共查询到20条相似文献,搜索用时 15 毫秒
1.
P. Cubiotti 《Journal of Optimization Theory and Applications》1992,72(3):577-582
In this paper, given a nonempty closed convex setX ? n , a functionf: X→? n , and a multifunction Γ:X→2X, we deal with the problem of finding a point \(\hat x\) ∈X such that $$\hat x \in \Gamma (\hat x) and \langle f(\hat x), \hat x - y\rangle \leqslant 0, for all y \in \Gamma (\hat x).$$ For such problem, we establish a result where, in particular, the functionf is not assumed to be continuous. More precisely, we extend to the present setting a finite-dimensional version of a result by Ricceri on variational inequalities (Ref. 1). 相似文献
2.
Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X * there exists δ > 0 such that $$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\|<\delta. $$ Then every continuous convex function g with $g \leqslant f$ on X is generically Fréchet differentiable. If, in addition, $\lim\limits_{\|x\|\rightarrow\infty}f(x)=\infty$ , then X is an Asplund space. 相似文献
3.
Zhan Tao 《数学学报(英文版)》1989,5(1):37-47
The well-known Bombieri-A. I. Vinogradov theorem states that (1) $$\sum\limits_{q \leqslant x^{\tfrac{1}{2}} (\log x)^{ - s} } {\mathop {\max }\limits_{(a,q) = 1} \mathop {\max }\limits_{y \leqslant x} } \left| {\psi (y,q;a) - \frac{y}{{\varphi (q)}}} \right| \ll \frac{x}{{(\log x)^A }},$$ whereA is an arbitrary positive constant,B=B(A)>0, and as usual, $$\psi (x,q;a) = \sum\limits_{\mathop {n \leqslant x}\limits_{n = a(q)} } {\Lambda (n),}$$ Λ being the Von Mangoldt's function. The problem of finding a result analogous to (1) for short intervals was investigated by many authors. Using Heath-Brown's identity and the approximate functional equation for DirichletL-functions, A. Perelli, J. Pintz and S. Salerno in 1985 established the following extension of Bombieri's theorem: Theorem 1. (2) $$\sum\limits_{q \leqslant Q} {\mathop {\max }\limits_{(a,q) = 1} \mathop {\max }\limits_{h \leqslant y} \mathop {\max }\limits_{\frac{x}{2}< \approx \leqslant x} } \left| {\psi (z + h,q;a) - \psi (z,q;a) - \frac{h}{{\varphi (q)}}} \right| \ll \frac{y}{{(\log x)^A }}$$ where A>0 is an arbitrary constant,y=x θ $$\frac{7}{{12}}< \theta \leqslant 1, Q = x^{\frac{1}{{40}}} .$$ ,Q=x 1/40. By improving the basic lemma which A. Perelli, J. Pintz and S. Salerno used as the main tool to prove Theorem 1, we obtain Theorem 2.Under the same condition as in Theorem 1,for Q=x 1/38.5, (2)still holds. 相似文献
4.
5.
M. B. Khripunova 《Mathematical Notes》1996,64(3):394-400
It is proved that if ?(n) is a multiplicative function taking a valueζ on the set of primes such thatζ 3 = 1,ζ ≠ 1 and? 3(p r)=1 forr≥2, then there exists aθ ∈ (0, 1), for which $$|\sum\limits_{p \leqslant x} f (p + 1)| \leqslant \theta \pi (x)$$ , where $$\pi (x) = \sum\limits_{p \leqslant x} 1$$ . 相似文献
6.
LISONG 《高校应用数学学报(英文版)》1996,11(3):361-368
For Szasz-Durrmeyer operators Ln (f,z), 1< p≤∞, we prove that, forsome m, w^2φ(f,1/√n)p ≤(≤M(││Ln,f,x) - ,f││p ││Lmn(f, x) -f││p),where φ(x)^2 =x, M >0,w^2φ(f,t)p is Ditzian-Totik modulus of smoothness. 相似文献
7.
