共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
Я. Л. Геронимус 《Analysis Mathematica》1977,3(2):95-108
Рассматривается сис тема ортогональных м ногочленов {P n (z)} 0 ∞ , удовлетворяющ их условиям $$\frac{1}{{2\pi }}\int\limits_0^{2\pi } {P_m (z)\overline {P_n (z)} d\sigma (\theta ) = \left\{ {\begin{array}{*{20}c} {0,m \ne n,P_n (z) = z^n + ...,z = \exp (i\theta ),} \\ {h_n > 0,m = n(n = 0,1,...),} \\ \end{array} } \right.} $$ где σ (θ) — ограниченная неу бывающая на отрезке [0,2π] функция с бесчисленным множе ством точек роста. Вводится последовательность параметров {аn 0 ∞ , независимых дру г от друга и подчиненных единств енному ограничению { ¦аn¦<1} 0 ∞ ; все многочлены {Р n (z)} 0/∞ можно найти по формуле $$P_0 = 1,P_{k + 1(z)} = zP_k (z) - a_k P_k^ * (z),P_k^ * (z) = z^k \bar P_k \left( {\frac{1}{z}} \right)(k = 0,1,...)$$ . Многие свойства и оце нки для {P n (z)} 0 ∞ и (θ) можн о найти в зависимости от этих параметров; например, условие \(\mathop \Sigma \limits_{n = 0}^\infty \left| {a_n } \right|^2< \infty \) , бо лее общее, чем условие Г. Cerë, необходимо и достато чно для справедливости а симптотической форм улы в области ¦z¦>1. Пользуясь этим ме тодом, можно найти также реш ение задачи В. А. Стекло ва. 相似文献
3.
Т. КОВАЛЬСКИ 《Analysis Mathematica》1988,14(1):49-63
In this paper we consider the behaviour of partial sums of Fourier—Walsh—Paley series on the group62-01. We prove the following theorems: Theorem 1. Let {n k } k =1/∞ be some increasing convex sequence of natural numbers such that $$\mathop {\lim sup}\limits_m m^{ - 1/2} \log n_m< \infty $$ . Then for anyf∈L ∞(G) $$\left( {\frac{1}{m}\sum\limits_{j = 1}^m {|Sn_j (f;0)|^2 } } \right)^{1/2} \leqq C \cdot \left\| f \right\|_\infty $$ . Theorem 2. Let {n k } k =1/∞ be a lacunary sequence of natural numbers,n k+1/n k≧q>1. Then for anyfεL ∞(G) $$\sum\limits_{j = 1}^m {|Sn_j (f;0)| \leqq C_q \cdot m^{1/2} \cdot \log n_m \cdot \left\| f \right\|_\infty } $$ . Theorems. Let µ k =2 k +2 k-2+2 k-4+...+2α 0,α 0=0,1. Then $$\begin{gathered} \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in L^\infty (G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = 0(m)^2 \} .} \hfill \\ \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = o(m)^2 \} = } \hfill \\ = \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} \hfill \\ \end{gathered} $$ . Theorem 4. {{S 2 k(f: 0)} k =1/∞ ,f∈L ∞(G)}=m. $$\{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = c. \{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} = c_0 $$ . 相似文献
4.
V. V. Napalkov 《Mathematical Notes》1972,12(6):849-855
Letf (z) be an entire function λn(n=0,1,2,...) complex numbers, such that the system f(λn n=0 ∞ is not complete in the circle ¦z¦n(z) have the form \(\sum\nolimits_{k = 0}^{p_n } {\alpha _{nk} } f(\lambda _k \cdot z)\) . We study the properties of the limit function of the sequence Qn(z) in the case when $$f(z) = 1 + \sum\nolimits_{n = 1}^\infty {\frac{{z^n }}{{P(1)P(2)...P(n)}}} ,$$ . where P(z) is a polynomial having at least one negative integral root. 相似文献
5.
6.
LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW 2 (x) are finite. Let {p n (W 2;x)} 0 ∞ denote the sequence of orthonormal polynomials with respect to the weightW 2, and let {α n } 1 ∞ and {β n } 1 ∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ?1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec n 1 ∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR. 相似文献
7.
8.
A. N. Kozhevnikov 《Mathematical Notes》1977,22(5):882-888
The spectral problem in a bounded domain Ω?Rn is considered for the equation Δu= λu in Ω, ?u=λ?υ/?ν on the boundary of Ω (ν the interior normal to the boundary, Δ, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues {λ j 0 } j=1 ∞ and {λ j ∞ } j=1 ∞ , converging respectively to 0 and +∞. It is also established that $$N^0 (\lambda ) = \sum\nolimits_{\operatorname{Re} \lambda _j^0 \geqslant 1/\lambda } {1 \approx const} \lambda ^{n - 1} , N^\infty (\lambda ) \equiv \sum\nolimits_{\operatorname{Re} \lambda _j^\infty \leqslant \lambda } {1 \approx const} \lambda ^{n/1} .$$ The constants are explicitly calculated. 相似文献
9.
