共查询到20条相似文献,搜索用时 31 毫秒
1.
If γ(x)=x+iA(x),tan ?1‖A′‖∞<ω<π/2,S ω 0 ={z∈C}| |argz|<ω, or, |arg(-z)|<ω} We have proved that if φ is a holomorphic function in S ω 0 and \(\left| {\varphi (z)} \right| \leqslant \frac{C}{{\left| z \right|}}\) , denotingT f (z)= ∫?(z-ζ)f(ζ)dζ, ?f∈C 0(γ), ?z∈suppf, where Cc(γ) denotes the class of continuous functions with compact supports, then the following two conditions are equivalent:
- T can be extended to be a bounded operator on L2(γ);
- there exists a function ?1 ∈H ∞(S ω 0 ) such that ?′1(z)=?(z)+?(-z), ?z∈S ω 0 ?z∈S w 0 .
2.
E. A. Sevost’yanov 《Mathematical Notes》2011,90(3-4):431-438
We consider the solvability problem for the equation $f_{\bar z} $ = v(z, f(z))f z , where the function v(z,w) of two variables may be close to unity. Such equations are called quasilinear Beltrami-type equations with ellipticity degeneration. We prove that, under some rather general conditions on v(z,w), the above equation has a regular homeomorphic solution in the Sobolev classW loc 1,1 . Moreover, such solutions f satisfy the inclusion f ?1 ∈ W loc 1,2 . 相似文献
3.
N. I. Nagnibida 《Mathematical Notes》1974,15(1):40-42
In this note we find sufficient conditions for uniqueness of expansion of any two functionsf(z) and g(z) which are analytic in the circle ¦ z ¦ < R (0 < R <∞) in series $$f(z) = \sum\nolimits_{n = 0}^\infty {(a_n f_2 (z) + b_n g_n (z))}$$ and $$g_i (z) = \sum\nolimits_{n = 0}^\infty {a_n \lambda _n f_n (z)} + b_n \mu _n f_n (x)),$$ which are convergent in the compact topology, where (f n {z} n=0 ∞ and {g} n=0 ∞ are given sequences of functions which are analytic in the same circle while {λ n } n=0 ∞ and {μ n } n=0 ∞ are fixed sequences of complex numbers. The assertion obtained here complements a previously known result of M. G. Khaplanov and Kh. R. Rakhmatov. 相似文献
4.
N. A. Shirokov 《Journal of Mathematical Sciences》1987,37(5):1306-1322
Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, leta ε int Γ, and let Rn(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the pointa. Let Λα(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γn(z) > 0 on Γ such that f∈Λα(Γ) if and only if for any nonnegative integer n there exists a function Rn, Rn ε Rn(a) such that ¦f(z) ? Rn(z)¦ ≤ cf·γn(z), z ε Γ. It is proved that the weights γn cannot be expressed simply in terms of ρ 1 + /n(z) and ρ 1 - /n(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on \(\mathbb{C}\backslash \mathbb{D}\) . 相似文献
5.
S Thangavelu 《Proceedings Mathematical Sciences》1990,100(1):9-20
Let Pk denote the projection of L2(R R ) onto the kth eigenspace of the operator (-δ+?x?2 andS N α =(1/A N α )Σ k N =0A N?k α P k . We study the multiplier transformT N α for the Weyl transform W defined byW(T N αf )=S n αW(f) . Applications to Laguerre expansions are given. 相似文献
6.
Peter L. Polyakov 《Journal of Geometric Analysis》1996,6(2):233-276
We construct integral operatorsR r andH r on the spaces of differential forms of the type (o, r) withr <q on a regularq-concave CR manifoldM such that $$f(z) = \bar \partial _M R_r (f)(z) + R_{r + 1} (\bar \partial _M f)(z) + H_r (f)(z),$$ for a differential formf ∈ L (0,r) s (M) and forz ∈ M′ ?M, whereH r is compact andR r admits sharp estimates. 相似文献
7.
For functionsf(z) ? 0, holomorphic in the unit disk u, infinitely differentiable in u, and belonging to a given Gevrey class on ?u, we establish sufficient conditions characterizing the sets K f ∞ = (z: ¦z¦ = 1,f (k) (z) = 0,k = 0, 1, 2, ... }. These conditions are close to the necessary condition due to L. Carleson and substantially more precise than the conditions given byA.-M. Chollet (see [1, 2]). 相似文献
8.
