On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ _t μ = L ^* μ for bounded Borel measures on ℝ ^d  × (0, T), where L is a second order elliptic operator, for example, Lu = D_xu + ( b,?_xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity

On the Lyapunov Exponent of a Multidimensional Stochastic Flow

Let X _t be a reversible and positive recurrent diffusion in ℝ^d described by

Properties of Riesz fractional derivatives for Korteweg–de Vries solitons

Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (?Δ) ^α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (−Δ) ^α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u ₀(X) with X = x − 4t, these derivatives, u _α (X) = D ^α u ₀(X), and their Hilbert transforms, v _α (X) = −HD ^α u ₀(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ₊(s, a) and ζ₋(s, a), respectively. New properties are established for u _α (X) and v _α (X). It is proved that the functions w _α (X) = u _α (X) + iv _α (X) with α > −1 are solutions of the differential equation

-\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      4. Uniform dimension results for Gaussian random fields      DongSheng Wu  YiMin Xiao 《中国科学A辑(英文版)》2009,52(7):1478-1496 Let X = {X(t), t ∈ ℝ N } be a Gaussian random field with values in ℝ d defined by ((1)) . The properties of space and time anisotropy of X and their connections to uniform Hausdorff dimension results are discussed. It is shown that in general the uniform Hausdorff dimension result does not hold for the image sets of a space-anisotropic Gaussian random field X. When X is an (N, d)-Gaussian random field as in (1), where X 1,...,X d are independent copies of a real valued, centered Gaussian random field X 0 which is anisotropic in the time variable. We establish uniform Hausdorff dimension results for the image sets of X. These results extend the corresponding results on one-dimensional Brownian motion, fractional Brownian motion and the Brownian sheet.       5. On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data      Ryosuke Hyakuna  Masayoshi Tsutsumi 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(3):309-327 We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      6. Viability for nonlinear multi-valued reaction–diffusion systems      Daniela Roşu 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(4):479-496 In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system { lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      7. Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion      M. N. Mishra  B. L. S. Prakasa Rao 《Statistical Inference for Stochastic Processes》2011,14(2):101-109 Consider a stochastic process {X t , 0 ≤ t ≤ T} governed by a stochastic differential equation given by dXt = S(Xt)   dt + e  dWtH,    X0=x0,    0 ￡ t ￡ T dX_t= S(X_t) \;dt + \epsilon \; dW_t^H,\quad X_0=x_0,\quad 0 \leq t \leq T      8. Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps      Feng Dai  Heping Wang 《Constructive Approximation》2010,31(1):1-36 Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbSd\mathbb{S}^{d} of ℝ d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbSdx,y\in\mathbb{S}^{d} . Given a spherical cap B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      9. Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors      Damir Z. Arov  Mikael Kurula  Olof J. Staffans 《Complex Analysis and Operator Theory》2011,5(2):331-402 This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L2([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L2([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C1([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and ([(x)\dot](t),x(t),w(t)) ? V,    t ? [0,￥), (\dot x(t),x(t),w(t))\in V,\quad t \in [0,\infty),      10. The Size of Exponential Sums on Intervals of the Real Line      Tamás Erdélyi  Kaveh Khodjasteh  Lorenza Viola 《Constructive Approximation》2012,35(1):123-136 We prove that there is a constant c>0 depending only on M≥1 and μ≥0 such that òyy+a |g(t)|  dt 3 exp(-c/(ad)),     ad ? (0,p],\int_y^{y+a}{ \bigl|g(t)\bigr| \, dt} \geq \exp \bigl(-c/(a\delta)\bigr), \quad a\delta \in (0,\pi],      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Uniform dimension results for Gaussian random fields

Let X = {X(t), t ∈ ℝ ^N } be a Gaussian random field with values in ℝ ^d defined by

