￠—半压缩算子和￠—强增殖算子方程的迭代

设E是任意实Banach空间,K是E的非空闭凸子集,T：K→K是一致连续￠-半压缩映像且值域有界。设{an},{bn},{cn},{a'n},{b'n}和{c'n}是[0,1]中的序列且满足条件：Ⅰ）an bn cn=a'n b'n c'n=1,任意n≥0;Ⅱ）limbn=limb'n=limc'n=0;Ⅲ）∑n=0^∞bn=∞;Ⅳ）cn=o(bn).对任意给定的x0,u0,v0∈K,定义Ishikawa迭代{xn}如下：{xn 1=anxn bnTyn cnun,yn=a'nxn b'nTxn c'nvn(任意n≥0),其中{un}和{vn}是K中两个有界序列。则{xn}强收敛于T的唯一不动点。最后研究了￠-强增殖算子方程解的Ishikawa迭代收敛性。     

ON FUNCTIONAL TRANSFORMATION OF THE SEMI-HYPONORMAL OPERATORS

Let H be a complex separable Hilbert space. If T=UP is a bounded operator and \(P - PU{P^ * } = D(T) \ge 0\), where we always Suppose that U is nnitary and \(P \ge 0\)，then T is called semi-hypormal. For the bounded closed set E of the real line R we set \（S(E) = \left\{ {\left. \varphi \right|{K_\varphi }({x_1},{x_2}) = \frac{{\varphi ({x_1}) - \varphi ({x_2})}}{{{x_1} - {x_2}}}{\rm{is semipositive kernel of an integral operator in }}{L^2}(E)} \right\}\）, For the closed set E of the unit circle G±, we set \[\psi '(E) = \left\{ {\left. \varphi \right|{H_\varphi }(\xi ,\eta ) = \frac{{1 - \varphi (\xi )\bar \varphi (\eta )}}{{1 - \xi \bar \eta }} is semi-positive kernel of an integral operator in {L^2}(E)} \right\}\],We haye proved Theorem 1. Let T = UP be semi-hyponormal, \(\varphi \in \psi '(\sigma (U))\)，then \(\tilde T = \varphi (U)\) is also semi-Hyponormal； Theorem 2. Let T=UP be semi-hyponormal operator, \(\psi \in S([0,\left\| T \right\|])\) and \(\psi \) be positive valued. Then \(\tilde T = U\psi (P)\) is also semi-hyponormal. Theorem 3. Let T = UP be Semi-hypormal operator and \(\psi \) be soalar-funotion on \([0,\infty )\) .If \(\tilde T = U\psi (P)\) is а]яо Semi-hypomormal, then we have \[\begin{array}{l} \sigma (\tilde T) = {l_\psi }(\sigma (T)),\\left\| {D(T)} \right\| \le \frac{1}{\pi }{\psi ^{ - 1}}(r)d\theta \end{array}\] Theorem 4. Let T=UP be semi-hypormal operator and soalar-funotion \(\psi \in S([0,\left\| T \right\|])\)，then\(\tilde T = U\psi (P)\)is semi-hypormal and (1), (2)are valid.     

Riesz basis, Paley-Wiener class and tempered splines

The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ∮2m(T,R)∩L2(R) which is the space of polynomial splines with irregularly distributed nodes T={tj}j∈Z, where {tj}j∈Z is a real sequence such that {eitξ}j∈Z constitutes a Riesz basis for L2([-π,π]). From these results, the asymptotic relation E(f,Bπ,2)2=lim E(f,∮2m(T,R)∩L2(R))2 is proved, where Bπ,2 denotes the set of all functions from L2(R) which can be continued to entire functions of exponential type ≤π, i.e. the classical Paley-Wiener class.     

