共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
Denote by span {f 1,f 2, …} the collection of all finite linear combinations of the functionsf 1,f 2, … over ?. The principal result of the paper is the following. Theorem (Full Müntz Theorem in Lp(A) for p ∈ (0, ∞) and for compact sets A ? [0, 1] with positive lower density at 0). Let A ? [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ j ) j=1 ∞ is a sequence of distinct real numbers greater than ?(1/p). Then span {x λ1,x λ2,…} is dense in Lp(A) if and only if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ . Moreover, if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ , then every function from the Lp(A) closure of {x λ1,x λ2,…} can be represented as an analytic function on {z ∈ ? \ (?∞,0] : |z| < rA} restricted to A ∩ (0, rA) where $r_A : = \sup \left\{ {y \in \mathbb{R}:\backslash ( - \infty ,0]:\left| z \right|< r_A } \right\}$ (m(·) denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Müntz, Szász, Clarkson, Erdös, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of {x λ1,x λ2,…} are also considered. 相似文献
3.
On approximation of a function and its derivatives by a polynomial and its derivatives 总被引:3,自引:0,他引:3
Let g∈C~q[-1, 1] be such that g~((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomialof degree at most n, such that P_n~((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivativesP_n~((k)) for k≤q well approximate g and its respective derivatives, provided only that P_n well approxi-mates g itself in the weighted norm ‖g(x)-P_n(x) (1-x~2)~(1/2)~q‖This result is easily extended to an arbitrary f∈C~q[-1, 1], by subtracting from f the polynomial ofminnimal degree which interpolates f~((0))…,f~((q)) at±1. As well as providing easy criteria for judging the simultaneous approximation properties of a givenPolynomial to a given function, our results further explain the similarities and differences betweenalgebraic polynomial approximation in C~q[-1, 1] and trigonometric polynomial approximation in thespace of q times differentiable 2π-periodic functions. Our proofs are elementary and basic in character,permitting the construction of actual error estimates for simultaneous approximation proedures for smallvalues of q. 相似文献
4.
Letk n be the smallest constant such that for anyn-dimensional normed spaceX and any invertible linear operatorT ∈L(X) we have $|\det (T)| \cdot ||T^{ - 1} || \le k_n |||T|^{n - 1} $ . LetA + be the Banach space of all analytic functionsf(z)=Σ k≥0 a kzk on the unit diskD with absolutely convergent Taylor series, and let ‖f‖A +=σ k≥0 |ακ|; define ? n on $\overline D ^n $ by $ \begin{array}{l} \varphi _n \left( {\lambda _1 ,...,\lambda _n } \right) \\ = inf\left\{ {\left\| f \right\|_{A + } - \left| {f\left( 0 \right)} \right|; f\left( z \right) = g\left( z \right)\prod\limits_{i = 1}^n {\left( {\lambda _1 - z} \right), } g \in A_ + , g\left( 0 \right) = 1 } \right\} \\ \end{array} $ . We show thatk n=sup {? n(λ1,…, λ n ); (λ1,…, λ n )∈ $\overline D ^n $ }. Moreover, ifS is the left shift operator on the space ?∞:S(x 0,x 1, …,x p, …)=(x 1,…,x p,…) and if Jn(S) denotes the set of allS-invariantn-dimensional subspaces of ?∞ on whichS is invertible, we have $k_n = \sup \{ |\det (S|_E )|||(S|_E )^{ - 1} ||E \in J_n (S)\} $ . J. J. Schäffer (1970) proved thatk n≤√en and conjectured thatk n=2, forn≥2. In factk 3>2 and using the preceding results, we show that, up to a logarithmic factor,k n is of the order of √n whenn→+∞. 相似文献
5.
