共查询到20条相似文献,搜索用时 187 毫秒
1.
А. Л. Лукашов 《Analysis Mathematica》1998,24(1):111-130
The Chebyshev-Markov problem about real algebraic functions of the form $A_n (x) = \frac{{x^n + c_1 x^{n - 1} + ... + c_n }}{{\left( {\prod {_{i = 1}^{2n} } \left( {1 - a_{i,n} x} \right)} \right)^{1/2} }}$ deviated least from zero on a system of intervals $\left[ {b_1 ;b_2 } \right] \cup ... \cup \left[ {b_{2p - 1} ;b_{2p} } \right], - \infty< b_1 \leqslant b_2< ...< b_{2p - 1} \leqslant b_{2p}< + \infty $ is considered. The expression under the square root above is a real polynomial of degree less than 2n, which is positive on [b 1;b 2p ]. The solution of this problem is given in a parametric form in terms of automorphic Schottky-Burside functions. Similar functions were first used by N. I. Akhyeser in the approximation theory. 相似文献
2.
L. P. Il'ina 《Mathematical Notes》1973,13(3):215-218
For the coefficients bn of an odd function \(f(z) = z + \sum\nolimits_{k = 1}^\infty {{}^bk^{z^{2k + 1} } } \) , regular in the unit disk, we obtain the estimate $$|b_n | \leqslant \frac{1}{{\sqrt 2 }}\sqrt {1 + |b_1 |^2 } \exp \frac{1}{2}\left( {\delta + \frac{1}{2}|b_1 |^2 } \right),where \delta = 0.312,$$ (1) from which it follows that ¦bn¦≤1, if ¦b1¦≤0.524. It follows from (1) that the coefficients cn, n = 3, 4,..., of a regular function \(f(2) = z + \sum\nolimits_{k = 2}^\infty {{}^ck^{z^k } } \) , univalent in the unit desk, satisfy $$|c_n | \leqslant \frac{1}{2}\left( {1 + \frac{{|c_2 |^2 }}{4}} \right)n\exp \left( {\delta + \frac{{|c_2 |^2 }}{8}} \right),where \delta = 0.312,$$ in particular, ¦cn¦≤n, if ¦c2¦≤1.046. 相似文献
3.
Let \({\frak {e}}\subset {\mathbb {R}}\) be a finite union of ?+1 disjoint closed intervals, and denote by ω j the harmonic measure of the j left-most bands. The frequency module for \({\frak {e}}\) is the set of all integral combinations of ω 1,…,ω ? . Let \(\{\tilde{a}_{n}, \tilde{b}_{n}\}_{n=-\infty}^{\infty}\) be a point in the isospectral torus for \({\frak {e}}\) and \(\tilde{p}_{n}\) its orthogonal polynomials. Let \(\{a_{n},b_{n}\}_{n=1}^{\infty}\) be a half-line Jacobi matrix with \(a_{n} = \tilde{a}_{n} + \delta a_{n}\), \(b_{n} = \tilde{b}_{n} +\delta b_{n}\). Suppose and \(\sum_{n=1}^{N} e^{2\pi i\omega n} \delta a_{n}\), \(\sum_{n=1}^{N} e^{2\pi i\omega n} \delta b_{n}\) have finite limits as N→∞ for all ω in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to ω, then for z∈???, \(p_{n}(z)/\tilde{p}_{n}(z)\) has a limit as n→∞. Moreover, we show that there are non-Szeg? class J’s for which this holds.
相似文献
$\sum_{n=1}^\infty \lvert \delta a_n\rvert ^2 + \lvert \delta b_n\rvert ^2 <\infty $
4.