有限个非扩张非自映像隐迭代格式的逼近

假设E为一致凸Banach空间,K为E的非空闭凸子集且为E的非扩张收缩,P为非扩张收缩映像.{Ti:i=1,2,…,N}:K→E为非扩张映像且F(T)=∩ from i=1 to N F(Ti)≠■.定义{xn}如下:x0∈K,xn=P(αnxn-1+(1-αn)TnP[βnxn-1+(1-βn)Tnxn]),n≥1,这里{αn},{βn}为[δ,1-δ]中的实序列,其中δ∈(0,1).若{Ti:i=1,2,…,N}满足条件(B),则{xn}强收敛于x*∈F(T).     

THE COMPLEX FORM AND SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR ELLIPTIC SYSTEMS OF SECOND ORDER

The present paper is concerned with the nonlinear elliptic system of second order. Firstly, we shall establish a complex form of the system. Secondly .we shall consider the solvability of some boundary value problems for tbe complex equation of second order. let (1) \[{\Phi _j}(x,y,U,V,{U_x},{U_y},...,{U_{xx}},{U_{yy}},{V_{xx}},{V_{xy}},{V_{yy}}) = 0,j = 1,2\] be the I. G. Petrowkii’s nonlinear elliptic system of second Qrder in the botinded domain G, where \[{\Phi _j}(x,y,{z_1},...,{z_{12}})(j = 1,2)\]) are continuous real functions of the variables \[x,y[(x,y) \in G],{z_1},...,{z_{12}} \in R\], (the real axis), and contiriupusly differentiable for \[{z_1},...,{z_{12}} \in R\]. The solutions \[[U(x,y),V(x,y)]\], F(a?, y)] of the system are understood in the generalized sense. THEOBEM I. i) If the I. G. Petrovskii;s nonlinear system of equations (1) satisfies the M. I. visik-D. Xiagi’s uniformly elliptic condition for any solutions U(x,y),V(x,y) of (1) in G, then it can be written as the following complex equation? (2)\[{W_{z\overline z }} = F(z,W,{W_z},\overline {{W_z}} ,{W_{zz}},{\overline W _{zz}})\] where W=U+iV, z=x+iy, \[{W_z} = \frac{1}{2}[{W_x} - i{W_y}],...,\], ii) If the I. G. Petrovskii's nonlinear elliptic system (1) satisfies the condition that there exist two positive constants \[\delta \] and K, such that (3) \[|{\Phi _{j{U_{xx}}}}|,|{\Phi _{j{U_{xy}}}}|,|{\Phi _{j{U_{yy}}}}|,|{\Phi _{j{V_{xx}}}}|,|{\Phi _{j{V_{xy}}}}|,|{\Phi _{j{V_{yy}}}}| \leqslant K,j = 1,2\] \[|det(A)| \geqslant \delta > 0\], in G, then by a suitable linear trans-formation of the variables (x,y)into variables \[(\xi ,\eta )\], system (1) can be written as the following coinplex equation ⑷ \[{W_{\xi \xi }} = F(\xi ,W,{W_\xi },{\overline W _\xi },{W_{\xi \xi }},{\overline W _{\xi \xi }}),\varsigma = \xi + i\eta \] In the following section, we discuss the complex equation (2) of the following form: ,We^B(z9 Wee)x .