A. A. Mirolyubov 《Mathematical Notes》1972,12(6):843-848
Let πn(u) be a sequence of polynomials with a biorthogonal system, and let {? n (z)} be functions defined in the singly connected domain D. We consider the problem of the completeness of the system $$A(z,\lambda _n ) = \sum\nolimits_{s = 0}^\infty {P_\mathfrak{s} } (z)\pi _s (\lambda _n ),n = 1,2,...,$$ in the class of functions F(z) having the representation $$F(z) = \sum\nolimits_{k = 0}^\infty d _k P_k (z).$$ . 相似文献
10.
H. Б. пОгОсьН 《Analysis Mathematica》1985,11(2):139-177
Let {? k } k=0 ∞ be a numerical sequence satisfying the conditions $$\varrho _k \downarrow 0(k \to \infty )and\mathop \sum \limits_{k = 0}^\infty \varrho _k^2 = + \infty .$$ It is proved that there exists a trigonometric series $$\mathop \sum \limits_{k = 0}^\infty \varrho '_k \cos 2\pi (kx + \theta _k )$$ where ¦?′ k ¦≦? k ,k=0, 1, 2, ..., possessing the following property. For each measurable and a.e. finite functionF(x), x?[0, 1], the numbersδ k =0 or 1,k=0, 1, ..., may be chosen in such a way that the series $$\mathop \sum \limits_{k = 0}^\infty \delta _k \varrho '_k \cos 2\pi (kx + \theta _k )$$ converges toF(x) a.e. on [0,1]. In addition, ifF(x)=0, then \(\delta _{k_0 } \varrho '_{k_0 } \ne 0\) for at least onek 0≧0. Certain generalizations are discussed, too. 相似文献
11.
We investigate the boundedness nature of positive solutions of the difference equation $$ x_{n + 1} = max\left\{ {\frac{{A_n }} {{X_n }},\frac{{B_n }} {{X_{n - 2} }}} \right\},n = 0,1,..., $$ where {A n } n=0 ∞ and {B n } n=0 ∞ are periodic sequences of positive real numbers. 相似文献
12.
А. X. гЕРМАН 《Analysis Mathematica》1980,6(2):121-135
LetD be a simply connected domain, the boundary of which is a closed Jordan curveγ; \(\mathfrak{M} = \left\{ {z_{k, n} } \right\}\) , 0≦k≦n; n=1, 2, 3, ..., a matrix of interpolation knots, \(\mathfrak{M} \subset \Gamma ; A_c \left( {\bar D} \right)\) the space of the functions that are analytic inD and continuous on \(\bar D; \left\{ {L_n \left( {\mathfrak{M}; f, z} \right)} \right\}\) the sequence of the Lagrange interpolation polynomials. We say that a matrix \(\mathfrak{M}\) satisfies condition (B m ), \(\mathfrak{M}\) ∈(B m ), if for some positive integerm there exist a setB m containingm points and a sequencen p p=1 ∞ of integers such that the series \(\mathop \Sigma \limits_{p = 1}^\infty \frac{1}{{n_p }}\) diverges and for all pairsn i ,n j ∈{n p } p=1 ∞ the set \(\left( {\bigcap\limits_{k = 0}^{n_i } {z_{k, n_i } } } \right)\bigcap {\left( {\bigcup\limits_{k = 0}^{n_j } {z_{k, n_j } } } \right)} \) is contained inB m . The main result reads as follows. {Let D=z: ¦z¦ \(\Gamma = \partial \bar D\) and let the matrix \(\mathfrak{M} \subset \Gamma \) satisfy condition (Bm). Then there exists a function \(f \in A_c \left( {\bar D} \right)\) such that the relation $$\mathop {\lim \sup }\limits_{n \to \infty } \left| {L_n \left( {\mathfrak{M}, f, z} \right)} \right| = \infty $$ holds almost everywhere on γ. 相似文献
13.
S. Thianwan 《Mathematical Notes》2011,89(3-4):397-407
Let X be a real uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T 1, T 2: C → X be two uniformly L-Lipschitzian, generalized asymptotically quasi-nonexpansive non-self-mappings of C satisfying condition A′ with sequences {k n (i) } and {δ n (i) } ? [1, ∞),, i = 1, 2, respectively such that Σ n=1 ∞ (k n (i) ? 1) < ∞, Σ n=1 (i) δ n (i) < ∞, and F = F(T 1) ∩ F(T 2) ≠ ?. For an arbitrary x 1 ∈ C, let {x n } be the sequence in C defined by $$ \begin{gathered} y_n = P\left( {\left( {1 - \beta _n - \gamma _n } \right)x_n + \beta _n T_2 \left( {PT_2 } \right)^{n - 1} x_n + \gamma _n v_n } \right), \hfill \\ x_{n + 1} = P\left( {\left( {1 - \alpha _n - \lambda _n } \right)y_n + \alpha _n T_1 \left( {PT_1 } \right)^{n - 1} x_n + \lambda _n u_n } \right), n \geqslant 1, \hfill \\ \end{gathered} $$ where {α n }, {β n }, {γ n } and {λ n } are appropriate real sequences in [0, 1) such that Σ n=1 ∞ ] γ n < ∞, Σ n=1 ∞ λ n < ∞, and {u n }, }v n } are bounded sequences in C. Then {x n } and {y n } converge strongly to a common fixed point of T 1 and T 2 under suitable conditions. 相似文献
14.
