THROREMS OF DISTORTION FOR SCHLICHT FUNCTIONS

Let \[f(z) = z + \sum\limits_{n = 1}^\infty {{a_n}{z^n} \in S} {\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \log \frac{{f(z) - f(\xi )}}{{z - \xi }} - \frac{{z\xi }}{{f(z)f(\xi )}} = \sum\limits_{m,n = 1}^\infty {{d_{m,n}}{z^m}{\xi ^n},} \], we denote \[{f_v} = f({z_v})\] , \[\begin{array}{l} {\varphi _\varepsilon }({z_u}{z_v}) = {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\frac{1}{{(1 - {z_u}{{\bar z}_v})}},\g_m^\varepsilon (z) = - {F_m}(\frac{1}{{f(z)}}) + \frac{1}{{{z^m}}} + \varepsilon {{\bar z}^m}, \end{array}\], where \({F_m}(t)\) is a Faber polynomial of degree m. Theorem 1. If \[f(z) \in S{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{u,v = 1}^N {{A_{u,v}}{x_u}{{\bar x}_v} \ge 0} \] and then \[\begin{array}{l} \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\exp \{ \alpha {F_l}({z_u},{z_v})\} \ \le \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} \varphi _\varepsilon ^\alpha ({z_u}{z_v})l = 1,2,3, \end{array}\], where \[\begin{array}{l} {F_1}({z_u},{z_v}) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} g_n^\varepsilon ({z_u})\bar g_n^\varepsilon ({z_v}),\{F_2}({z_u},{z_v}) = \frac{1}{{1 + {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}),\{F_3}({z_u},{z_v}) = \frac{1}{{1 - {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}). \end{array}\] The \[F({z_u},{z_v}) = \frac{1}{2}{g_1}({z_u}){{\bar g}_2}({z_v})\] is due to Kungsun. Theorem 2. If \(f(z) \in S\) ,then \[P(z) + \left| {\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {{\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}\frac{{{z_u}{z_v}}}{{{f_u}{f_v}}}} \right|}^\varepsilon }} \right| \le \sum\limits_{u,v = 1}^N {{\lambda _u}{{\bar \lambda }_v}} \frac{1}{{1 - {z_u}{{\bar z}_v}}}\], where \[\begin{array}{l} P(z) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} {G_n}(z),\{G_n}(z) = {\left| {\left| {\sum\limits_{n = 1}^N {{\beta _u}({F_n}(\frac{1}{{f({z_u})}}) - \frac{1}{{z_u^n}})} } \right| - \left| {\sum\limits_{n = 1}^N {{\beta _u}z_u^n} } \right|} \right|^2}, \end{array}\], \(P(z) \equiv 0\) is due to Xia Daoxing.     