F. Móricz 《Analysis Mathematica》1983,9(1):57-67
Основной целью работ ы является обобщение одного результата Кратца и Т раутнера [4], известного для одном ерных функциональны х рядов, на кратные ряды. Этот рез ультат касается суммируемо сти функционального ряда почти всюду при слабых пред положениях. В частности, он примен им к суммируемости по Чезаро и по Риссу. Мы рассматриваемd-кр атный ряд $$\mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x), \mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty c_{k_1 ,...,k_d }^2< \infty $$ и предполагается, что функции \(f_{k_1 ,...,k_d } (x)\) интегрируе мы по пространству с полож ительной мерой и имеют почти вс юду ограниченные фун кции Лебега для метода суммирова ния Т. Метод Т определяетсяd-мерной матрицей \(T = \{ a_{m_1 ,...,m_d ;k_1 ,...,k_d } \} \) сл едующим образом: $$t_{m_1 ,...,m_d } (x) = \mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty a_{m_1 ,...,m_d ;k_1 ,...,k_d } c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x).$$ Эти средние существу ют, поскольку мы предп олагаем, что \(a_{m_1 ,...,m_d ;k_1 ,...,k_d } = 0\) ,если max(k 1,...,k d) достаточно вели к (в зависимости, конеч но, отm 1,...,m d). При некоторых дополнительных усло виях на матрицуТ (см. (7)– (9) в разделе 3) устанавлива ется почти всюду регулярная схо димость средних \(t_{m_1 ,...,m_d } (x) \user2{} \user2{(}m_1 \user2{,}...\user2{,}m_d \user2{)} \to \infty \) . Как вспомогательный результат, в работе об общается теорема Алексича [1] о сх одимости почти всюду некоторы х подпоследовательн остей частных сумм функцио нального ряда. 相似文献
5.
6.
In this paper, we first consider difference equations with several delays in the neutral term of the form * $$\Delta\left(y_{n}+\sum_{i=1}^{L}p_{i}y_{n-{k_{i}}}-\sum_{j=1}^{M}r_{j}y_{n-{\rho_{j}}}\right)+q_{n}y_{n-\tau}=0\quad \mbox{for}\ n\in\mathbb{Z}^{+}(0),$$ study various cases of coefficients in the neutral term and obtain the asymptotic behavior for non-oscillatory solution of (*) under some hypotheses. Moreover, we consider reaction-diffusion difference equations with several delays in the neutral term of the form $$\begin{array}{l}\Delta_{1}\left(u_{n,m}+\displaystyle \sum_{i=1}^{L}p_{i}u_{n-{k_{i}},m}-\displaystyle \sum_{j=1}^{M}r_{j}u_{n-{\rho_{j}},m}\right)+q_{n,m}u_{n-\tau,m}\\[18pt]\quad {}=a^{2}\Delta_{2}^{2}u_{n+1,m-1}\end{array}$$ for (n,m)∈?+(0)×Ω, study various cases of coefficients in the neutral term and obtain the asymptotic behavior for non-oscillatory solution under some hypotheses. 相似文献
7.
S. S. Khloponin 《Mathematical Notes》1976,20(5):933-938
We have proved that if the partial numerators of the continued fraction f(c)=1/1+c2/l+c3/l+... are all nonzero and for at least some number n?1 satisfy the inequalities $$p_n \left| {1 + c_n + c_{n + 1} } \right| \ge p_{n - 2} p_n \left| {c_n } \right| + \left| {c_{n + 1} } \right|(n \ge 1,p_{ - 1} = p_0 = c_1 = 0,p_n \ge 0),$$ then f(c) converges in the wide sense if and only if at least one of the series $$\begin{array}{l} \sum\nolimits_{n = 1}^\infty {\left| {c_3 c_5 \ldots c_{2n - 1} /(c_2 c_4 \ldots c_{2n} )} \right|} , \\ \sum\nolimits_{n = 1}^\infty {\left| {c_2 c_4 \ldots c_{2n} /(c_3 c_5 \ldots c_{2n + 1} )} \right|} \\ \end{array}$$ 相似文献
8.
I. E. Egorov 《Mathematical Notes》1978,23(3):211-217
Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W 2 1 (QT) and W 2 2 (QT). 相似文献
9.
