On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.     

On convergence of function series

, a _n f _n (x) . .     

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 - . , .     

Ein Störungssatz für abgeschlossene,lineare Operatoren

Zusammenfassung In vorliegender Note wird ein Satz von Kato [7] über die Störung eines abgeschlossenen, normal auflösbaren OperatorsT mit endlichem Null-defekt (T) durch einen streng singulären Operator verallgemeinert. Zu diesem Zweck wird für jedes 0 mit Hilfe des Kuratowskischen Nichtkompaktheitsmaßes eine KlasseC von beschränkten, linearen Operatoren eingeführt, welche sowohl die streng singulären Operatoren als auch die OperatorenS mit S enthält.Das erzielte Resultat steht in engem Zusammenhang mit den Untersuchungen von Gol'denteinn, Gohberg und Markus [5] und von Gol'denteienn und Markus [6].     

Finitely Valued f-Modules, An Addendum

In an -group M with an appropriate operator set it is shown that the -value set (M) can be embedded in the value set (M). This embedding is an isomorphism if and only if each convex -subgroup is an -subgroup. If (M) has a.c.c. and M is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets ₁ and ₂ and the corresponding -value sets and . If R is a unital -ring, then each unital -module over R is an f-module and has exactly when R is an f-ring in which 1 is a strong order unit.     

Mixed-norm estimates for a class of integral operators

(, ) — R ^m ×R ⁿ . f R ^m ×R ⁿ f_p,q, f L _p (R ^m) x y, L^q(Rⁿ). ׃ _q,r cƒ _p,r , ׃ R ^m ×R ⁿ , , , q r . , ( ¦¦) K ₀ (y); p, g r , K ₀.     

Arithmetic of the Modular Function j1,8

Since the genus of the modular curve X_1 (8) = _1 (8) * is zero, we find a field generator j _1,8(z) = ₃(2z)/₃(4z) (₃(z) := _n ein ^2z ) such that the function field over X ₁(8) is (j _1,8). We apply this modular function j _1,8 to the construction of some class fields over an imaginary quadratic field K, and compute the minimal polynomial of the singular value of the Hauptmodul N(j _1,8) of (j _1,8).     

Investigation of Haar and Franklin series in Hardy spaces

(L ¹,H) (, ) , ; H — . , , L ¹ . [13] , . , , , .     

Nicht endlich erzeugte Primideale in Steinschen Algebren

In this note we exhibit a closed prime idealF in the ring Ó(³) of all holomorphic functions on ³ which is not finitely generated.F is the ideal of a certain irreducible curve Y³, obtained as the image of a proper holomorphic map f³.

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Herrn Karl Stein gewidmet     

L 2-approximation by the translates of a function and related attenuation factors

Summary Let be thek-dimensional subspace spanned by the translates (·–2j/k),j=0, 1, ...,k–1, of a continuous, piecewise smooth, complexvalued, 2-periodic function . For a given functionfL ²(–, ), its least squares approximantS _kf from can be expressed in terms of an orthonormal basis. Iff is continuous,S _kf can be computed via its discrete analogue by fast Fourier transform. The discrete least squares approximant is used to approximate Fourier coefficients, and this complements the works of Gautschi on attenuation factors. Examples of include the space of trigonometric polynomials where is the de la Valleé Poussin kernel, algebraic polynomial splines where is the periodic B-spline, and trigonometric polynomial splines where is the trigonometric B-spline.     

Minimax filtration of linear transformations of stationary sequences

We will consider the problem of determining a linear, mean-square optimal estimate of the transformation of a stationary random sequence (k) with density f() from observations of the sequence (k) + n(k) withk0, where (k) is a stationary sequence not correlated with (k) with density g(). The least favorable spectral densities and minimax (robust) spectral characteristics of an optimal estimate A for different classes of densities are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 92–99, January, 1991.     

On term by term dyadic differentiability of Walsh series

. . . . : {ja _j },j=1,2,... — , f(x) , , f ^[1](x) — f .     

On the angular derivative and univalence

f . , , — , A f f(). , , f() 0 . , , ,A , f . , f() - f() . , , . (1976) ( ¦f(z)¦<1) . . (1969) ( ).     

On the asymptotic behavior of an unstable solution of an inhomogeneous stochastic diffusion equation

Sufficient conditions are obtained for the normalized trajectories of an unstable solution of the one-dimensional Itô stochastic differential equation with coefficientsa(t, x) and(t, x) to coincide with the normalized trajectories of the solution of the equation with coefficientsa(x) and(x) ast assuming that the coefficientsa(t, x) and(t, x) have a certain average closeness to the coefficientsa(x) and(x) over time ast.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 15, pp. 3–10, 1987.     

On depth first search strees inm-out digraphs

We consider depth first search (DFS for short) trees in a class of random digraphs: am-out model. Let _i be thei ^th vertex encountered by DFS andL(i, m, n) be the height of _i in the corresponding DFS tree. We show that ifi/n asn, then there exists a constanta(,m), to be defined later, such thatL(i, m, n)/n converges in probability toa(,m) asn. We also obtain results concerning the number of vertices and the number of leaves in a DFS tree.     

Theorem on a convergence condition in the spaces

For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if f_n(x) f(x) and f _n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f _n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.     

On the summability of Fourier series with the method of lacunary arithmetic means

. f- ,S _n (f)— . {n _k }, n _k+1/n _k >1+ck ^– ,— , 0<1/2, f ₀, .     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

Some Classes of L 1-Convergence of Fourier Series

In this paper equivalent classes of the classes M and S _pr , p > 1, > 0. r { 0,1,2, ... ,[]} defined by Shuyun [3] are obtained. Then, it is shown that the class S _pr , 1 > p 2, 0, r {0,1,2,...,[]} is a subclass of BVC _r , where S _pr is the equivalent class of the Shuyun's class S _pr , BV is the class of null sequences of bounded variation and C _r is the extension of the Garrett--Stanojevic class. As a corollary of this result, we have obtained the theorem, proved in [7].     

Determinants in net subgroups

Suppose R is a commutative ring with 1, =( _ij ) is a fixedD-net of ideals of R of ordern, and G is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for a homomorphismdet of the subgroup G() into a certain Abelian group (). Let I be the index set {1...,n}. For each subset I let ()= _{ij ji} , wherei, ranges over all indices in and j independently over the indices in the complement I ((I) is the zero ideal). Letdet (a) denote the principal minor of order ||n of the matrixa G () corresponding to the indices in , and let' () be the Cartesian product of the multiplicative groups of the quotient rings R/() over all subsets I. The homomorphismdet is defined as follows: It is proved that if R is a semilocal commutative Bezout ring, then the kernelKer det coincides with the subgroup E() generated by all transvections in G(). For these R is also definedTm det .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 37–49, 1982.