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1.
Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t], and define a random vector Z n: C[0, t] → R n+1 by \({Z_n}\left( x \right) = \left( {x\left( 0 \right) + a\left( 0 \right),\int_o^{{t_1}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_1}} \right),...,\int_0^{{t_n}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_n}} \right)} } } \right)\), where aC[0, t], hL 2[0, t], and 0 < t 1 <... < t nt is a partition of [0, t]. Using simple formulas for generalized conditional Wiener integrals, given Z n we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions F in a Banach algebra which corresponds to Cameron-Storvick’s Banach algebra S. Finally, we express the generalized analytic conditional Feynman integral of F as a limit of the non-conditional generalized Wiener integral of a polygonal function using a change of scale transformation for which a normal density is the kernel. This result extends the existing change of scale formulas on the classical Wiener space, abstract Wiener space and the analogue of the Wiener space C[0, t].  相似文献   

2.
Let {F t : t ≥ 0} be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F 0(x) ? F t (x) exist for xK and t > 0, then D t F t (x) = (?1)F t ((?1)G(x)) for xK and t ≥ 0, where D t F t (x) denotes the derivative of F t (x) with respect to t and $G(x) = \mathop {\lim }\limits_{s \to 0} {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} \mathord{\left/ {\vphantom {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} {\left( { - s} \right)}}} \right. \kern-0em} {\left( { - s} \right)}}$ for xK.  相似文献   

3.
Let (X, Λ) be a pair of random variables, where Λ is an Ω (a compact subset of the real line) valued random variable with the density functiong(Θ: α) andX is a real-valued random variable whose conditional probability function given Λ=Θ is P {X=x|Θ} withx=x 0, x1, …. Based onn independent observations ofX, x (n), we are to estimate the true (unknown) parameter vectorα=(α 1, α2, ...,αm) of the probability function ofX, Pα(X=∫ΩP{X=x|Θ}g(Θ:α)dΘ. A least squares estimator of α is any vector \(\hat \alpha \left( {X^{\left( n \right)} } \right)\) which minimizes $$n^{ - 1} \sum\limits_{i = 1}^n {\left( {P_\alpha \left( {x_i } \right) - fn\left( {x_i } \right)} \right)^2 } $$ wherex (n)=(x1, x2,…,x n) is a random sample ofX andf n(xi)=[number ofx i inx (n)]/n. It is shown that the least squares estimators exist as a unique solution of the normal equations for all sufficiently large sample size (n) and the Gauss-Newton iteration method of obtaining the estimator is numerically stable. The least squares estimators converge to the true values at the rate of \(O\left( {\sqrt {2\log \left( {{{\log n} \mathord{\left/ {\vphantom {{\log n} n}} \right. \kern-0em} n}} \right)} } \right)\) with probability one, and has the asymptotically normal distribution.  相似文献   

4.
Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: Forf∈C r[?1,1],mN, and anyn≥max{m+r?1, 2r+1}, an algebraic polynomialP n of degree ≤n exists that satisfies $$\left| {f^{\left( k \right)} \left( x \right) - P_n^{\left( k \right)} \left( {f,x} \right)} \right| \leqslant C\left( {r,m} \right)\Gamma _{nrmk} \left( x \right)^{r - k} \omega ^m \left( {f^{\left( r \right)} ,\Gamma _{nrmk} \left( x \right)} \right),$$ for 0≤k≤r andx ∈ [?1,1], where ωυ(f(k),δ) denotes the usual vth modulus of smoothness off (k), and Moreover, for no 0≤k≤r can (1?x 2)( r?k+1)/(r?k+m)(1/n2)(m?1)/(r?k+m) be replaced by (1-x2)αkn2αk-2, with αk>(r-k+a)/(r-k+m).  相似文献   

5.
Let fC[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ n + 2(f, x). In this paper we study the estimate of Δ n + 2(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if fC r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$   相似文献   