Let (α) denote the class of locally univalent normalized analytic functions f in the unit disk |z| < 1 satisfying the condition $Re\left( {1 + \frac{{zf''(z)}} {{f'(z)}}} \right) < 1 + \frac{\alpha } {2}for|z| < 1 $ and for some 0 < α ≤ 1. We firstly prove sharp coefficient bounds for the moduli of the Taylor coefficients a n of f ∈ (α). Secondly, we determine the sharp bound for the Fekete-Szegö functional for functions in (α) with complex parameter λ. Thirdly, we present a convolution characterization for functions f belonging to (α) and as a consequence we obtain a number of sufficient coefficient conditions for f to belong to (α). Finally, we discuss the close-to-convexity and starlikeness of partial sums of f ∈ (α). In particular, each partial sum s n (z) of f ∈ (1) is starlike in the disk |z| ≤ 1/2 for n ≥ 11. Moreover, for f ∈ (1), we also have Re(s′ n (z)) > 0 in |z| ≤ 1/2 for n ≥ 11. 相似文献
9.
For q ∈ (0, 1) let the q-difference operator be defined as follows $$\partial _q f(z) = \frac{{f(qz) - f(z)}} {{z(q - 1)}} (z \in \mathbb{U}),$$ where \(\mathbb{U}\) denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R q λ f is defined. Applying R q λ f a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated. 相似文献
10.
Z. D. Usmanov 《Mathematical Notes》1996,59(2):196-200
We prove that the equation $$2\bar z\partial _{\bar z} \bar w = 0_1 z \in G,$$ in whichB(z) ∈C ∞(G),B 0(z)=O(|z})α),α>0,z → 0, and $$b(\varphi ) = \sum\limits_{k = - m_o }^m {b_k e^{ik\varphi } } $$ does not have nontrivial solutions in the classC ∞(G). 相似文献
11.
B. I. Golubov 《Mathematical Notes》1974,15(1):20-25
For periodic functions of the Hölder class H 2 α (0 < α≤1) defined in the two-dimensional space D2, we find the asymptotic form as R → + ∞ of the quantity $$\mathop {\sup }\limits_{f \in H_2^\alpha } \parallel S_R^\delta (x,f) - f(x)\parallel _{C(E_2 )} \left( {\delta > \frac{1}{2} + \alpha } \right),$$ where S R δ is the Riesz spherical mean of orderδ of the Fourier series of the functionf(x). 相似文献
12.
V. V. Goryainov 《Mathematical Notes》1975,18(5):967-971
Let SM, M > 1, be the class of functionsf(z) which are regular and univalent in the disk ¦z¦ < 1 and satisfy the conditionsf(0) = 0,f'(0) = 1, and ¦f(z)¦ < M. In the present note we will obtain an exact estimate for the argument of the derivative of a function of the class SM. 相似文献
13.
К. Ю. Осипенко 《Analysis Mathematica》1987,13(3):199-210
Для класса ? аналитич еских в единичном кру ге функций, ограниченны х по модулю единицей, погрешност ью наилучшего прибли жения в точкеz 0 по значениям в точкахz 1,..., zn, заданным с погрешнос тьюδ, называется вели чинаr(z 0, z1 z..., zn, α)=inf sup sup ¦f(z0)-S(f1, ...fn)¦, где нижняя грань бере тся по всевозможным ф ункциям S: Сn→С. ДляE~((?1,1) иz 0∈ ∈(-1,1)Е рассматривается задача о нахождении п орядка информативности мно жестваЕ, т.е. минимальногоп, на котором достигается нижняя грань в равенстве $$R(z_0 ,\delta ,E) = \mathop {\inf }\limits_n {\text{ }}\mathop {\inf }\limits_{z_1 , \ldots ,z_n \in E} {\text{ }}r(z_0 ,z_1 , \ldots ,z_n ,\delta ).$$ Кроме того, приδ, близ ких к 1, решена задача о нахождении величины $$r_n (\delta ,E) = \mathop {\inf }\limits_{z_1 , \ldots ,z_n \in Ez_0 \in E} \sup r(z_0 ,z_1 , \ldots ,z_n ,\delta )$$ и найдены узлы, на кото рых достигается нижн яя грань. 相似文献
14.