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2     

THE GENERALIZED RIEMANN-HILBERT BOUNDARY VALUE PROBLEM FOR A SYSTEM OF SECOND ORDER QUASI-LINEAR ELLIPTIC EQUATIONS

In this paper, we consider the generalized Riemann-Hilberij problem for second order quasi-linear elliptic complex equation \[\begin{array}{l} \frac{{{\partial ^2}w}}{{\partial {{\bar z}^2}}} + {q_1}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}w}}{{\partial {z^2}}} + {q_2}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}\bar w}}{{\partial z\partial \bar z}}\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {q_3}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}w}}{{\partial z\partial \bar z}} + {q_4}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}\bar w}}{{\partial z\partial \bar z}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1)\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \gamma (z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}}),z \in G \end{array}\] satifying the boundary condition \[{\mathop{\rm Re}\nolimits} \left[ {{{\bar \lambda }_1}(z)\frac{{\partial w}}{{\partial \bar z}}} \right] = {\gamma _1}(z),{\mathop{\rm Re}\nolimits} \left[ {{{\bar \lambda }_2}(z)\frac{{\partial w}}{{\partial \bar z}}} \right] = {\gamma _2}(z),z \in \gamma {\kern 1pt} {\kern 1pt} {\kern 1pt} (2)\] Many authors (see that papers 1, 4-6) have studied the Diriohlet problem and Riemann-Hilbert problem for linear elliptic complex equation. In our papers 2, 3 we also considered the generalized Riemann-Hilbert problem of the general second order linear elliptic complex equation. We obtained the existence theorem, the explicit form of generalized solution and the sufficient and necessary conditions for the solvability of the above mentioned boundary value problem. Based on these results and applying the property of the introduced integral operators and Schauder's fixed-point principle, it can be proved that the analogous deductions in 3 also hold for the generalized Riemann-Hilber problem (1), (2) of the quasi-linear complex equation, i, e., we have the following theorem: Theorem, If the coefficients of second order quasi-linear elliptic complex equation (1) satifies some conditions then i) When index \({n_1} \ge 0,{n_2} \ge 0\), the boundary value problem (1), (2) is always solvable and the solution depends on 2 \(2({n_1} + {n_2} + 1)\) arbitrary real constants. ii) When index \({n_1} \ge 0,{n_2} < 0{\kern 1pt} {\kern 1pt} {\kern 1pt} (or{\kern 1pt} {\kern 1pt} {\kern 1pt} {n_1} < 0,{n_2} \ge 0{\kern 1pt} )\), the sufficient and necessary condition for the solvability of the above mentioned boundary value problem (1),(2) consists of \( - 2{n_2} - 1{\kern 1pt} {\kern 1pt} {\kern 1pt} ( - 2n, - 1)\) real equalities, if and only if the equalities are satisfied, the boundary value problem is solvable and the solution depends on \(2{n_1} + 1{\kern 1pt} {\kern 1pt} (2{n_2} + 1)\) arbitrary real constants. iii)When index \({n_1} < 0,{n_2} < 0\), the sufficient and necessary condition for the solvability of the above mentioned boundary value problem (1) , (2) consists of \( - 2({n_1} + {n_2} + 1)\) real equalities, if and only if the equalitieis are satisfied, the boundary-value problem is solvable. Finally, in the similar way, we may farther extend the result to the case of the nonlinear uniform elliptic complex equation.     

ON LINEAR AND NONLINEAR RIEMANN-HILBERT PROBLEMS FOR REGULAR FUNCTION WITH VALUES IN A CLIFFORD ALGEBRA

This paper deals with the boundary value problems for regular function with valuesin a Clifford algebra: ()W=O, x∈R~n\Г, w~+(x)=G(x)W~-(x)+λf(x, W~+(x), W~-(x)), x∈Г; W~-(∞)=0,where Г is a Liapunov surface in R~n the differential operator ()=()/()x_1+()/()x_2+…+()/()x_ne_n, W(x) =∑_A, ()_AW_A(x) are unknown functions with values in a Clifford algebra ()_n Undersome hypotheses, it is proved that the linear baundary value problem (where λf(x, W~+(x),W~-(x)) =g(x)) has a unique solution and the nonlinear boundary value problem has atleast one solution.     

UNIFORM STRONG CONVERGENCE RATE OF NEAREST NEIGHBOR DENSITY ESTIMATION

Based on [3] and [4],the authors study strong convergence rate of the k_n-NNdensity estimate f_n(x)of the population density f(x),proposed in [1].f(x)>0 and fsatisfies λ-condition at x(0<λ≤2),then for properly chosen k_nlim sup(n/(logn)~(λ/(1 2λ))丨_n(x)-f(x)丨C a.s.If f satisfies λ-condition,then for propeoly chosen k_nlim sup(n/(logn)~(λ/(1 3λ)丨_n(x)-f(x)丨C a.s.,where C is a constant.An order to which the convergence rate of 丨_n(x)-f(x)丨andsup 丨_n(x)-f(x)丨 cannot reach is also proposed.     

QUALITATIVE METHOD IN ANALYSIS OF A P-L-L WITH TANGENT CHARACTERISTIC AND FREQUENCY MODULATION INPUT

In analysis of p-L-L with tangent characteristic and frequency modulation input, we have obtained the following two types of the phase looked loop equation. \[\begin{array}{l} \frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + \alpha \frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t + {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (I)\\frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + (\alpha + \eta {\sec ^2}\varphi )\frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t - {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (II) \(\alpha > 0,\gamma > 0,\eta > 0,{\beta _1} > 0,{\beta _2} > 0,{\Omega _M} > 0) \end{array}\] In this paper, our aim is to explain the usual qualitative method and Lyapunov's function method, by which the existence of a periodic solution of (I), (II) is established. In addition, we especially point out: How is to construct the Lyapunovas function for the nonlinear and nonairtoiiomous system? This is a very important problem.     