6.
The nonparametric regression problem has the objective of estimating conditional expectation. Consider the model $$Y = R(X) + Z$$ , where the random variableZ has mean zero and is independent ofX. The regression functionR(x) is the conditional expectation ofY givenX = x. For an estimator of the form $$R_n (x) = \sum\limits_{i = 1}^n {Y_i K{{\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} \mathord{\left/ {\vphantom {{\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} {\sum\limits_{i = 1}^n {K\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{i = 1}^n {K\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} }}} $$ , we obtain the rate of strong uniform convergence $$\mathop {\sup }\limits_{x\varepsilon C} \left| {R_n (x) - R(x)} \right|\mathop {w \cdot p \cdot 1}\limits_ = o({{n^{{1 \mathord{\left/ {\vphantom {1 {(2 + d)}}} \right. \kern-\nulldelimiterspace} {(2 + d)}}} } \mathord{\left/ {\vphantom {{n^{{1 \mathord{\left/ {\vphantom {1 {(2 + d)}}} \right. \kern-\nulldelimiterspace} {(2 + d)}}} } {\beta _n \log n}}} \right. \kern-\nulldelimiterspace} {\beta _n \log n}}),\beta _n \to \infty $$ . HereX is ad-dimensional variable andC is a suitable subset ofR d .  相似文献   

7.
Estimates are given for the measure of a section of an arbitrary straight line of the set $$E_\delta = \left\{ {z:\left| {P' {{\left( z \right)} \mathord{\left/ {\vphantom {{\left( z \right)} {\left( {nP \left( z \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {nP \left( z \right)} \right)}} \leqslant \delta } \right|} \right\} \left( {\delta > 0} \right)$$ where P (z) is a polynomial of degree n. THEOREM. Suppose P (x) = (x ? x1) ... (x ? xn) is a polynomial with real zeros. Then, for any δ > 0, on any intervala ?x ?b, containing all of the xk (k=1, 2, ..., n), outside an exceptional set Eδ?[a,b] such that $$mes E_\delta \leqslant \left( {\sqrt {1 + \delta ^2 \left( {b - a} \right)^2 } - 1} \right)/\delta $$ , we have the inequality $$\left| {P' {{\left( x \right)} \mathord{\left/ {\vphantom {{\left( x \right)} {\left( {nP \left( x \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {nP \left( x \right)} \right)}}} \right| > \delta $$ . A similar estimate is given for polynomials whose roots lie either in Imz ? 0 or in Imz ? 0.  相似文献   

8.
Let Ω be an arbitrary open set in R n , and let σ(x) and g i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W p r (Ω, σ, $ \vec g $ ), where rN, p ≥ 1, and $ \vec g $ (x) = (g 1(x), g 2(x), ..., g n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W p r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation.  相似文献   

9.
Suppose thatx(t) ∈ C [a,b (n)] and has n zeros at the pointsa and b. It is shown that if x(n)(t) preserves sign on [a, b], then $$\left| {x\left( t \right)} \right| \geqslant \frac{{p_0 }}{{n - 1}}\mathop {\left[ {\mathop {\sup }\limits_{\tau \in \left( {a, b} \right)} \frac{{\left| {x\left( \tau \right)} \right|}}{{\left( {\tau - a} \right)^{p - 1} \left( {b - \tau } \right)^{q - 1} }}} \right]}\limits_{\left( {a< t< b} \right),} \left( {t - a} \right)^p \left( {b - t} \right)^q $$ where p and q are the multiplicities of the zeros of x(t) ata and b, respectively, and po=min{p,q}. Two-sided estimates of the Green's function for a two-point interpolation problem for the operator Lx ≡ x(n) are established in the proof. As an application, new conditions for the solvability of de la Vallée Poussin's two-point boundary problems are obtained.  相似文献   

10.
In this paper we consider the following Dirichlet problem for elliptic systems: $$\begin{array}{*{20}c} {\overline {DA\left( {x,u\left( x \right),Du\left( x \right)} \right)} = B\left( {x,u\left( x \right),Du\left( x \right)} \right), x \in \Omega ,} \\ {u\left( x \right) = 0, x\partial \Omega } \\ \end{array}$$ where D is a Dirac operator in Euclidean space, u(x) is defined in a bounded Lipschitz domain Ω in ? n and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the above-mentioned systems in the space W 0 1,p(x) (Ω,C? n ) under appropriate assumptions.  相似文献   

11.
A finite subsetX of thed-dimensional unit sphereS d-1 is called a sphericalt-design, if and only if $$\frac{1}{{\left| {S^{d - 1} } \right|}}\int_{S^{d - 1} } {f(x)d\omega (x)} = \frac{1}{{\left| x \right|}}\sum\limits_{x \in X} {f(x)} $$ holds for all polynomialsf(x) =f(x 1,x 2,...,x d ) of degree at mostt. In 1984 Seymour and Zaslavsky proved the existence of sphericalt-designs for anyt andd, but for sufficiently large |X|. Since spherical designs can be used for numerical integration, it is of interest to give explicit constructions. Mimura gave a construction fort = 2,d ∈ ? and |X| ≥n 2 for somen 2 ∈ ? (n 2 is sharp). Here we will give an explicit construction fort = 4 and 5,d ∈ ? and |X| ≥n 4 for somen 4 ∈ ?.  相似文献   