On the Discontinuous Infinite-Dimensional Generalized Quasivariational Inequality Problem 总被引:1,自引:0,他引:1
In this paper, we deal with the following generalized quasivariational inequality problem: given a real normed space E with topological dual E* and two multifunctions G: X→2 X and F: X→2 E*, find $\left( {\hat x,\hat \phi } \right)$ ∈X × E* such that $\hat x \in G\left( {\hat x} \right),{\text{ }}\hat \phi \in F\left( {\hat x} \right),{\text{ }}\left\langle {\hat \phi ,\hat x - y} \right\rangle \leqslant 0,{\text{for all }}y \in G\left( {\hat x} \right).$ We extend to such infinite-dimensional setting some existence results which have been obtained recently for the special case where E is finite dimensional. In particular, our assumptions do not imply any kind of continuity for the multifunction F. 相似文献
8.
V. E. Katznel'son 《Journal of Mathematical Sciences》1981,16(3):1095-1101
Letf be an entire function (in Cn) of exponential type for whichf(x)=0(?(x)) on the real subspace \(\mathbb{R}^w (\phi \geqslant 1,{\mathbf{ }}\mathop {\lim }\limits_{\left| x \right| \to \infty } \phi (x) = \infty )\) and ?δ>0?Cδ>0 $$\left| {f(z)} \right| \leqslant C_\delta \exp \left\{ {h_s (y) + S\left| z \right|} \right\},z = x + iy$$ where h, (x)=sup〈3, x〉, S being a convex set in ?n. Then for any ?, ?>0, the functionf can be approximated with any degree of accuracy in the form p→ \(\mathop {\sup }\limits_{x \in \mathbb{R}^w } \frac{{\left| {P(x)} \right|}}{{\varphi (x)}}\) by linear combinations of functions x→expi〈λx〉 with frequenciesX belonging to an ?-neighborhood of the set S. 相似文献
9.
Dr. Evelyn Brinitzer 《Monatshefte für Mathematik》1977,84(1):13-19
Let γ(n) be the largest squarefree divisor of a natural numbern. In this note we give an asymptotic formula for \(\sum\limits_{n \leqslant x} {\gamma (n)/n^k } \) wherek is an arbitrary integer. 相似文献
10.
In this paper, the authors give the boundedness of the commutator [b, ????,?? ] from the homogeneous Sobolev space $\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)$ to the Lebesgue space L p (? n ) for 1 < p < ??, where ????,?? denotes the Marcinkiewicz integral with rough hypersingular kernel defined by $\mu _{\Omega ,\gamma } f\left( x \right) = \left( {\int_0^\infty {\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega \left( {x - y} \right)}} {{\left| {x - y} \right|^{n - 1} }}f\left( y \right)dy} } \right|^2 \frac{{dt}} {{t^{3 + 2\gamma } }}} } \right)^{\frac{1} {2}} ,$ , with ?? ?? L 1(S n?1) for $0 < \gamma < min\left\{ {\frac{n} {2},\frac{n} {p}} \right\}$ or ?? ?? L(log+ L) ?? (S n?1) for $\left| {1 - \frac{2} {p}} \right| < \beta < 1\left( {0 < \gamma < \frac{n} {2}} \right)$ , respectively. 相似文献
11.
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
12.
Roman Ger 《Results in Mathematics》1994,26(3-4):281-289
Let X, Y be two linear spaces over the field ? of rationals and let D ≠ ? be a (?—convex subset of X. We show that every function ?: D → Y satisfying the functional equation $${\mathop\sum^{n+1}\limits_{j=0}}(-1)^{n+1-j}\Bigg(^{n+1}_{j}\Bigg)f\Bigg((1-{j\over {n+1}})x+{j\over{n+1}}y\Bigg)=0,\ \ \ x,y\in\ D,$$ admits an extension to a function F: X → Y of the form $$F(x)=A^o+A^1(x)+\cdot\cdot\cdot+A^n(x),\ \ \ x\in\ X,$$ where A o ∈ Y, Ak(x) ? Ak(x,…,x), x ∈ X, and the maps A k: X k → Y are k—additive and symmetric, k ∈ {1,…, n}. Uniqueness of the extension is also discussed. 相似文献
13.