Ulrich Huckenbeck 《manuscripta mathematica》1987,59(2):147-173
In this paper we prove the following theorem: Suppose that n≥3 and 1≤jn $$(\forall a,b) d(a,b) : = \sum\limits_{\nu = 1}^j { (a_\nu - b_\nu )^2 - \sum\limits_{\nu = j + 1}^n { (a_\nu - b_\nu )^2 .} }$$ If a function f:?n→?n satisfies the condition: (*) $$(\forall x,y \in \mathbb{R}^n ) d(f(x),f(y)) = 0 \Leftrightarrow d(x,y) = 0,$$ then f is affine. Moreover, f preserves distances up to a constant factor C≠0, i.e. d(f(x),f(y))=C·d(x,y) for every x,y. In contrast to Alexandrov's result [1] we do not assume that f is bijective, and we also do not assume that j=n?1. A very important part of our proof will be the discussion of a functional equation. 相似文献
10.
Let \(0< \rho <1\) and let \(\{a_n, b_n\}_{n=1}^\infty \) be a sequence of integers with bounded from upper and lower. Associated with them there exists a unique Borel probability measure \(\mu _{\rho , \{0, a_n, b_n\}}\) generated by the following infinite convolution product in the weak convergence, where \(\delta _E=\frac{1}{\# E}\sum _{e \in E} \delta _e\) and \(\hbox {gcd}(a_n, b_n)=1\) for all \(n \in {{\mathbb {N}}}\). In this paper, we show that \(L^2(\mu _{\rho , \{0, a_n, b_n\}})\) admits an exponential orthonormal basis if and only if \(\rho ^{-1} \in 3{{\mathbb {N}}}\) and \(\{a_n, b_n\} \equiv \{1, 2\} \ (\mathrm {mod} \ 3)\) for all \(n \in {{\mathbb {N}}}\).
相似文献
$$\begin{aligned} \mu _{\rho , \{0, a_n, b_n\}}=\delta _{\rho \{0, a_1, b_1\}} *\delta _{\rho ^2 \{0, a_2, b_2\}} *\delta _{\rho ^3 \{0, a_3, b_3\}} *\cdots \end{aligned}$$
11.
Stevo Stević 《Journal of Applied Mathematics and Computing》2007,24(1-2):295-303
In this paper we investigate the boundedness character of the positive solutions of the rational difference equation of the form $$x_{n + 1} = \frac{{a_0 + \sum\nolimits_{j = 1}^k {a_j x_{n - j + 1} } }}{{b_0 + \sum\nolimits_{j = 1}^k {b_j x_{n - j + 1} } }}, n = 0,1,...$$ where k ε N, andaj,bj, j = 0,1,…, k, are nonnegative numbers such thatb 0+∑ j=1 k b j x n-j+1>0 for everyn ∈N ∪{0}. In passing we confirm several conjectures recently posed in the paper: E. Camouzis, G. Ladas and E. P. Quinn, On third order rational difference equations (part 6). 相似文献
12.
B. P. Osilenker 《Mathematical Notes》2007,82(3-4):366-379
We study discrete Sobolev spaces with symmetric inner product $$\left\langle {f,g} \right\rangle _\alpha = \int_{ - 1}^1 {f g d\mu _\alpha } + M[f(1)g(1) + f( - 1)g( - 1)] + K[f'(1)g'(1) + f'( - 1)g'( - 1)]$$ , where M ≥ 0, k ≥ 0, and $$d\mu _\alpha (x) = \frac{{\Gamma (2\alpha + 2)}}{{2^{2\alpha + 1} \Gamma ^2 (\alpha + 1)}}(1 - x^2 )^\alpha dx, \alpha > - 1$$ , is the Gegenbauer probability measure. We obtain the solution of the following extremal problem: Calculate $$\mathop {\inf }\limits_{a_0 ,a_1 ,...,a_{N - r} } \left\{ {\langle P_N^{(r)} ,P_N^{(r)} \rangle _\alpha ,1 \leqslant r \leqslant N - 1, P_N^{(r)} (x) = \sum\limits_{j = N - r + 1}^N {a_j^0 x^j } + \sum\limits_{j = 0}^{N - r} {a_j x^j } } \right\}$$ , where the a j 0 , j = N ? r + 1, N ? r + 2, ..., N ? 1, N, a N 0 > 0, are fixed numbers, and find the extremal polynomial. 相似文献
13.