ZheFeng Xu 《中国科学 数学(英文版)》2013,56(8):1597-1606
For any real constants λ 1, λ 2 ∈ (0, 1], let $n \geqslant \max \{ [\tfrac{1} {{\lambda _1 }}],[\tfrac{1} {{\lambda _2 }}]\} $ , m ? 2 be integers. Suppose integers a ∈ [1, λ 1 n] and b ∈ [1, λ 2 n] satisfy the congruence b ≡ a m (mod n). The main purpose of this paper is to study the mean value of (a ? b)2k for any fixed positive integer k and obtain some sharp asymptotic formulae. 相似文献
6.
S. D. Fisher 《Constructive Approximation》1996,12(4):463-480
Let Ω be a finitely-connected planar domain and μ be a positive measure with compact supportE in Ω. LetA p be the unit ball of the Hardy spaceH p. The main result of this paper is that Kolmogorov, Gelfand, and linearn-widths ofA p inL q are comparable in size to each other and to the sampling error ifq≤p. Moreover, ifp=q=2 andE is small enough, then all these quantities are equal. 相似文献
7.
Let e λ (x) be an eigenfunction with respect to the Laplace-Beltrami operator Δ M on a compact Riemannian manifold M without boundary: Δ M e λ = λ 2 e λ . We show the following gradient estimate of e λ : for every λ ≥ 1, there holds ${\lambda\|e_\lambda\|_\infty/C\leq \|\nabla e_\lambda\|_\infty\leq C\lambda\|e_\lambda\|_\infty}$ , where C is a positive constant depending only on M. 相似文献
8.
Consider an exponential familyP λ which is maximal, smooth, and has uniformly bounded standardized fourth moments. Consider a sequenceX 1,X 2,... of i.i.d. random variables with parameter λ. LetQ nsk be the law ofX 1,...,X k given thatS n=X 1+...+X n=s. Choose λ so thatE λ(X 1)=s/n. Ifk andn→∞ butk/n→0, then $$\parallel Q_{nsk} - P_\lambda ^k \parallel = \gamma \frac{k}{n} + o\left( {\frac{k}{n}} \right)$$ where γ=1/2E{|1?Z 2|} andZ isN(0,1). The error term is uniform ins, the value ofS n. Similar results are given fork/n→θ and for mixtures of theP λ k . Versions of de Finetti's theorem follow. 相似文献
9.
Letk t(G) be the number of cliques of ordert in the graphG. For a graphG withn vertices let \(c_t (G) = \frac{{k_t (G) + k_t (\bar G)}}{{\left( {\begin{array}{*{20}c} n \\ t \\ \end{array} } \right)}}\) . Letc t(n)=Min{c t(G)∶?G?=n} and let \(c_t = \mathop {\lim }\limits_{n \to \infty } c_t (n)\) . An old conjecture of Erdös [2], related to Ramsey's theorem states thatc t=21-(t/2). Recently it was shown to be false by A. Thomason [12]. It is known thatc t(G)≈21-(t/2) wheneverG is a pseudorandom graph. Pseudorandom graphs — the graphs “which behave like random graphs” — were inroduced and studied in [1] and [13]. The aim of this paper is to show that fort=4,c t(G)≥21-(t/2) ifG is a graph arising from pseudorandom by a small perturbation. 相似文献
10.
E. G. Goluzina 《Journal of Mathematical Sciences》1987,38(4):2061-2064
LetP κ,n (λ,β) be the class of functions \(g(z) = 1 + \sum\nolimits_{v = n}^\infty {c_\gamma z^v }\) , regular in ¦z¦<1 and satisfying the condition $$\int_0^{2\pi } {\left| {\operatorname{Re} \left[ {e^{i\lambda } g(z) - \beta \cos \lambda } \right]} \right|} /\left( {1 - \beta } \right)\cos \lambda \left| {d\theta \leqslant \kappa \pi ,} \right.z = re^{i\theta } ,$$ , 0 < r < 1 (κ?2,n?1, 0?Β<1, -π<λ<π/2;M κ,n (λ,β,α),n?2, is the class of functions \(f(z) = z + \sum\nolimits_{v = n}^\infty {a_v z^v }\) , regular in¦z¦<1 and such thatF α(z)∈P κ,n?1(λ,β), where \(F_\alpha (z) = (1 - \alpha )\frac{{zf'(z)}}{{f(z)}} + \alpha (1 + \frac{{zf'(z)}}{{f'(z)}})\) (0?α?1). Onr considers the problem regarding the range of the system {g (v?1)(z?)/(v?1)!}, ?=1,2,...,m,v=1,2,...,N ?, on the classP κ,1(λ,β). On the classesP κ,n (λ,β),M κ,n (λ,β,α) one finds the ranges of Cv, v?n, am, n?m?2n-2, and ofg(?),F ?(?), 0<¦ξ¦<1, ξ is fixed. 相似文献
11.