\[(5)\left\{ \begin{gathered} {W_{zz}} = F(z,W,{W_z},{\overline W _z},{W_{zz}},{\overline W _{zz}}) \hfill \ F = {Q_1}{W_{zz}} + {Q_2}\overline {{W_{\overline z \overline z }}} + {Q_4}{W_{zz}} + {A_1}{W_z} + {A_2}{\overline W _{\overline z }} \hfill \ + {A_3}\overline {{W_z}} + {A_4}{W_{\bar z}} + {A_5}W + {A_6}\bar W + {A_7}, \hfill \ {Q_j} = {Q_j}(z,W,{W_{\bar z}},{\overline W _{\bar z}},{W_{zz}},{\overline W _{zz}}),j = 1,...,4 \hfill \ {A_j} = {A_j}(z,W,{W_z},{\overline W _z}),j = 1,...,7 \hfill \\ \end{gathered} \right.\] 1) \[{Q_j}(z,W,{W_z},{\overline W _z},U,V),j = 1,...,4.{A_j} = (z,W,{W_z},{\overline W _z}),j = 1,...,7\] are measurable functions of z for any continuously differentiable functions W(z) and measurable functions U(z), V(z) in G, Furthermore they satisfy (6)\[{\left\| {{A_j}} \right\|_{{L_p}(\overline {G)} }} \leqslant {K_0},j = 1,2,{\left\| {{A_j}} \right\|_{{L_p}(\overline {G)} }} \leqslant {K_1},j = 3,...,7\] where\[{K_0},{K_1}( \leqslant {K_0}),p( > 2)\] are constants: 2) Qj, Aj are continuous for \[W,{W_z},{\overline W _z} \in E\](the whole plane) and the continuity is uniform with respect to almost every point \[z \in G\] and \[U,V \in E\] 3) \[F(z,W,{W_z},{\overline W _z},U,V)\] satisfies the following Lipschitz's condition, i.e. for almost every point \[z \in G\], and for all \[W,{W_z},{\overline W _z}{U_1},{U_2},{V_1},{V_2} \in E\], the inequality (7)\[\begin{gathered} |F(z,W,{W_z},{\overline W _z},{U_1},{V_1}) - F(z,W,{W_z},{\overline W _z},{U_2},{V_2})| \hfill \ \leqslant {q_0}|{U_1} - {U_2}| + q_0^'|{V_1} - {V_2}|,{q_0} + q_0^' < 1 \hfill \\ \end{gathered} \] holds where \[{q_0},q_0^'\] are two nonnegative constants. In this paper, let G be a simply connected domain with boundary \[\Gamma \in C_\mu ^2(0 < \mu < 1)\]； without loss of geaerality, we may assume that G is the unit disk |z|<1. Now we, describe the results of the solvability of Riemann-Hilbert botindary value problem (Problem R-H) and the oblique derivative problem (Problem P) for Eq. (5) in the unit disk G: |z| <1. Problem R-H. We try to find a solution W（z）of Eq. (5) which is continuonsly differentiable on \[G\], and satisfies the boundary conditions: (8) \[\operatorname{Re} [{{\bar z}^{{\chi _1}}},{W_z}] = {r_1}(z),Re[{{\bar z}^{{\chi _2}}}\overline {W(z)} ] = {r_2}(z),z \in \Gamma \]? where \[{\chi _1},{\chi _2}\] are two integers, and \[{r_j} \in C_v^{j - 1}(\Gamma ),j = 1,2,\frac{1}{2} < v < 1\] Problem P. we try to find a solution