12.
This work is concerned with the inverse problem of determining initial value of the Cauchy problem for a nonlinear diffusion process with an additional condition on free boundary. Considering the flow of water through a homogeneous isotropic rigid porous medium, we have such desire: for every given positive constants K and T 0, to decide the initial value u 0 such that the solution u(x, t) satisfies $\mathop {\sup }\limits_{x \in H_u (T_0 )} |x| \geqslant K$ , where H u(T 0) = {x, ?N: u(x, T 0) > 0}. In this paper, we first establish a priori estimate u t ? C(t)u and a more precise Poincaré type inequality $\left\| \phi \right\|_{L^2 (B_\varrho )}^2 \leqslant \varrho \left\| {\nabla \phi } \right\|_{L^2 (B_\varrho )}^2 $ , and then, we give a positive constant C 0 and assert the main results are true if only $\left\| {u_0 } \right\|_{L^2 (\mathbb{R}^N )} \geqslant C_0 $ .  相似文献   

13.
14.
LetQ(x) denote a quadratic form over the rational integers in four variables (x=(x1,...,x4)). ThenQ is representable as a symmetric matrix. Assume this matrix to be non-singular modp(p≠2 prime); then the “inverse” quadratic formQ ?1 modp can be defined. Letf:?4→? be defined such that the Fourier transformf exists and the sum $$\sum\limits_{x \in \mathbb{Z}^4 } {f(c x), c \in \mathbb{R}, c \ne 0} $$ is convergent. Furthermore, letm=p 1...p k be the product ofk distinct primes withm>1, 2×m; let $$\varepsilon = \prod\limits_{i = 1}^k {\left( {\frac{{\det Q}}{{p_i }}} \right)} \ne 0$$ for the Legendre symbol $$\left( {\frac{ \cdot }{p}} \right)$$ ; define $$B_i (Q,x) = \left\{ {\begin{array}{*{20}c} {1 for Q(x) \equiv 0\bmod p_i } \\ , \\ {0 for Q(x)\not \equiv 0\bmod p_i } \\ \end{array} } \right.$$ and forr∈?,r>0, $$F(Q,f,r) = \sum\limits_{x \in \mathbb{Z}^4 } {\left( {\prod\limits_{i = 1}^k {\left( {B_i (Q,x) - \frac{1}{{p_i }}} \right)} } \right)f(r^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} x)} $$ Then we have $$F(Q,f,m) = \varepsilon F(Q^{ - 1} ,\hat f,m)$$   相似文献   

15.
We consider the problem of the rate of approximation of continuous 2π-periodic functions of class WrH[ω]C by trigonometric polynomials of order n on sets of total measure. We prove that when r≥0,ω(δ)δ ?1 → ∞ (δ → 0) there exists a function f ε WrH[ω]C such thatf ε WrH[ω]C and for any sequence {tn n=1 we have almost everywhere on [0, 2π] $\begin{array}{l} \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0, \\ \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {\tilde f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0. \\ \end{array}$   相似文献   

16.
In this paper, we obtain bounds for the decay rate in the L r (? d )-norm for the solutions of a nonlocal and nonlinear evolution equation, namely, $$u_t \left( {x,t} \right) = \int_{\mathbb{R}^d } {K\left( {x,y} \right)\left| {u\left( {y,t} \right) - u\left( {x,t} \right)} \right|^{p - 2} \left( {u\left( {y,t} \right) - u\left( {x,t} \right)} \right)dy, x \in \mathbb{R}^d , t > 0.}$$ . We consider a kernel of the form K(x, y) = ψ(y?a(x)) + ψ(x?a(y)), where ψ is a bounded, nonnegative function supported in the unit ball and a is a linear function a(x) = Ax. To obtain the decay rates, we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $$T\left( u \right) = - \int_{\mathbb{R}^d } {K\left( {x,y} \right)\left| {u\left( y \right) - u\left( x \right)} \right|^{p - 2} \left( {u\left( y \right) - u\left( x \right)} \right)dy, 1 \leqslant p < \infty .}$$ . The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole space ? d : $$\lambda _{1,p} \left( {\mathbb{R}^d } \right) = 2\left( {\int_{\mathbb{R}^d } {\psi \left( z \right)dz} } \right)\left| {\frac{1} {{\left| {\det A} \right|^{1/p} }} - 1} \right|^p .$$ Moreover, we deal with the p = ∞ eigenvalue problem, studying the limit of λ 1,p 1/p as p→∞.  相似文献   