15.
16.
Tejinder Singh Neelon 《Analysis Mathematica》2007,33(2):123-134
Let {M k } be a logconvex sequence satisfying the differentiability condition $$\sup (M_{n + 1} /M_n )^{1/n} < \infty $$ . It is shown that the Carleman class C{k! M k } contains all C ∞ roots of its nonflat elements, i.e., if f ∈ C{k! M k } and α > 0, then $$f^\alpha \in C\{ k!M_k \} whenever f^\alpha \in C^\infty $$ . If {M k } also satisfies the additional condition M n 1/n → ∞, then the Beurling class C(k! M k ) is also contains all C ∞ roots of its nonflat elements. 相似文献
17.
Ralph de Laubenfels 《Semigroup Forum》1986,33(1):257-263
We show that a linear operator (possibly unbounded), A, on a reflexive Banach space, X, is a scalar-type spectral operator, with non-negative spectrum, if and only if the following conditions hold.
- A generates a uniformly bounded holomorphic semigroup {e?zA}Re(z)≥0.
- If \(F_N (s) \equiv \int_{ - N}^N {\tfrac{{\sin (sr)}}{r}} e^{irA} dr\) , then {‖FN‖} N=1 ∞ is uniformly bounded on [0,∞) and, for all x in X, the sequence {FN(s)x} N=1 ∞ converges pointwise on [0, ∞) to a vector-valued function of bounded variation.
18.
A. A. Ryabinin 《Journal of Mathematical Sciences》1994,68(4):577-584
For series of random variables $\sum\limits_{k = 1}^\infty {a_k x_k }$ ,a K ∈R 1, {X K } K=1 ∞ being an Ising system, i.e., for each n ≥ 2 the joint distribution of {X K } K=1 n has the form $$P_n (t_1 ,...,t_n ) = ch^{ - (n - 1)} J \cdot exp(J\sum\limits_{k - 1}^{n - 1} {t_k t_{k + 1} )\prod\limits_{k = 1}^n {\frac{1}{2}\delta (t_{k^{ - 1} }^2 ),J > 0} }$$ one obtains a criterion for almost everywhere convergence: $\sum\limits_{k = 1}^\infty {a_k^2< \infty }$ . The relation between the asymptotic behavior of large deviations of the sum and the rate of decrease of the sequence {ak} of the coefficients is investigated. 相似文献
19.
S. N. Poleščuk 《Analysis Mathematica》1981,7(4):265-275
Пусть {f n } n=1 ∞ — после довательность измер имых функций, а {ω(n)} n=1 ∞ — неу бывающая последовательность положительных чисел. Система {f n }∈W(uc, ω (n)), есл и всякий ряд (1) $$\mathop \Sigma \limits_{n = 1}^\infty a_n f_n \left( t \right)$$ после любой перестан овки членов сходится почти всюду, как только $$\mathop \Sigma \limits_{n = 1}^\infty a_n^2 \omega \left( n \right)< \infty $$ то есть {ω (n)} является множителем Вейля для безусловной сходимо сти рядов вида (1). Если {f n }∈W(uc, ω (n)), но для л юбой последовательн остиγ(n)=o(ω(n)) приn→∞ система {f n }?W(uc, γ (n)) то {ω (n)} называют точным множителем Вейля для безусловной сходимости рядов вид а (1). Основной результат: с уществует полная ортонормированная с истема, которая имеет точный множитель Вей ля для безусловной сх одимости. 相似文献
20.
G. A. Karagulian 《Analysis Mathematica》1992,18(4):249-259
В статье доказываетс я Теорема.Какова бы ни была возрастающая последовательность натуральных чисел {H k } k = 1 ∞ c $$\mathop {\lim }\limits_{k \to \infty } \frac{{H_k }}{k} = + \infty$$ , существует функцияf∈L(0, 2π) такая, что для почт и всех x∈(0, 2π) можно найти возраст ающую последовательность номеров {nk(x)} k=1 ∞ ,удовлетворяющую усл овиям 1) $$n_k (x) \leqq H_k , k = 1,2, ...,$$ 2) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t} (x)} (x,f) = + \infty ,$$ 3) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t - 1} (x)} (x,f) = - \infty$$ . 相似文献