Wolfgang Müller 《Monatshefte für Mathematik》1989,108(4):301-323
LetK be a quadratic number field with discriminantD and denote byF(n) the number of integral ideals with norm equal ton. Forr≥1 the following formula is proved $$\sum\limits_{n \leqslant x} {F(n)F(n + r) = M_K (r)x + E_K (x,r).} $$ HereM k (r) is an explicitly determined function ofr which depends onK, and for every ε>0 the error term is bounded by \(|E_K (x,r)|<< |D|^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2} + \varepsilon } x^{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6} + \varepsilon } \) uniformly for \(r<< |D|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} x^{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6}} \) Moreover,E k (x,r) is small on average, i.e \(\int_X^{2X} {|E_K (x,r)|^2 dx}<< |D|^{4 + \varepsilon } X^{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2} + \varepsilon } \) uniformly for \(r<< |D|X^{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-0em} 4}} \) . 相似文献
14.
M. A. Chahkiev 《Analysis Mathematica》2008,34(3):177-185
Let $ \mathcal{P}_n $ denote the set of algebraic polynomials of degree n with the real coefficients. Stein and Wpainger [1] proved that $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \leqslant C_n , $$ where C n depends only on n. Later A. Carbery, S. Wainger and J. Wright (according to a communication obtained from I. R. Parissis), and Parissis [3] obtained the following sharp order estimate $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \sim \ln n. $$ . Now let $ \mathcal{T}_n $ denote the set of trigonometric polynomials $$ t(x) = \frac{{a_0 }} {2} + \sum\limits_{k = 1}^n {(a_k coskx + b_k sinkx)} $$ with real coefficients a k , b k . The main result of the paper is that $$ \mathop {\sup }\limits_{t( \cdot ) \in \mathcal{T}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{it(x)} }} {x}dx} } \right| \leqslant C_n , $$ with an effective bound on C n . Besides, an analog of a lemma, due to I. M. Vinogradov, is established, concerning the estimate of the measure of the set, where a polynomial is small, via the coefficients of the polynomial. 相似文献
15.
Define , $S_{k,n} = \Sigma _{1 \leqslant i_1< \cdot \cdot \cdot< l_k \leqslant n} X_{i_1 } \cdot \cdot \cdot X_{i_k } ,n \geqslant k \geqslant {\text{1}}$ where {X, X n ,n≥1} are i.i.d. random variables withEX=0,EX 2=1 and letH k (·) denote the Hermite polynomial of degreek. By establishing an LIL for products of correlated sums of i.i.d. random variables, the a.s. decomposition $$\begin{gathered} k!S_{k,n} = n^{k/2} H_k (S_{1n} /n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ) - \left( {\begin{array}{*{20}c} k \\ 2 \\ \end{array} } \right)S_{1.n}^{k - 2} \sum\limits_{i = 1}^n {(X_i^2 - 1)} \hfill \\ + O(n^{(k - 1)/2} (\log \log n)^{(k - 3/2} ) \hfill \\ \end{gathered} $$ valid whenEX 4<∞, elicits an LIL forη k,n =k!S k,n ?n k/2 H k (S 1n /n 1/2) under a reduced normalization. Moreover, whenE|X| p <∞ for somep in [2, 4], a Marcinkiewicz-Zygmund type strong law forη k,n is obtained, likewise under a reduced normalization. 相似文献
16.