ON THE NECESSARY AND SUFFICIENT CONDITION OF THE EXISTENCE OF QUASI INVARIANT MEASURES

If E is a separable type-2 Banach space and Esub<0>sub is a linear subspace of E, then the following are equivalent: (a) There exists a probability measure \[\mu \] on E, Which is \[{E_{\text{0}}}\]-quasi-invariant. (b) There exists a sequence \[({X_n}) \subset E\] such that \[\sum {{e_n}(\omega ){X_n}} \] converges a.s., where \[{{e_n}(\omega )}\] are indepondend identically distributed symmetric stable random variables of index 2,i,e.\[E(\exp (it{\kern 1pt} {\kern 1pt} {e_n}(\omega ))) = exp( - \frac{{{t^2}}}{2})\]for all real t, and \[{E_{\text{0}}} \subset \{ x,x = \sum {{\lambda _n}{X_n}} ,\forall ({\lambda _n}) \in {l_2}\} \] In this note we prove that \[\sum {{\lambda _n}{X_n}} \] is convergent.     

ON SMOOTHNESS AND DIFFERENTIABILITY OF FUNCTIONS

Let \(f(x)\) be a bounded real function on [-1,1],we define the modulus of continuity of f as \[\omega (f,\delta ) = \mathop {\sup }\limits_{x,y \in [ - 1,1],\left| {x - y} \right| \le \delta } \left| {f(x) - f(y)} \right|\] and the modulus of smoothness of f as \[{\omega _2}(f,\delta ) = \mathop {\sup }\limits_{x \pm h \in [ - 1,1],\left| h \right| \le \delta } \left| {f(x + h) + f(x - h) - 2f(x)} \right|\] Functions \(f(x)\), continuous on [-1,1] and \({\omega _2}(f,\delta ) = o(\delta )\) ,are called uniformly smooth functions. It is well known that there is a uniformly smooth functions whose derivative exisits on a null-set only. It would is of interest to discuss what condition should be added on the nonnegative function \(\varphi (\delta )\), \(\left( {0 \le \delta \le \frac{1}{2}} \right)\),in order that every bounded function f satisfying\[{\omega _2}(f,\delta ) = O(\varphi (\delta ))\] possess continous (or finite) derivative. the main result of this paper are the following two theorems. Theorem 1 let \(\varphi (\delta )\),\(\left( {0 \le \delta \le \frac{1}{2}} \right)\) ,be a nonnegative function, then, in order that every bounded function \(f(x)\) satisfying condition \[{\omega _2}(f,\delta ) = O(\varphi (\delta ))\] possess continous (or finite) derivative \(f'(x)\) on [-1,1],it is necessary and sufficient that the following condition hold \[\int_0^{\frac{1}{2}} {\frac{{\tilde \varphi (t)}}{t}} dt < \infty \] where \[\tilde \varphi (\delta ) = {\delta ^2}\mathop {\inf }\limits_{0 \le \eta \le \delta } \left\{ {{\eta ^{ - 2}}\mathop {\inf }\limits_{\eta \le \xi \le 1/2} \varphi (\xi )} \right\}\] Theorm 2 Let \(f(x)\) be a bounded function with \[\int_0^{\frac{1}{2}} {\frac{{{\omega _2}(f,t)}}{{{t^2}}}} dt < \infty \] then \(f'(x)\) is a continous function and \[{\omega _2}(f',\delta ) = O\left\{ {\int_0^\delta {\frac{{{\omega _2}(f,t)}}{{{t^2}}}} dt} \right\}\].     