Jean-Pierre Gabardo 《Monatshefte für Mathematik》1993,116(3-4):197-229
Given a finite intervalI?R, a characterization is given for those discrete sets of real numbers Λ and associated sequences {c λ}λ∈Λ, withc λ>0, having the properties that every functionf∈L 2(I) can be expanded inL 2(I) as the unconditionally convergent series $$f = \sum\limits_{\lambda \in \Lambda } {\hat f} (\lambda )c_\lambda e^{2\pi i\lambda x} $$ and that the range of the mappingL 2(I)→L μ 2 :f→f has finite codimension inL μ 2 , iff denotes the Fourier transform off and μ is the measure μ = ∑λ∈Λ c λ δλ. 相似文献
12.
Wu Pengcheng 《数学学报(英文版)》1996,12(2):191-204
This paper proves the following result: Letf(z) be a meromorphic function in thez-plane with a deficient value, and δ(θ k )(k=1,2, ...,q;0≤θ 1<θ2<...<θ q<θ q+1=θ 1+2π) beq rays (1≤q<∞) starting at the origin, and letn≥3 be an integer such that for any given positive numberε,0<ε<π/2, $$\overline {\mathop {\lim }\limits_{r \to \infty } } \frac{{\log ^ + n\left\{ { \cup _{k = 1}^q \Omega \left( {\theta _k + \varepsilon ,\theta _{k + 1} - \varepsilon ,r} \right),f\prime f^n = 1} \right\}}}{{\log r}} \leqslant v< \infty ,$$ whereΝ is a constant independent ofε. IfΜ<∞, then we have $$\lambda \leqslant \frac{\pi }{\omega } + v,$$ whereΜ andλ denote the lower order and order off(z), respectively,Ω=minθ k+1 ?θ k ;1≤k≤q, andn(E, f=a) is the number of zeros off(z)?a inE with multiple zeros being counted with their multiplicities. 相似文献
13.
Я. С. Бугров 《Analysis Mathematica》1979,5(2):119-133
Получены новые оценк иL-нормы тригонометр ических полиномов $$T_n (t) = \frac{{\lambda _0 }}{2} + \mathop \sum \limits_{k = 1}^n \lambda _k \cos kt$$ в терминах коэффицие нтовλ k и их разностейΔλ k=λ k?λ k?1: (1) $$\mathop \smallint \limits_{ - \pi }^\pi |T_n (t)|dt \leqq \frac{c}{n}\mathop \sum \limits_{k = 0}^n |\lambda _\kappa | + c\left\{ {x(n,\varphi )\mathop \sum \limits_{k = 0}^n \Delta \lambda _\kappa \mathop \sum \limits_{l = 0}^n \Delta \lambda _l \delta _{\kappa ,l} (\varphi )} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,$$ где $$\kappa (n,\varphi ) = \mathop \smallint \limits_{1/n}^\pi [t^2 \varphi (t)]^{ - 1} dt, \delta _{k,1} (\varphi ) = \mathop \smallint \limits_0^\infty \varphi (t)\sin \left( {k + \frac{1}{2}} \right)t \sin \left( {l + \frac{1}{2}} \right)t dt,$$ a ?(t) — произвольная фун кция ≧0, для которой опр еделены соответствующие инт егралы. Из (1) следует, что методы $$\tau _n (f;t) = (N + 1)^{ - 1} \mathop \sum \limits_{k = 0}^{\rm N} S_{[2^{k^\varepsilon } ]} (f;t), n = [2^{N\varepsilon } ],$$ являются регулярным и для всех 0<ε≦1/2. ЗдесьS m (f, x) частные суммы ряда Фу рье функцииf(x). В статье исследуется многомерный случай. П оказано, что метод суммирования (о бобщенный метод Рисса) с коэффиц иентами $$\lambda _{\kappa ,l} = (R^v - k^\alpha - l^\beta )^\delta R^{ - v\delta } (0 \leqq k^\alpha + l^\beta \leqq R^v ;\alpha \geqq 1,\beta \geqq 1,v< 0)$$ является регулярным, когда δ > 1. 相似文献
14.