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      6. Viability for nonlinear multi-valued reaction–diffusion systems      Daniela Roşu 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(4):479-496 In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system { lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      7. Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion      M. N. Mishra  B. L. S. Prakasa Rao 《Statistical Inference for Stochastic Processes》2011,14(2):101-109 Consider a stochastic process {X t , 0 ≤ t ≤ T} governed by a stochastic differential equation given by dXt = S(Xt)   dt + e  dWtH,    X0=x0,    0 ￡ t ￡ T dX_t= S(X_t) \;dt + \epsilon \; dW_t^H,\quad X_0=x_0,\quad 0 \leq t \leq T      8. Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps      Feng Dai  Heping Wang 《Constructive Approximation》2010,31(1):1-36 Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbSd\mathbb{S}^{d} of ℝ d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbSdx,y\in\mathbb{S}^{d} . Given a spherical cap B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      9. Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors      Damir Z. Arov  Mikael Kurula  Olof J. Staffans 《Complex Analysis and Operator Theory》2011,5(2):331-402 This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L2([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L2([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C1([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and ([(x)\dot](t),x(t),w(t)) ? V,    t ? [0,￥), (\dot x(t),x(t),w(t))\in V,\quad t \in [0,\infty),      10. The Size of Exponential Sums on Intervals of the Real Line      Tamás Erdélyi  Kaveh Khodjasteh  Lorenza Viola 《Constructive Approximation》2012,35(1):123-136 We prove that there is a constant c>0 depending only on M≥1 and μ≥0 such that òyy+a |g(t)|  dt 3 exp(-c/(ad)),     ad ? (0,p],\int_y^{y+a}{ \bigl|g(t)\bigr| \, dt} \geq \exp \bigl(-c/(a\delta)\bigr), \quad a\delta \in (0,\pi],      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Viability for nonlinear multi-valued reaction–diffusion systems

In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system

{ lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      7. Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion      M. N. Mishra  B. L. S. Prakasa Rao 《Statistical Inference for Stochastic Processes》2011,14(2):101-109 Consider a stochastic process {X t , 0 ≤ t ≤ T} governed by a stochastic differential equation given by dXt = S(Xt)   dt + e  dWtH,    X0=x0,    0 ￡ t ￡ T dX_t= S(X_t) \;dt + \epsilon \; dW_t^H,\quad X_0=x_0,\quad 0 \leq t \leq T      8. Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps      Feng Dai  Heping Wang 《Constructive Approximation》2010,31(1):1-36 Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbSd\mathbb{S}^{d} of ℝ d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbSdx,y\in\mathbb{S}^{d} . Given a spherical cap B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      9. Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors      Damir Z. Arov  Mikael Kurula  Olof J. Staffans 《Complex Analysis and Operator Theory》2011,5(2):331-402 This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L2([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L2([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C1([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and ([(x)\dot](t),x(t),w(t)) ? V,    t ? [0,￥), (\dot x(t),x(t),w(t))\in V,\quad t \in [0,\infty),      10. The Size of Exponential Sums on Intervals of the Real Line      Tamás Erdélyi  Kaveh Khodjasteh  Lorenza Viola 《Constructive Approximation》2012,35(1):123-136 We prove that there is a constant c>0 depending only on M≥1 and μ≥0 such that òyy+a |g(t)|  dt 3 exp(-c/(ad)),     ad ? (0,p],\int_y^{y+a}{ \bigl|g(t)\bigr| \, dt} \geq \exp \bigl(-c/(a\delta)\bigr), \quad a\delta \in (0,\pi],      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion

Consider a stochastic process {X _t , 0 ≤ t ≤ T} governed by a stochastic differential equation given by

Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps

Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbS^d\mathbb{S}^{d} of ℝ ^d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbS^dx,y\in\mathbb{S}^{d} . Given a spherical cap

B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      9. Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors      Damir Z. Arov  Mikael Kurula  Olof J. Staffans 《Complex Analysis and Operator Theory》2011,5(2):331-402 This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L2([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L2([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C1([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and ([(x)\dot](t),x(t),w(t)) ? V,    t ? [0,￥), (\dot x(t),x(t),w(t))\in V,\quad t \in [0,\infty),      10. The Size of Exponential Sums on Intervals of the Real Line      Tamás Erdélyi  Kaveh Khodjasteh  Lorenza Viola 《Constructive Approximation》2012,35(1):123-136 We prove that there is a constant c>0 depending only on M≥1 and μ≥0 such that òyy+a |g(t)|  dt 3 exp(-c/(ad)),     ad ? (0,p],\int_y^{y+a}{ \bigl|g(t)\bigr| \, dt} \geq \exp \bigl(-c/(a\delta)\bigr), \quad a\delta \in (0,\pi],      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors

This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L²([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L²([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C¹([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and

The Size of Exponential Sums on Intervals of the Real Line

We prove that there is a constant c>0 depending only on M≥1 and μ≥0 such that