On the strong summation of Fourier series with variable exponents

Пустьf 2π-периодическ ая суммируемая функц ия, as _k (x) еë сумма Фурье порядк аk. В связи с известным ре зультатом Зигмунда о сильной суммируемости мы уст анавливаем, что если λn→∞, то сущес твует такая функцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _{2n} } } \right\}^{1/\lambda _{2n} } = \infty .$$ Отсюда, в частности, вы текает, что если λn?∞, т о существует такая фун кцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } } \right\}^{1/\lambda _n } = \infty .$$ Пусть, далее, ω-модуль н епрерывности и $$H^\omega = \{ f:\parallel f(x + h) - f(x)\parallel _c \leqq K_f \omega (h)\} .$$ . Мы доказываем, что есл и λ _n ?∞, то необходимым и достаточным условие м для того, чтобы для всехf∈H ^ω выполнялос ь соотношение $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _n } } \right\}^{1/\lambda _n } = 0(x \in [0;2\pi ])$$ является условие $$\omega \left( {\frac{1}{n}} \right) = o\left( {\frac{1}{{\log n}} + \frac{1}{{\lambda _n }}} \right).$$ Это же условие необхо димо и достаточно для того, чтобы выполнялось соотнош ение $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } = 0(f \in H^\omega ,x \in [0;2\pi ]).$$      

LIMIT THEOREM FOR A CLASS OF SEQUENCE OF WEAKLY DEPENDENT RANDOM VARIABLES

In this paper, we consider a central limit theorem for the sequence of stationary m-dependent random variables, the variance of which is possibly infinite. Theorem. Let {Xn, n=l, 2,...} be a sequence of stationary m-dependent random variables with means zero. The following conditions are satisfied. (i) \[{M^2}\int_{{\text{|}}{X_1}| > M} {dP} /\int_{{X_1}| < M} {X_1^2} dP \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} (M \to \infty )\] (ii) \[\int_{\{ {X_1}| < M,|{X_i}| < M} {X_1^{}} {X_i}dP/\int_{|{X_1}| < M} {X_1^2} dP \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (M \to \infty )\] then there are constants Bsubsub>0, such that \[\frac{1}{{{B_n}}}\sum\limits_{i = 1}^n {{X_1}} \] converges in distribution N(0,1).     

On the strong approximation by the (C, α)-means of Fourier series. II

стАтьь ьВльЕтсь пРОД ОлжЕНИЕМ пРЕДыДУЩЕИ ОДНОИМЕННОИ РАБОты АВтОРА, гДЕ ИжУ ЧАлсь пОРьДОк ВЕлИЧИН пРИ УслОВИьх, ЧтО α>-1/2, Рα >- 1 И ЧтО МАтРИцАt _nk УДОВлЕтВОРьЕт НЕкОт ОРОМУ УслОВИУ РЕгУльРНОстИ. жДЕсь ДОкАжыВАЕтсь, Ч тО ЕслИf∈H ^Ω, тО ВыпОлНь Етсь ОцЕНкА $$\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left| {\sigma _k^\alpha \left( x \right) - f\left( x \right)} \right|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} = O\left( {\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left( {\frac{1}{k}\mathop \smallint \limits_{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}}^{2\pi } \frac{{\omega \left( t \right)}}{{t^2 }}dt} \right)^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} + \left( {\frac{{\lambda _n }}{n}} \right)^\alpha \omega \left( {\frac{1}{n}} \right)} \right)$$ (λ₁=1, λ_n+1-λ_n≦1), А тАкжЕ ЧтО Ёт А ОцЕНкА ОкОНЧАтЕльН А В сВОИх тЕРМИНАх; пОДОБ НыИ РЕжУль-тАт спРАВЕДлИВ тАкжЕ И Дль сОпРьжЕННОИ ФУНкцИИ . ДОкАжыВАЕтсь, ЧтО Усл ОВИьα>?1/2 Иpα>?1, кОтОРыЕ Б ылИ НАлОжЕНы В УпОМьНУтО И ВышЕ ЧАстИ I, сУЩЕстВЕН Ны.     