Dhruv Mubayi 《Combinatorica》2013,33(5):591-612
For various k-uniform hypergraphs F, we give tight lower bounds on the number of copies of F in a k-uniform hypergraph with a prescribed number of vertices and edges. These are the first such results for hypergraphs, and extend earlier theorems of various authors who proved that there is one copy of F. A sample result is the following: Füredi-Simonovits [11] and independently Keevash-Sudakov [16] settled an old conjecture of Sós [29] by proving that the maximum number of triples in an n vertex triple system (for n sufficiently large) that contains no copy of the Fano plane is p(n)=( 2 ?n/2? )?n/2?+( 2 ?n/2? ?n/2?). We prove that there is an absolute constant c such that if n is sufficiently large and 1 ≤ q ≤ cn 2, then every n vertex triple system with p(n)+q edges contains at least $6q\left( {\left( {_4^{\left\lfloor {n/2} \right\rfloor } } \right) + \left( {\left\lceil {n/2} \right\rceil - 3} \right)\left( {_3^{\left\lfloor {n/2} \right\rfloor } } \right)} \right)$ copies of the Fano plane. This is sharp for q≤n/2–2. Our proofs use the recently proved hypergraph removal lemma and stability results for the corresponding Turán problem. 相似文献
15.
T. A. Leont'eva 《Analysis Mathematica》1988,14(1):99-109
Пусть $$f_n (z) = \exp \{ \lambda _n z\} [1 + \psi _n (z)], n \geqq 1$$ гдеψ n (z) — регулярны в н екоторой односвязно й областиS, λ n — нули целой функц ии экспоненциальног о ростаL(λ) с индикатрис ой ростаh(?), причем $$|L\prime (\lambda _n )| > C(\delta )\exp \{ [h(\varphi _n ) - \varepsilon ]|\lambda _n |\} \varphi _n = \arg \lambda _n , \forall \varepsilon > 0$$ . Предположим, что на лю бом компактеK?S $$|\psi _n (z)|< Aq^{|\lambda |_n } , a< q< 1, n \geqq 1$$ гдеA иq зависит только отK. Обозначим через \(\bar D\) со пряженную диаграмму функцииL(λ), через \(\bar D_\alpha \) — смещение. \(\bar D\) на векторα. Рассмотр им множестваD 1 иD 2 так ие, чтоD 1 иD 2 и их вьшуклая обо лочкаE принадлежатS. Пусть \(\bar D_{\alpha _1 } \subset D_1 , \bar D_{\alpha _2 } \subset D_2 \) Доказывается, что сущ ествует некоторая об ластьG?E такая, что \(\mathop \cup \limits_{\alpha \in [\alpha _1 ,\alpha _2 ]} \bar D_\alpha \subset G\) и дляz∈G верна оценка $$\sum\limits_{v = 1}^n {|a_v f_v (z)|} \leqq B\max (M_1 ,M_2 ), M_j = \mathop {\max }\limits_{t \in \bar D_j } |\sum\limits_{v = 1}^n {a_v f_v (t)} |$$ , где константаB не зав исит от {a v }. 相似文献
16.