G. I. Arkhipov 《Mathematical Notes》1978,23(6):431-433
We give a simple proof of a mean value theorem of I. M. Vinogradov in the following form. Suppose P, n, k, τ are integers, P≥1, n≥2, k≥n (τ+1), τ≥0. Put $$J_{k,n} (P) = \int_0^1 \cdots \int_0^1 {\left| {\sum\nolimits_{x = 1}^P {e^{2\pi i(a_1 x + \cdots + a_n x^n )} } } \right|^{2k} da_1 \ldots da_n .} $$ Then $$J_{k,n} \leqslant n!k^{2n\tau } n^{\sigma n^2 u} \cdot 2^{2n^2 \tau } P^{2k - \Delta } ,$$ where $$\begin{gathered} u = u_\tau = min(n + 1,\tau ), \hfill \\ \Delta = \Delta _\tau = n(n + 1)/2 - (1 - 1/n)^{\tau + 1} n^2 /2. \hfill \\ \end{gathered} $$ 相似文献
14.
Under the Keller?COsserman condition on ${\Sigma_{j=1}^{2}f_{j}}$ , we show the existence of entire positive solutions for the semilinear elliptic system ${\Delta u_{1}+|\nabla u_{1}|=p_{1}(x)f_{1}(u_{1},u_{2}), \Delta u_{2}+|\nabla u_{2}|=p_{2}(x)f_{2}(u_{1},u_{2}),x \in \mathbb{R}^{N}}$ , where ${p_{j}(j=1, 2):\mathbb{R}^{N} \rightarrow [0,\infty)}$ are continuous functions. 相似文献
15.
P. Z. Mkrtychyan 《Journal of Mathematical Sciences》1985,28(5):742-750
In a bounded domain of the n -dimensional (n?2) space one considers a class of degenerate quasilinear elliptic equations, whose model is the equation $$\sum\limits_{i = 1}^n {\frac{{\partial F}}{{\partial x_i }}} (a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i - 2} u_{x_i } ) = f(x),$$ where x =(x1,..., xr), li?0, mi>1, the function f is summable with some power, the nonnegative continuous function a(u) vanishes at a finite number of points and satisfies \(\frac{{lim}}{{\left| u \right| \to \infty }}a(u) > 0\) . One proves the existence of bounded generalized solutions with a finite integral $$\int\limits_\Omega {\sum\limits_{i = 1}^n {a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i } dx} }$$ of the Dirichlet problem with zero boundary conditions. 相似文献
16.
We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure μ with even continuous density and sections are of arbitrary dimension n ? k, 1 ≤ k < n. If K is a generalized k-intersection body, then $$\mu(K)\,\leq\,\frac{n}{n-k}c_{n,k} \max_{H} \mu(K\cap H) {\rm Vol}_n(K)^{k/n}.$$ Here ${c_{n,k} = |B_2^n|^{(n-k)/n}/|B_2^{n-k}| <1 ,{ }|B_2^n|}$ is the volume of the unit Euclidean ball, and maximum is taken over all (n ? k)-dimensional subspaces of ${\mathbb{R}^{n}}$ . The constant is optimal, and for each intersection body the inequality holds for every k. We also prove a stronger “difference” inequality. The proof is based on stability in the lower dimensional Busemann–Petty problem for arbitrary measures in the following sense. Let ${\varepsilon >0 ,\ 1\le k < n}$ . Suppose that K and L are origin-symmetric star bodies in ${\mathbb{R}^{n}}$ , and K is a generalized k-intersection body. If for every (n ? k)-dimensional subspace H of ${\mathbb{R}^{n}}$ $$\mu(K\cap H)\leq \mu(L\cap H)+\varepsilon,$$ then $$\mu(K)\leq \mu(L) +\frac{n}{n-k}c_{n,k} {\rm Vol}_n(K)^{k/n} \varepsilon.$$ 相似文献
17.