UNUNIQUENESS OF SOME HOLOMORPHIC FUNCTIONS

In the present paper, we show that there exist a bounded, holomorphic function $\[f(z) \ne 0\]$ in the domain $\[\{ z = x + iy:\left| y \right| < \alpha \} \]$ such that $\[f(z)\]$ has a Dirichlet expansion $\[\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ in the halfplane $\[x > {x_f}\]$ if and only if $\[\frac{a}{\pi }\log r - \sum\limits_{{u_n} < r} {\frac{2}{{{u_n}}}} \]$ has a finite upperbound on $\[[1, + \infty )\]$, where $\[\alpha \]$ is a positive constant,$\[{x_f}( < + \infty )\]$ is the abscissa of convergence of $\[\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ and the infinite sequence $\[\{ {u_n}\} \]$ satisfies $\[\mathop {\lim }\limits_{n \to + \infty } ({u_{n + 1}} - {u_n}) > 0\]$. We also point out some necessary conditions and sufficient ones Such that a bounded holomorphic function in an angular(or half-band) domain is identically zero if an infinite sequence of its derivatives and itself vanish at some point of the domain. Here some result are generalizations of those in [4].     

Low Mach number limit of multidimensional steady flows on the airfoil problem

Given $$\alpha >0$$, we establish the following two supercritical Moser–Trudinger inequalities $$\begin{aligned} \mathop {\sup }\limits _{ u \in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( (\alpha _n + |x|^\alpha ) |u|^{\frac{n}{n-1}} \big ) dx < +\infty \end{aligned}$$and $$\begin{aligned} \mathop {\sup }\limits _{ u\in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( \alpha _n |u|^{\frac{n}{n-1} + |x|^\alpha } \big ) dx < +\infty , \end{aligned}$$where $$W^{1,n}_{0,\mathrm{rad}}(B)$$ is the usual Sobolev spaces of radially symmetric functions on B in $${\mathbb {R}}^n$$ with $$n\ge 2$$. Without restricting to the class of functions $$W^{1,n}_{0,\mathrm{rad}}(B)$$, we should emphasize that the above inequalities fail in $$W^{1,n}_{0}(B)$$. Questions concerning the sharpness of the above inequalities as well as the existence of the optimal functions are also studied. To illustrate the finding, an application to a class of boundary value problems on balls is presented. This is the second part in a set of our works concerning functional inequalities in the supercritical regime.     

SINGULAR PERTURBATION PROBLEMS FOR A CLASS OF ELLIPTIC EQUATIONS OF SECOND ORDER AS DEGENERATED OPERATOR HAS SINGULAR POINTS

In this paper we study the first and tiie third boundary value problems for the elliptic equation \[\begin{array}{l} \varepsilon \left( {\sum\limits_{i,j = 1}^m {{d_{i,j}}(x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} + \sum\limits_{i = 1}^m {{d_i}(x)\frac{{\partial u}}{{\partial {x_i}}} + d(x)u} } } \right) + \sum\limits_{i = 1}^m {{a_i}(x)\frac{{\partial u}}{{\partial {x_i}}} + b(x) + c} \ = f(x),x \in G(0 < \varepsilon \le 1), \end{array}\] as the degenerated operator bas singular points, where \[\sum\limits_{i,j = 1}^m {{d_{i,j}}(x){\xi _i}{\xi _j}} \ge {\delta _0}\sum\limits_{i = 1}^m {\xi _i^2} ,({\delta _0} > 0,x \in G).\] The uniformly valid asymptotic solutions of boundary value problems have been obtained under the condition of \[\sum\limits_{i = 1}^m {{a_i}(x){n_i}(x){|_{\partial G}} > 0,or} \sum\limits_{i = 1}^m {{a_i}(x){n_i}(x){|_{\partial G}} < 0} ,\] where \(n = ({n_1}(x),{n_2}(x), \cdots ,{n_m}(x))\) is the interior normal to \({\partial G}\).     

ON THE GROWTH OF SOME RANDOM HYPERDIRICHLET SERIES

The paper considers the random L-Dirichlet seriesf(s,ω)=sum from n=1 to ∞ P_n(s,ω)exp(-λ_ns)and the random B-Dirichlet seriesψτ_0(s,ω)=sum from n=1 to ∞ P_n(σ iτ_0,ω)exp(-λ_ns),where {λ_n} is a sequence of positive numbers tending strictly monotonically to infinity, τ_0∈R is a fixed real number, andP_n(s,ω)=sum from j=1 to m_n ε_(nj)a_(nj)s~ja random complex polynomial of order m_n, with {ε_(nj)} denoting a Rademacher sequence and {a_(nj)} a sequence of complex constants. It is shown here that under certain very general conditions, almost all the random entire functions f(s,ω) and ψ_(τ_0)(s,ω) have, in every horizontal strip, the same order, given byρ=lim sup((λ_nlogλ_n)/(log A_n~(-1)))whereA_n=max |a_(nj)|.Similar results are given if the Rademacher sequence {ε_(nj)} is replaced by a steinhaus seqence or a complex normal sequence.     