Donoho et al. in 1996 have made almost perfect achievements in wavelet estimation for a density function f in Besov spaces Bsr,q(R). Motivated by their work, we define new linear and nonlinear wavelet estimators flin,nm, fnonn,m for density derivatives f(m). It turns out that the linear estimation E(‖flinn,m-f(m)‖p) for f(m) ∈ Bsr,q(R) attains the optimal when r≥ p, and the nonlinear one E(‖fnonn,m-f(m)‖p) does the same if r≤p/2(s+m)+1 . In addition, our method is applied to Sobolev spaces with non-negative integer exponents as well. 相似文献
15.
I. I. Sharapudinov 《Mathematical Notes》2013,94(1-2):281-293
We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x 2)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S n ?1 (f) = S n ?1 (f, x) is the projection onto the subspace of algebraic polynomials p n = p n (x) of degree at most n, i.e., S n (p n ) = p n ; in addition, S n ?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S n ?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ n (x) of the partial sums S n ?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ . 相似文献
16.
L. G. Rybnikov 《Functional Analysis and Its Applications》2006,40(3):188-199
We construct a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U (
) of a semisimple Lie algebra
. This family is parameterized by finite sequences μ, z
1, ..., z
n
, where μ ∈
* and z
i
∈ ℂ. The construction presented here generalizes the famous construction of the higher Gaudin Hamiltonians due to Feigin,
Frenkel, and Reshetikhin. For n = 1, the corresponding commutative subalgebras in the Poisson algebra S(
) were obtained by Mishchenko and Fomenko with the help of the argument shift method. For commutative algebras of our family,
we establish a connection between their representations in the tensor products of finite-dimensional
-modules and the Gaudin model.
__________
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 40, No. 3, pp. 30–43, 2006
Original Russian Text Copyright ? by L. G. Rybnikov 相似文献
17.
S. Yu. Favorov 《Mathematical Notes》1975,18(3):853-857
The class of functions Φ(z, t) defined for z∈ Cn and t ≥0 such that the functions Φ(z, ¦w¦), w∈C, are plurisubharmonic in Cn+1 is called the classD. A typical example of functions of the classB are functions of the form \(\ln M_g (z,t) = \mathop {\ln \sup |}\limits_{|w| = t} g(z,w)|\) where g(z, w), z∈Cn, w∈C, is an entire function in Cn+1. In this note it is proved under certain restrictions on the function Φ(z, t)εB that its lower order relative to the variable t is the same for all z∈Cn except, possibly, for the points z of a set of zero Γ capacity. See [5]. 相似文献
18.
Violeta Petkova 《Central European Journal of Mathematics》2013,11(3):561-573
We study multipliers M (bounded operators commuting with translations) on weighted spaces L ω p (?), and establish the existence of a symbol µ M for M, and some spectral results for translations S t and multipliers. We also study operators T on the weighted space L ω p (?+) commuting either with the right translations S t , t ∈ ?+, or left translations P + S ?t , t ∈ ?+, and establish the existence of a symbol µ of T. We characterize completely the spectrum σ(S t ) of the operator S t proving that $\sigma (S_t ) = \{ z \in \mathbb{C}:|z| \leqslant e^{t\alpha _0 } \} ,$ where α 0 is the growth bound of (S t ) t≥0. A similar result is obtained for the spectrum of (P + S ?t ), t ≥ 0. Moreover, for an operator T commuting with S t , t ≥ 0, we establish the inclusion , where $\mathcal{O}$ = {z ∈ ?: Im z < α 0}. 相似文献
19.
S. V. Shvedenko 《Mathematical Notes》1974,15(1):56-61
In this article we study, for a Hilbert spaceB of analytic functions in the open unit disk, the dependence of the structure of the space of sequencesB(Z)={{f(zk)} k=1 ∞ :f∈B} on the choice of the sequence Z={zk} k=1 ∞ of distinct points of the unit disk [6]. 相似文献
20.
The aim of this paper is to study the binomial coefficients ( n x ), the factorial polynomials [x]n and [x]n, the Stirling numbers of first and second kind, namely s(n,k) and S(n,k), in the case that n ∈ ? is replaced by real α ∈ ?. In the course of the paper, the Vandermonde convolution formula is presented in an infinite series frame, the binomial coefficient function ( a x ), α ∈ ?, is sampled in terms of the binomial coefficients ( k x ) for k ∈ ?o, Bell numbers of fractional orders are introduced. Emphasis is placed on the fractional order Stirling numbers s(α,k) and S(α,k), first studied here. Some applications of the S(α,k) are given. 相似文献