W（z) of Eq. (5) which is continuously diffierentiabfe on \[\overline G \] and satisfies the boundaory conditions： (9) \[\operatorname{Re} [{{\bar z}^{{\chi _1}}}{W_z}] = {r_1}(z),Re[{{\bar z}^{{\chi _2}}}\overline {W(z)} ] = {r_2}(z),z \in \Gamma \], Where \[{\chi _1},{\chi _2},{r_1}(z),{r_2}(z)\] are the same as in (8), but \[{r_2}(z) \in {C_v}(\Gamma )\]. Theorem II. Suppose that Eq. (5) satisfies the condition C and the constants \[q_0^'\] and K1 are adequately small; then the solvability of Problem R-H is as follows： 1) When \[{\chi _1} \geqslant 0,{\chi _2} \geqslant 0\] Problem R-H is solvable; 2) When \[{\chi _1} < 0,{\chi _2} \geqslant 0(or{\kern 1pt} {\kern 1pt} {\chi _1} \geqslant 0,{\chi _2} < 0){\kern 1pt} \] there are \[2|{\chi _1}| - 1(or2|{\chi _2}| - 1)\] solvable conditions for Problem R-H; 3) WHen \[{\chi _1} < 0,{\chi _2} < 0\], there are \[2(|{\chi _1}| + |{\chi _2}| - 1)\] solvable conditions for Problem R-H. Theorem III Let Eq (5) satisf the condition C and the constants \[q_0^'\] and \[{K_1}\] are adequately small, then tbe solvability of Problem P is as follows： 1) When \[{\chi _1} \geqslant 0,{\chi _2} \geqslant 0\] Problem P is solvable; 2) When \[{\chi _1} < 0,{\chi _2} \geqslant 0(or{\kern 1pt} {\kern 1pt} {\chi _1} \geqslant 0,{\chi _2} < 0){\kern 1pt} {\kern 1pt} {\kern 1pt} \], there are \[2|{\chi _1}| - 1(or2|{\chi _2}| - 1)\] solvable conditions for Problem P; 3) When \[{\chi _1} < 0,{\chi _2} < 0\]; there are \[2|{\chi _1}|{\text{ + }}|{\chi _2}| - 1)\] solvable conditions for Problem P. Furthermore, the solution W(z) of Problem P for Eq. (5) may be expressed as \[{g_j}(\xi ,z) = \left\{ \begin{gathered} \int_0^z {\frac{{{z^{2{\chi _j} + 1}}}}{{1 - \bar \xi z}}dz,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\chi _j} \geqslant 0} \hfill \ \int_0^z {\frac{{{\xi ^{ - 2{\chi _j} - 1}}}}{{1 - \bar \xi z}}dz,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\chi _j} < 0} \hfill \\ \end{gathered} \right.j = 1,2\] where \[{\Phi _0}(z) = a + ib\] is a complex constant,and \[{\Phi _1}(z),{\Phi _2}(z)\] are two analytic functions. The proofs of the above stated theorems are based on a prior estimates for the bounded solutes of these boundary value problems and Leray-Schander theorem. Besides, we have considered also the solvability of Problem R-H and Problem P for Eq. (6) in the multiply connected domain.     