D. S. Lubinsky 《Constructive Approximation》1988,4(1):321-339
Letf(z):=Σ j=0 ∞ a j z j , where aj ≠ 0,j large enough, and for someq ε C such that ¦q¦ $$q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q,j \to \infty .$$ Define for m,n = 0,1,2,..., the Toeplitz determinant $$D(m/n): = \det (a_{m - j + k} )_{j,k = 1}^n .$$ Given ? > 0, we show that form large enough, and for everyn = 1,2,3,..., $$(1 - \varepsilon )^n \leqslant \left| {{{D(m/n)} \mathord{\left/ {\vphantom {{D(m/n)} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right| \leqslant (1 + \varepsilon )^n .$$ We apply this to show that any sequence of Padé approximants {[m k /n k ]} 1 ∞ tof, withm k →∞ ask→ ∞, converges locally uniformly in C. In particular, the diagonal sequence {[n/n]} 1 ∞ converges throughout C. Further, under additional assumptions, we give sharper asymptotics forD(m/n). 相似文献
18.
Rym Chemmam 《Mediterranean Journal of Mathematics》2013,10(3):1259-1272
We consider in this paper the existence and the asymptotic behavior of positive ground state solutions of the boundary value problem $${-}\Delta u = a_{1}(x)u^{\alpha_{1}} + a_{2}(x) u^{\alpha_{2}}\,\, {\rm in}\,\, \mathbb{R}^{n}, \lim_{|x| \rightarrow \infty} u(x) = 0$$ , where α 1, α 2 < 1 and a 1, a 2 are nonnegative functions in ${C^{\gamma}_{loc}} (\mathbb{R}^{n})$ , ${0 < \gamma < 1}$ , satisfying some appropriate assumptions related to Karamata regular variation theory. 相似文献
19.
V. A. Sadovnichii 《Mathematical Notes》1973,14(2):717-723
We investigate the question of the regularized sums of part of the eigenvalues zn (lying along a direction) of a Sturm-Liouville operator. The first regularized sum is $$\sum\nolimits_{n = 1}^\infty {(z_n - n - \frac{{c_1 }}{n} + \frac{2}{\pi } \cdot z_n arctg \frac{1}{{z_n }} - \frac{2}{\pi }) = \frac{{B_2 }}{2} - c_1 \cdot \gamma + \int_1^\infty {\left[ {R(z) - \frac{{l_0 }}{{\sqrt z }} - \frac{{l_1 }}{z} - \frac{{l_2 }}{{z\sqrt z }}} \right]} } \sqrt z dz,$$ where the zn are eigenvalues lying along the positive semi-axis, z n 2 =λn, $$l_0 = \frac{\pi }{2}, l_1 = - \frac{1}{2}, l_2 = - \frac{1}{4}\int_0^\pi {q(x) dx,} c_1 = - \frac{2}{\pi }l_2 ,$$ , B2 is a Bernoulli number, γ is Euler's constant, and \(R(z)\) is the trace of the resolvent of a Sturm-Liouville operator. 相似文献
20.
Fixed Points and Stability for Quadratic Mappings in β–Normed Left Banach Modules on Banach Algebras
The goal of the present paper is to investigate some new stability results by applying the alternative fixed point of generalized quadratic functional equation $$\begin{array}{ll}f\left(\sum\limits_{i=1}^{n}a_ix_i\right)+{\sum\limits_{i=1}^{n-1}}{\sum\limits_{j=i+1}^{n}}\left[f(a_ix_i+a_jx_j)+2f(a_ix_i-a_jx_j)\right]\\ \qquad \quad = (3n-2){\sum\limits_{i=1}^{n}}a^2_{i}f(x_{i})\end{array}$$ in β–Banach modules on Banach algebras, where ${a_{1},\dots,a_{n}\in \mathbb{Z}{\setminus}\{0\}}$ and some ${\ell\in\{1 , 2 ,\dots, n-1\},}$ a ? ?≠ ±1 and a n ?=?1, where n is a positive integer greater or at least equal to two. 相似文献