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1.
Let G and {kj} be the domain and the array mentioned in the title (the boundary of the domain is assumed to be rectifiable). One describes a general scheme for the approximation of fonctionsf in the domain G, representable in the form f(z)=(2i)–1g()(–z)–1d, where g Lz (G), by a sequence of rational fractions. The characteristic feature of this scheme is the fact that the poles k of the fraction lie in the k-th row of the array {kj}. There is given a condition on {kj}, necessary and sufficient in order that each functionf, of the kind described above, should admit a uniform approximation inside G with the aid of the indicated scheme. In the case when this condition is not satisfied and \G.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 254–273, 1989.  相似文献   

2.
LetZ be a compact set of the real space with at leastn + 2 points;f,h1,h2:Z continuous functions,h1,h2 strictly positive andP(x,z),x(x 0,...,x n ) n+1,z , a polynomial of degree at mostn. Consider a feasible setM {x n+1z Z, –h 2(z) P(x, z)–f(z)h 1(z)}. Here it is proved the null vector 0 of n+1 belongs to the compact convex hull of the gradients ± (1,z,...,z n ), wherez Z are the index points in which the constraint functions are active for a givenx* M, if and only ifM is a singleton.This work was partially supported by CONACYT-MEXICO.  相似文献   

3.
Certain Convolution Operators for Meromorphic Functions   总被引:3,自引:0,他引:3  
Let (p N) be the class of functions analytic in 0 < |z| < 1. A convolution operator Lp(a, c) on p is introduced. This paper gives some sharp inequalities for f(z) satisfying Re{(1 – )zpLp(a, c) f(z) + zpLp(a + 1, c) f(z)} > , where 0, < 1, a > 0 and c 0, –1, –2,....AMS Subject Classification (1991) 30C45 30A10  相似文献   

4.
Let be a domain of N . We study the infimum 1(h) of the functional |u| p +h p V(x)|u| p dx in W 1,p () for ||u|| LP()= 1 where h > 0 tends to zero and V is a measurable function on . When V is bounded, we describe the behaviour of 1(h), in particular when V is radial and 'slowly' decaying to zero. We also study the limit of 1(h) when h 0 for more general potentials and show a necessary and sufficient condition for 1(h) to be bounded. A link with the tunelling effect is established. We end with a theorem of existence for a first eigenfunction related to 1(h).  相似文献   

5.
The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K 1(A[z,z-1]) of a twisted Laurent polynomial extension A[z,z-1] of a ring A is generalized to a decomposition of the Whitehead group K 1(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z-1]. The decomposition involves a summand W1(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a1z+a2z2+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )ab W1(A, ) is not an isomorphism.  相似文献   

6.
The sequence spaceH P (z)={{f (zh)}:f H p} is defined for a fixed sequence Z={zk} of different points of the open unit disk and the Hardy class HP of analytic functions in the disk. For an arbitrary p[1, ) is constructed a point sequence Z= {zk} such that 1h p(z), but r hp (Z) for r > 1. It follows from a well-known result of L. Carleson that the inclusions r h (Z) for all r[1,] are equivalent.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 503–508, April, 1977.  相似文献   

7.
One considers the class G of holomorphic functions in a domain G, whose values are contractions in a separable Hilbert space. It is proved that if T(·) G , T(z0) is a weak contraction, its singular part Ts(z0) is complete, and the increments T(z)–T(z0) are not too large (for example, finite-dimensional), then the operator Ts(z0) is complete for almost all zG. If, however, T(z0) is, in addition, completely nonunitary and satisfies definite smoothness conditions, then in the nontrivial case the spectrum [z] of the contraction Ts(z) (zG) is a thin set: The proof of the mentioned results is based on the investigation of the formulas obtained in the paper, connecting the characteristic functions of the contractions T(z) for different values of zG.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 30–44, 1987.  相似文献   

8.
Let be an open set in the complex plane and let be a holomorphic function on . Let K be a compact subset of with nonempty interior such that 0 K. Let be the Borel measure of R 4 C 2 given by(E = K E(z, (z))|z|–2 d(z)where 0 < 2 and d(x 1 + ix 2) = dx 1 dx 2 denotes the Lebesgue measure on C. Let T be the convolution operator T f = * f. In this paper we characterize the type set E associated to T .  相似文献   

9.
We prove that a surjective morphism :XX between ruled surfaces is finite and it descends to a finite morphism :CC between base curves of X and X. When is restricted to the fibres of X, it has a constant degree, say a, and then deg=a deg. In addition, we have several properties on the inverse image of a minimal section and a fibre of X as well as on the direct images. We also investigate precisely the case when both C and C are elliptic and X is the fibre product C ×C X especially.  相似文献   

10.
In this paper we are concerned with the summability of the geometric series by matrix methods. We prove the following theorem: Suppose Mo:={z:|z|<1}, M1, M2, is a collection of countably many Lebesgue measureable, disjoint sets. For k=1,2, let fk be a prescribed function, analytic on . Then there exists a triangular matrix , such that the V-transform {n(z)} of the geometric series has the following properties: {n(z)} converges compactly to on Mo; for k=1,2, there are sets Bk, such that has Lebesgue-measure zero and n(z)fk(z) for zBk; if there is a set B*, such that B*M* has Lebesgue-measure zero and {n(z)} diverges for zB*.  相似文献   