V. V. Napalkov 《Mathematical Notes》1974,15(1):92-95
Three convolution-type equations are considered in the space of entire functions with topology ofd uniform convergence: $$\begin{gathered} M{_{\mu}{_1}} [f] \equiv \smallint _C f(z + t)d\mu _1 = 0, \hfill \\ M{_\mu{_1}} [f] \equiv \smallint _C f(z + t)d\mu _2 = 0, \hfill \\ M_\mu [f] \equiv \smallint _C f(z + t)d\mu = 0 \hfill \\ \end{gathered}$$ with respective characteristic functions L1(λ), L2(λ), L(λ)=L1(λ)· L2(λ), suppμ ?c, suppμ 1 ?c, suppμ 2 ?c. The necessary and sufficient conditions are found that every solutionf(z) of the equation Mμ[f[ can be written as a sumf 1(z) +f 2(z), wheref 1(z) is the solution of the equation \(M{_\mu{_1}} [f] = 0\) ,f 2(z) is the solution of the equation \(M{_\mu{_2}} [f] = 0\) . 相似文献
17.
Eliyahu Beller 《Israel Journal of Mathematics》1975,22(1):68-80
The functionf(z), analytic in the unit disc, is inA p if \(\int {\int {_{\left| z \right|< 1} \left| {f(z)} \right|^p dxdy< \infty } } \) . A necessary condition on the moduli of the zeros ofA p functions is shown to be best possible. The functionf(z) belongs toB p if \(\int {\int {_{\left| z \right|< 1} \log ^ + \left| {f(z)} \right|)^p } } \) . Let {z n } be the zero set of aB p function. A necessary condition on |z n | is obtained, which, in particular, implies that Σ(1?|z n |)1+(1/p)+g <∞ for all ε>0 (p≧1). A condition on the Taylor coefficients off is obtained, which is sufficient for inclusion off inB p. This in turn shows that the necessary condition on |z n | is essentially the best possible. Another consequence is that, forq≧1,p<q, there exists aB p zero set which is not aB q zero set. 相似文献
18.
Björn G. Walther 《Arkiv f?r Matematik》1999,37(2):381-393
For Ξ∈R n ,t∈R andf∈S(R n ) define $\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$ . We determine the optimal regularitys 0 such that $\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$ holds whereC is independent off∈S(R n ) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11]. Our theorems can be generalized to the case where the exp(it|ξ|2) is replaced by exp(it|ξ|a),a≠2. The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL 2(R n ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity. 相似文献
19.
If an isometric embeddingl p m →l q n with finitep, q>1 exists, thenp=2 andq is an even integer. Under these conditions such an embedding exists if and only ifn?N(m, q) where $$\left( {\begin{array}{*{20}c} {m + q/2 - 1} \\ {m - 1} \\ \end{array} } \right) \leqslant N(m,q) \leqslant \left( {\begin{array}{*{20}c} {m + q - 1} \\ {m - 1} \\ \end{array} } \right).$$ To construct some concrete embeddings, one can use orbits of orthogonal representations of finite groups. This yields:N(2,q)=q/2+1 (by regular (q+2)-gon),N(3, 4)=6 (by icosahedron),N(3, 6)?11 (by octahedron), etc. Another approach is based on relations between embeddings, Euclidean or spherical designs and cubature formulas. This allows us to sharpen the above lower bound forN(m, q) and obtain a series of concrete values, e.g.N(3, 8)=16 andN(7, 4)=28. In the cases (m, n, q)=(3, 6, 10) and (3, 8, 15) some ε-embeddings with ε ~ 0.03 are constructed by the orbit method. The rigidness of spherical designs in Bannai's sense and a similar property for the embeddings are considered, and a conjecture of [7] is proved for any fixed (m, n, q). 相似文献
20.
K. Tandori 《Analysis Mathematica》1979,5(2):149-166
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.
- Если λn=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность {Vn (λ)(?;x)} расходится почти вс юду.
- Если λn=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