极大单调算子的一个邻近点算法

设E是自反的Banach空间,T∶E→2E是极大单调算子.T-10≠.令x0∈E,yn=(J λnT)-1xn en,xn 1=J-1(αnJxn (1-αn)Jyn),n≥0,λn>0,αn∈[0,1],本文研究了{xn}收敛性.     

准马尔可夫矩阵能扩充为转移矩阵的充要条件

<正> 一个取值于{0,1,2,…,N}的随机过程 Y(t)(t≥0) 称为 n 阶准马尔可夫链,如果对任意 i=1,2,…,N,T>0,在事件{Y(T)=i}和(?)_T={Y(s);0≤s≤T}的条件下,过程 Y(T+t) (t≥0) 的有限维分布仅依赖于 i 而不依赖于 T 和(?)_T(见[1]).当此性质对 i=0也成立,Y(t)就是通常的马尔可夫链.     

DMk OPERATORS AND SPECTRAL OPERATORS

1谱位于平面上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子 记号与[1,2]相同，不再一一赘述.设序列 {Mk}满足（M.1)，(M.2)，(M.3)即.对数凸性、非拟解析性、可微性[1]. 由{M（k）}我们可以 定义二元相关函数\[M({t_1},{t_2})\](详见[7])以及二元\[{\mathcal{D}_{ < {M_k} > }}\]空间 \[{\mathcal{D}_{ < {M_k} > }} = \{ \varphi |\varphi \in \mathcal{D};\exists \nu ,st{\left\| \varphi \right\|_\nu } = \mathop {\sup }\limits_\begin{subarray}{l} s \in {R^2} \\ {k_i} \geqslant 0 \\ (i = 1,2) \end{subarray} |\frac{{{\partial ^{{k_1} + {k_2}}}}}{{{\partial ^{{k_1}}}{s_1}\partial _{{s_2}}^{{k_2}}}}\varphi (s)|/{\nu ^k}{M_k} < + \infty \} \] 其中\[s = ({s_1},{s_2})k = {k_1} + {k_2}\].关于谱位于复平面上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子的定义及性质可 参看[3,4].设X为Banach空间，B(X)为X上有界线性算子的全体组成的环.当 \[T \in B(X)\]为\[{\mathcal{D}_{ < {M_k} > }}\]型算子时，有\[T = {T_1} + i{T_2};{T_1} = {U_{Ret}}{T_2}{\text{ = }}{U_{\operatorname{Im} {\kern 1pt} t}}\] ,此处U为T的谱超广义函数，t为复变量.由于supp(U)为紧集，故可将U延拓到\[{\varepsilon _{ < {M_k} > }}\]上且保持连续性. 经过简单的计算，若\[T \in B(X)\]为谱位于平面上的一个\[{\mathcal{D}_{ < {M_k} > }}\]型算子，则T的一个谱 超广义函数（1）U可表成 \[{U_\varphi } = \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{e^{i({t_1}{T_1} + {t_2}{T_2})}}\hat \varphi } } ({t_1},{t_2})d{t_1}d{t_2}\] 设\[T \in B(X)\]为谱算子，S、N、E（.)分别为T的标量部分、根部、谱测度.下面的定理给出了谱算子成为\[{\mathcal{D}_{ < {M_k} > }}\]型算子的一个充分条件： 定理1设T为谱算子适合下面的条件 \[\mathop {\sup }\limits_{k > 0} \mathop {\sup }\limits_\begin{subarray}{l} |{\mu _j}| < 1 \\ {\delta _j} \in \mathcal{B} \\ j = 1,2,...,k \end{subarray} {(\left\| {\frac{{{N^n}}}{{n!}}\sum\limits_{j = 1}^k {{\mu _j}E({\delta _j})} } \right\|{M_n})^{\frac{1}{n}}} \to 0(n \to \infty )\] 其中\[\mathcal{B}\]为平面本的Borel集类.则T为\[{\mathcal{D}_{ < {M_k} > }}\]型算子且它的一个谱广义函数可表为 \[{U_\varphi } = \sum\limits_{n = 0}^\infty {\frac{{{N^n}}}{{n!}}} \int {{\partial ^n}} \varphi (s)dE(s)\] 推论1设E（?），N满足 \[{(\frac{{{M_n}}}{{n!}} \vee ({N^n}E))^{\frac{1}{n}}} \to 0\] 则T为\[{\mathcal{D}_{ < {M_k} > }}\]型算子. 推论2设N为广义幂零算子，则对于任何与N可换的标量算子S,S+N为\[{\mathcal{D}_{ < {M_k} > }}\]型算子的充分必要条件是 \[{(\frac{{\left\| {{N^n}} \right\|}}{{n!}}{M_n})^{\frac{1}{n}}} \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (n \to \infty )\] 在[4]中称满足上式的算子为\[\{ {M_k}\} \]广义幂零算子.