost everywhere or is in ₀ ^,p , where –n<<1 and 1<p<.     

A generalization of the Heine limit for functions which converge on a base

. , , . . . .     

Radon—Nikodym theorems for multimeasures and transition multimeasures

— . , — . , .

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Research supported by N. S. F. Grant DMS-8802688.     

Exact error bound of approximation by interpolating splines inL-metric on the classesW p r (1≦p

Корнейчук  Н. П. 《Analysis Mathematica》1977,3(2):109-117

Comparison of Normal Linear Experiments by Quadratic Forms

Let X and Y be observation vectors in normal linear experiments =N(A, V) and F = N(B, W). We write > Fif for any quadratic form YGY there exists a quadratic formXHX such that E(XHX) = E(Y'GY) and var(X'HX) var(Y'GY).The relation > is characterized by the matrices A, B, V and W. Moreoversome connections with known orderings of linear experiments are given.