M. A. Sychev 《Doklady Mathematics》2013,87(3):334-337
Let L: Ω × R m × R m × n → R be a Caratheodory integrand with $c_1 |\nu |^{p(x)} + c_2 \leqslant L(x,u,\nu ) \leqslant c_3 |\nu |^{p(x)} + c_4 ,c_3 \geqslant c_1 > 0,n + \varepsilon \leqslant p( \cdot ) \leqslant p < \infty ,\varepsilon > 0.$ Under these assumptions the weak convergence theory holds for the integral functional $J(u): = \int\limits_\Omega {L(x,u(x),Du(x))dx} $ without further requirements. Weak convergence theory includes lower seraicontinuity with respect to the weak convergence of Sobolev functions, the convergence in energy property (weak convergence of Sobolev functions and convergence in energy imply the strong convergence of the functions), the integral representation for the relaxed energy and related questions. The results of the weak convergence theory follows from a characterization of gradient Young measures associated with these functionals. 相似文献
17.
Prof. Dr. Richard Warlimont 《Monatshefte für Mathematik》1979,86(4):333-336
Denote byf(k) the least natural numberm for which 1+mk is squarefree.Erdös gave the estimatef(k)?k 3/2(logk)?1. We derive (by elementary means) the asymptotic relation $$\sum\limits_{k \leqslant x} {(f(k))^\alpha = c(\alpha )x + 0(x} ) (x \to \infty ,0 \leqslant \alpha \leqslant 1)$$ 相似文献
18.
Interpolation with lagrange polynomials a simple proof of Markov inequality and some of its generalizations 总被引:1,自引:0,他引:1
The following theorem is provedTheorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying inthe interval[-1,1] and△'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}.If polynomial pP_n satisfies the inequalitythen for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality丨p~(k)(x)丨≤max{丨q~((k))(x)丨,丨1/k(x~2-1)q~(k+1)(x)+xq~((k))(x)丨}.This estimate leads to the Markov inequality for the higher order derivatives ofpolynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero.Some other results are established which gives evidence to the conjecture that under theconditions of Theorem 1 the inequality ‖p~((k))‖≤‖q~(k)‖holds. 相似文献
19.
I. A. Khadzhi 《Mathematical Notes》2012,91(5-6):857-867
For the equation of mixed elliptic-hyperbolic type $u_{xx} + (\operatorname{sgn} y)u_{yy} - b^2 u = f(x)$ in a rectangular domainD = {(x, y) | 0 < x < 1, ?α < y < β}, where α, β, and b are given positive numbers, we study the problem with boundary conditions $\begin{gathered} u(0,y) = u(1,y) = 0, - \alpha \leqslant y \leqslant \beta , \hfill \\ u(x,\beta ) = \phi (x),u(x,\alpha ) = \psi (x),u_y (x, - \alpha ) = g(x),0 \leqslant x \leqslant 1. \hfill \\ \end{gathered} $ . We establish a criterion for the uniqueness of the solution, which is constructed as the sum of the series in eigenfunctions of the corresponding eigenvalue problem and prove the stability of the solution. 相似文献
20.
Z. Ditzian 《Analysis Mathematica》1977,3(1):45-49
В РАБОтЕ ДОкАжАНО, ЧтО limk a *f(x)=f(x) пОЧтИ ВсУДУ, гДЕk a(t)=a?n k(a?1t), t?Rn, Для Дль ДОВОльНО шИРОкОг О клАссА ФУНкцИИk(t). ДАНы УслОВИь, пРИ кОтО Рых пОлУЧЕННыИ РЕжУл ьтАт РАспРОстРАНьЕтсь НА ФУНкцИУ $$k(x,y) = \gamma \frac{1}{{1 + |x|^\alpha }} \cdot \frac{1}{{1 + |y|^\beta }},$$ гДЕ α, β>1, А γ — НОРМИРУУЩ ИИ МНОжИтЕль тАкОИ, Чт О∫∫k(x, y) dx dy=1. 相似文献