ON THE PROBLEM OF BEST CONVERGENCE RATES OF DENSITY ESTIMATES

Let X_1,…,X_n be iid samples drawn from an m-dimensional population with a probabilitydensity f,belonging to the family C_(ka),i.e.the family of all densities whose partialderivatives of order k are bounded by a.It is desired to estimate the value of f at somepredetermined point a,for example a=0.Farrell obtained some results concerning the bestpossible convergence rates for all estimator sequence,from which it follows,for example,thatthere exists no estimator sequence{γ_n(0)=γ_n(X_1,…,X_n,0)}such that(?)E_f[γ_n(0)-f(0)]~2=o(n~(-2k/(2k m))).This article pursues this problem further and proves that there existsno estimator sequence{γ_n(0)}such thatn~(-k/(2k m))(γ_n(0)-f(0))(?)0,for each f∈C_(ka),where(?)denotes convergence in probability.     

CONSTRUCTIVE PROPERTIES OF A KIND OF UNIFORMLY ALMOST PERIODIC FUNCTIONS

In this paper,we have discussed constructive properties of a kind of uniformly almost periodic functions, of which the sequence of its Fourier exponents has unique limit point at infinity. \[\begin{gathered} f(x) \sim \sum\limits_{k = - \infty }^\infty {{A_k}} {e^{i{\Lambda _k}x}} \hfill \ {\Lambda _0} = \alpha ,0 < \alpha \leqslant {\Lambda _k} < {\Lambda _{k + 1}}(k = 0,1,2,...) \hfill \ \mathop {\lim }\limits_{k \to \infty } {\Lambda _k} = \infty ,{\Lambda _k} = - {\Lambda _k} \hfill \ |{\Lambda _k}| + |{\Lambda _{ - k}}| > 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (k \ne 0) \hfill \\ \end{gathered} \] Analogons to the approximation theory of periodic functioiis, we get some theorems similar to the Jackson theorem, Bernstein theorem and Zygmund theorem of periodio functions.     

On the divergence of de la Vallée Poussin means of Fourier series

Пусть {λ _{n 1} ^t8 — монотонн ая последовательнос ть натуральных чисел. Дл я каждой функции fεL(0, 2π) с рядом Фурье строятся обобщенные средние Bалле Пуссена $$V_n^{(\lambda )} (f;x) = \frac{{a_0 }}{2} + \mathop \sum \limits_{k = 1}^n (a_k \cos kx + b_k \sin kx) + \mathop \sum \limits_{k = n + 1}^{n + \lambda _n } \left( {1 - \frac{{k - n}}{{\lambda _n + 1}}} \right)\left( {a_k \cos kx + b_k \sin kx} \right).$$ Доказываются следую щие теоремы.

1. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность {V_n ^(λ)(?;x)} расходится почти вс юду.
2. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность $$\left\{ {\frac{1}{\pi }\mathop \smallint \limits_{ - \pi /\lambda _n }^{\pi /\lambda _n } f(x + t)\frac{{\sin (n + \tfrac{1}{2})t}}{{2\sin \tfrac{1}{2}t}}dt} \right\}$$ расходится почти всю ду

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ON THE JOINT SPECTRUM FOR N-TUPLE OF HYPONORMAL OPERATORS

Let A=(A_1,…,A,)be an n-tuple of double commuting hyponormal operators.It is-proved that:1.The joint spectrum of A has a Cartesian decomposition:Re[Sp(A)]=S_p(ReA),Im[Sp(A)]=Sp(ImA);2.The.joint resolvent of A satisfies the growth condition:‖()‖=1/(dist(z,Sp(A)));3.If 0σ(A_i),i=1,2,…,n,then‖A‖=γ_(sp)(A).