显然\[\{ {M_k}\} \]广义幂零算子必为通 常的广义幂零算子.下面的命题给出了\[\{ {M_k}\} \] 广义幂零算子的一些性质. 命题 设N为广义幂零算子，则下列事实等价： (i ) N为\[\{ {M_k}\} \]广义幂零算子； (ii)对于任给的\[\lambda > 0\],存在\[{B_\lambda } > 0\]使(1) \[\left\| {R(\xi ,N)} \right\| \leqslant {B_\lambda }{e^{{M^*}(\frac{\lambda }{{|\xi |}})}}\](\[{|\xi |}\]充分小)； (iii)对于任给的\[\mu > 0\],存在\[{A_\mu } > 0\]使 \[\left\| {{e^{izN}}} \right\| \leqslant {A_\mu }{e^{M(\mu |z|)}}\] 2谱位于实轴上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子本节讨论有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子T成为谱算子 的条件，这里假定\[{\mathcal{D}_{ < {M_k} > }}\]中的函数是一元的，于是Т的谱位于实轴上.X^*表示X的共轭 空间. 设\[f \in {\mathcal{D}^'}_{ < {M_k} > }\]，由[8, 9]，存在测度\[{\mu _n}(n \geqslant 0)\]使得对任何h>0,存在A>0适合 \[\sum\limits_{n = 0}^\infty {\frac{{{h^n}}}{{n!}}} {M_n}\int {|d{\mu _n}| \leqslant A} \]且 \[ < f,\varphi > = \sum\limits_{n = 0}^\infty {\frac{1}{{n!}}} \int {{\varphi ^{(n)}}} (t)d{\mu _n}(t)\] 一般说，上述\[{\mu _n}(n \geqslant 0)\]不是唯一的，为此我们引入 定义设\[{n_0}\]为正整，如果对一切\[n \geqslant {n_0}\]，存在测度\[{{\mu _n}}\]，它们的支集均包含在某一L 零测度闭集内，则称f是\[{n_0}\]奇异的，若\[{n_0}\] = 1，则称f是奇异的.设\[T \in B(X)\]为\[{\mathcal{D}_{ < {M_k} > }}\]型 算子，U为其谱超广义函数，如果对于任何\[x \in X{x^*} \in {X^*},{x^*}U\].x是\[{n_0}\]奇异的（奇异 的)，则称T是\[{n_0}\]奇异的(奇异的）\[{\mathcal{D}_{ < {M_k} > }}\]型算子. 经过若干准备，可以证明下面的 定理2 设X为自反的Banach空间，则\[T \in B(X)\]为奇异\[{\mathcal{D}_{ < {M_k} > }}\]型算子的充分必要 条件是T为满足下列条件的谱算子： (i)对每个\[x \in X\]及\[{x^*} \in X\]，\[\sup p({x^*}{N^n}E()x)\]包含在一个与\[n \geqslant 1\]无关的L零测 度闭集F内(F可以依赖于\[x{x^*}\])，此处E(?)、N分别是T的谱测度与根部； (ii)算子N是\[\{ {M_k}\} \]广义幂零算子. 推论 设X为自反的banach空间，\[T \in B(X)\]为奇异\[{\mathcal{D}_{ < {M_k} > }}\]型算子且\[\sigma (T)\]的测度 为零的充分必要条件是T为满足下列条件的谱算子： (i) E(?）的支集为L零测度集； (ii) 算子N是\[\{ {M_k}\} \]广义幂零算子.；     

Banach空间中一类混合映射不动点的Ishikawa迭代逼近问题

设E为一致光滑Banach空间,K为E的非空闭凸子集,T:K→K为Φ-强伪压缩映射.其中T=T1+T2,T1:K→K为Lipschitz映射,T2:K→K为具有有界值域映射.设{αn}n∞=0和{βn}n∞=0是[0,1]中满足一定条件的两实数列.则Ishikawa迭代序列{xn}∞n=0强收敛于T的唯一不动点.     

Banach空间中极大单调算子的一个邻近点算法

设E是Banach空间,T∶E→2E*是极大单调算子,T-10≠ф.令x0∈E,yn=(J λnT)-1xn en,xn 1=J-1(αnJxn (1-αn)Jyn),n0,λn>0,αn∈[0,1],文章研究了{xn}收敛性.     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

<正>August 10-14,2015Beijing,ChinaThe International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.     

Boundedness of commutators related to Marcinkiewicz integrals on hardy type spaces

In this paper, the authors study the boundedness of the operator [μΩ, b], the commutator generated by a function b ∈ Lipβ(Rn)(0 ＜β≤ 1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈ Lipα(Sn-1)(0 ＜α≤ 1).     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.