11.
Carleson measure characterization of Bloch functions   总被引:1,自引:0,他引:1  
We give several equivalences of Bloch functions and little Bloch functions. Using these results we obtain the generalized Carleson measure characterization of Bloch functions and the generalized vanishing Carleson measure characterization of little Bloch functions, that is,f B if and only if |D f(z)| p (1-|z|2)p-1 dm(z) is a generalized Carleson measure;f B 0 if and only if |D f(z)| p (1-|z|2)p-1 dm(z) is a generalized vanishing Carleson measure, whereD f( > 0) is the fractional derivative of analytic functionf of order, m denotes the normalised Lebesgue measure.Supported partly by the Young Teacher Natural Science Foundation of Shandong Province.  相似文献   

12.
Let Tr be the class of functionsf (z)=z+c2z2+..., regular in the disk ¦z¦ <1, real on the diameter-1f (z) · Im z>0 in the remainder of the disk ¦z¦ <1. Let z f be the solution off (z)= f (a) on Tr, where is any fixed complex number 0, 1, is any fixed real number, ¦¦< 1. We determine the region of values of the functional zf on the class Tr. Variation formulas for Stieltjes integrals due to G.M. Goluzin are used.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 41–52, July, 1971.  相似文献   

13.
Summary We find the complete set of continuous solutionsf, g of Wilson's functional equation n = 0 N – 1 f(x + wny) = Nf(x)g(y), x, y C, given a primitiveN th rootw of unity.Disregarding the trivial solutionf = 0 andg any complex function, it is known thatg satisfies a version of d'Alembert's functional equation and so has the formg(z) = g (z) = N–1 n = 0 N – 1 E(wnz) for some C2. HereE (1, 2)(x + iy) = exp( 1x + 2).For fixedg = g the space of solutionsf of Wilson's functional equation can be decomposed into theN isotypic subspaces for the action of Z N on the continuous functions on C. We prove that ther th component, wherer {0, 1, ,N – 1}, of any solution satisfies the signed functional equation n = 0 N – 1 f(x + wny)wnr = Ng(x)f(y), x, y C. We compute the solution spaces of each of these signed equations: They are 1-dimensional and spanned byz n = 0 N – 1 wnr E(wnz), except forg = 1 andr 0 where they are spanned by andz N – r. Adding the components we get the solution of Wilson's equation. Analogous results are obtained with the action ofZ N on C replaced by that ofSO(2).The case ofg = 0 in the signed equations is special and solved separately both for Z N andSO(2).  相似文献   

14.
The one-dimensional dyadic martingale Hardy spaces H p are introduced and it is proved that the maximal operator of the (C,) means of a Walsh—Fourier series is bounded from H p to L p (1/( + 1) < p < ) and is of weak type (L 1,L 1). As a consequence, we obtain the summability result due to Fine; more exactly, the (C,) means of the Walsh—Fourier series of a function f L 1 converge a.e. to the function in question. Moreover, we prove that the (C,) means are uniformly bounded on H p whenever 1/( + 1) < p < . We define the two-dimensional dyadic hybrid Hardy space H 1 and verify that the maximal operator of the (C,,) means of a two-dimensional function is of weak type H 1 ,L 1). Consequence, the Walsh—Fourier series of every function f H 1 is (C,,) summable to the function f.  相似文献   

15.
Let be the unit circle {z|z|=1} and n c n e in be a bounded measurable function on . Theslant Toeplitz operator A onL 2 ( ) is defined by A e n ,e m =c 2mn for allm, n wheree n (z)=z n , . In this paper, we continue the study initiated in [6] onA * , the adjoint ofA . Specifically, we will show that for a certain dense set of continuous functions on ,A * is similar to some constant multiple of either a shift, or a shift plus a rank one operator.  相似文献   

16.
Let D be a simply connected domain on the complex plane such that 0 D. For r > 0 , let D r be the connected component of D {z : |z| < r} containing the origin. For fixed r, we solve the problem on minimization of the conformal radius R(D r, 0) among all domains D with given conformal radius R(D, 0). This also leads to the solution of the problem on maximization of the logarithmic capacity of the local -extension E (a) of E among all continua E with given logarithmic capacity. Here, E (a) = E {z : |za| }, a E, > 0. Bibliography: 12 titles.  相似文献   

17.
We investigate the asymptotic behaviour of the summatory functions of z(n, ), k(n, ) z (n) and k(n, ) z (n).  相似文献   

18.
The vector epsilon algorithm (VEA) has many advantages as a method for accelerating the convergence of a sequence of vectors. A vector Padé approximantP (z)/Q(z) of type [n/2k] can be associated with each entry of the vector epsilon table. In the scalar case, it reduces to the Padé approximantp(z)/q(z) of type [n–k/k]. It is thought that the disadvantages of VEA are (indirectly) attributable to the positivity property ofQ(x), x , recalling that in the scalar case,Q(z)q(z) 2. In this paper, a specification of a polynomial (z) of degreek is given, such that (z)2Q(z). The coefficients of (z) specify an accelerator for a sequence of vectors which should avoid many of the numerical difficulties of VEA.This work was supported in part by the EC-HCM project ROLLS under contract CHRX-CT93-0416.  相似文献   

19.
Letr(z) be a rational approximation to cosz with only imaginary poles ±i 1 –1/2 , ±i 2 –1/2 , ..., ±i m –1/2 such that |cozzr(z)| C|z|2m+2 as |z| 0. If the degree of the numerator ofr(z) is less than or equal to 2m and i m/4,i=1, ...,m, then we show that |r(z)|1 for all realz.  相似文献   

20.
We establish sharp upper and lower bounds for the functionalJ (p)- Re{p (z)-zp'(z)/p(z), 1. where ¦z¦=r is fixed, 0 <r <1, p(z).Pn (A, B), –1 B < a 1, a certain class of regular functions in the disk with values in the right halfplane.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 686–689, May, 1990.  相似文献   

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