17.
Найдены методы восст ановления интеграла по информации $$I\left( f \right) = \left\{ {f^{(j)} \left( {x_i } \right)\left( {j = 0, ..., \gamma _i - 1; i = 1, ..., n; 1 \leqq \gamma _i \leqq r; \gamma _i + ... + \gamma _n \leqq N} \right.} \right\},$$ оптимальные на класс ахW p r ,r=1,2,...; 1≦p≦∞. Это позволило, в частност и, получить наилучшие для классаW p r квадратурные форму лы вида $$\mathop \smallint \limits_0^1 f\left( x \right)dx = \mathop \Sigma \limits_{i = 1}^n \mathop \Sigma \limits_{j = 1}^{\gamma _i - 1} a_{ij} f^{(j)} \left( {x_i } \right) + \mathop \Sigma \limits_{j = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} b_j f^{(2j - 1)} \left( 0 \right) + \mathop \Sigma \limits_{k = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} c_k f^{(2k - 1)} \left( 1 \right) + R\left( f \right)$$ И $$\mathop \smallint \limits_0^1 f\left( x \right)dx = af\left( 0 \right) + \mathop \Sigma \limits_{i = 1}^n \mathop \Sigma \limits_{j = 0}^{\gamma _i - 1} a_{ij} f^{(j)} \left( {x_i } \right) + bf\left( 1 \right) + \mathop \Sigma \limits_{j = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} b_j f^{(2j - 1)} \left( 0 \right) + \mathop \Sigma \limits_{k = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} c_k f^{(2k - 1)} \left( 1 \right) + R\left( f \right).$$   相似文献   

18.
In this paper, we examine, in a systematic fashion, some ill-posed problems arising in the theory of heat conduction. In abstract terms, letH be a Hilbert space andA: D (A)?H→H be an unbounded normal operator, we consider the boundary value problemü(t)=Au(t), 0<t<∞,u(0)=u 0D(A), \(\mathop {\lim }\limits_{t \to 0} \left\| {u\left( t \right)} \right\| = 0\) . The problem of recoveringu 0 whenu(T) is known for someT>0 is not well-posed. Suppose we are given approximationsx 1,x 2,…,x N tou(T 1),…,u(T N) with 0<T, <…<T N and positive weightsP i,i=1,…,n, \(\sum\limits_{i = 1}^N {P_i = 1} \) such that \(Q_2 \left( {u_0 } \right) = \sum\limits_{i = 1}^N {P_i } \left\| {u\left( {T_i } \right) - x_i } \right\|^2 \leqslant \varepsilon ^2 \) . If ‖u t(0)‖≤E for some a priori constantE, we construct a regularized solution ν(t) such that \(Q\left( {\nu \left( 0 \right)} \right) \leqslant \varepsilon ^2 \) while \(\left\| {u\left( 0 \right) - \nu \left( 0 \right)} \right\| = 0\left( {ln \left( {E/\varepsilon } \right)} \right)^{ - 1} \) and \(\left\| {u\left( t \right) - \nu \left( t \right)} \right\| = 0\left( {\varepsilon ^{\beta \left( t \right)} } \right)\) where 0<β(t)<1 and the constant in the order symbol depends uponE. The function β(t) is larger thant/m whent k andk is the largest integer such that \((\sum\limits_{k = 1}^N {P_i (T_i )} )< (\sum\limits_{k = 1}^N {P_i (T_i )} = m\) , which β(t)=t/m on [T k, m] and β(t)=1 on [m, ∞). Similar results are obtained if the measurement is made in the maximum norm, i.e.,Q (u 0)=max{‖u(T i)?x i‖, 1≤iN}.  相似文献   

19.
We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as
{ x  ?  X|t - q\textRea y1 ( tA )xt - q\textRea y2 ( tA )x ? L*p ( ( 0,¥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}}  相似文献   

20.
The instability property of the standing wave uω(t, x) = eiωtφ(x) for the Klein–Gordon– Hartree equation  相似文献   

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