GLOBAL EXISTENCE OF THE SOLUTION OF NONLINEAR SCHRDINGER EQUATIONS IN EXTERIOR DOMAINS

In this paper,we discuss the problem for the nonlinear Schr(?)dinger equation(?)where Ω is the exterior domain of a compact set in B~n,a_j(u)=O(|u|),b_j(u)=O(|u|)(1≤j≤n),c(u)=O(|u|~2)near u=0.If n≥5,some Sobolev norm of u_0(x)is sufficiently small,under suitableassumptions on smoothnessand and compatibility and the shape of Ω,we get that the problem has aunique global solution u(t,x),with the decay estimate‖u(t,·)‖_(L(?)(Ω))=O(t~(-n/4)),‖u(t,·)‖_(L~4(Ω))=O(t~(-n/4)),t→+∞.     

Lipschitz-Nikolski\imath constants and asymptotic simultaneous approximation on theM n -operatorsconstants and asymptotic simultaneous approximation on theM n -operators

This note is a study of approximation of classes of functions and asymptotic simultaneous approximation of functions by theM _n -operators of Meyer-König and Zeller which are defined by $$(M_n f)(x) = (1 - x)^{n + 1} \sum\limits_{k = 0}^\infty {f\left( {\frac{k}{{n + k}}} \right)} \left( \begin{array}{l} n + k \\ k \\ \end{array} \right)x^k , n = 1,2,....$$ Among other results it is proved that for 0<α≤1 $$\mathop {\lim }\limits_{n \to \infty } n^{\alpha /2} \mathop {\sup }\limits_{f \in Lip_1 \alpha } \left| {(M_n f)(x) - f(x)} \right| = \frac{{\Gamma \left( {\frac{{\alpha + 1}}{2}} \right)}}{{\pi ^{1/2} }}\left\{ {2x(1 - x)^2 } \right\}^{\alpha /2} $$ and if for a functionf, the derivativeD ^m+2 f exist at a pointx∈(0, 1), then $$\mathop {\lim }\limits_{n \to \infty } 2n[D^m (M_n f) - D^m f] = \Omega f,$$ where Ω is the linear differential operator given by $$\Omega = x(1 - x)^2 D^{m + 2} + m(3x - 1)(x - 1)D^{m + 1} + m(m - 1)(3x - 2)D^m + m(m - 1)(m - 2)D^{m - 1} .$$      

THROREMS OF DISTORTION FOR SCHLICHT FUNCTIONS

Let \[f(z) = z + \sum\limits_{n = 1}^\infty {{a_n}{z^n} \in S} {\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \log \frac{{f(z) - f(\xi )}}{{z - \xi }} - \frac{{z\xi }}{{f(z)f(\xi )}} = \sum\limits_{m,n = 1}^\infty {{d_{m,n}}{z^m}{\xi ^n},} \], we denote \[{f_v} = f({z_v})\] , \[\begin{array}{l} {\varphi _\varepsilon }({z_u}{z_v}) = {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\frac{1}{{(1 - {z_u}{{\bar z}_v})}},\g_m^\varepsilon (z) = - {F_m}(\frac{1}{{f(z)}}) + \frac{1}{{{z^m}}} + \varepsilon {{\bar z}^m}, \end{array}\], where \({F_m}(t)\) is a Faber polynomial of degree m. Theorem 1. If \[f(z) \in S{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{u,v = 1}^N {{A_{u,v}}{x_u}{{\bar x}_v} \ge 0} \] and then \[\begin{array}{l} \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\exp \{ \alpha {F_l}({z_u},{z_v})\} \ \le \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} \varphi _\varepsilon ^\alpha ({z_u}{z_v})l = 1,2,3, \end{array}\], where \[\begin{array}{l} {F_1}({z_u},{z_v}) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} g_n^\varepsilon ({z_u})\bar g_n^\varepsilon ({z_v}),\{F_2}({z_u},{z_v}) = \frac{1}{{1 + {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}),\{F_3}({z_u},{z_v}) = \frac{1}{{1 - {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}). \end{array}\] The \[F({z_u},{z_v}) = \frac{1}{2}{g_1}({z_u}){{\bar g}_2}({z_v})\] is due to Kungsun. Theorem 2. If \(f(z) \in S\) ,then \[P(z) + \left| {\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {{\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}\frac{{{z_u}{z_v}}}{{{f_u}{f_v}}}} \right|}^\varepsilon }} \right| \le \sum\limits_{u,v = 1}^N {{\lambda _u}{{\bar \lambda }_v}} \frac{1}{{1 - {z_u}{{\bar z}_v}}}\], where \[\begin{array}{l} P(z) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} {G_n}(z),\{G_n}(z) = {\left| {\left| {\sum\limits_{n = 1}^N {{\beta _u}({F_n}(\frac{1}{{f({z_u})}}) - \frac{1}{{z_u^n}})} } \right| - \left| {\sum\limits_{n = 1}^N {{\beta _u}z_u^n} } \right|} \right|^2}, \end{array}\], \(P(z) \equiv 0\) is due to Xia Daoxing.     

On semilinear dissipative systems of equations with a small parameter that arise in solution of the Navier-Stokes equations,equation of motion of the Oldroyd fluids,and equations of motion of the Kelvin-Voight fluids

Solutions of the two-dimensional initial boundary-value problem for the Navier-Stokes equations are approximated by solutions of the initial boundary-value problem 9 $$\begin{array}{*{20}c} {\frac{{\partial v}}{{\partial t}}^\varepsilon - v\Delta v^\varepsilon + v_k^\varepsilon v_{x_k }^\varepsilon + \frac{1}{2}v^\varepsilon div v^\varepsilon - \frac{1}{\varepsilon }grad div w^\varepsilon = f_1 ,} \\ {\frac{{\partial w^\varepsilon }}{{\partial t}} + \alpha w^\varepsilon = v^\varepsilon ,} \\ \end{array} $$ 10 $$v^\varepsilon \left| {_{t = 0} = v_0^\varepsilon (x), w^\varepsilon } \right|_{t = 0} = 0, x \in \Omega , v^\varepsilon \left| {_{\partial \Omega } = w^\varepsilon } \right|_{\partial \Omega } = 0, t \in \mathbb{R}^ + $$ . We study the proximity of the solutions of these problems in suitable norms and also the proximity of their minimal global B-attractors. Similar results are valid for two-dimensional equations of motion of the Oldroyd fluids (see Eqs. (38) and (41)) and for three-dimensional equations of motion of the Kelvin-Voight fluids (see Eqs. (39) and (43)). Bibliography: 17 titles.     

On the strong summation of Fourier series with variable exponents

Пустьf 2π-периодическ ая суммируемая функц ия, as _k (x) еë сумма Фурье порядк аk. В связи с известным ре зультатом Зигмунда о сильной суммируемости мы уст анавливаем, что если λn→∞, то сущес твует такая функцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _{2n} } } \right\}^{1/\lambda _{2n} } = \infty .$$ Отсюда, в частности, вы текает, что если λn?∞, т о существует такая фун кцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } } \right\}^{1/\lambda _n } = \infty .$$ Пусть, далее, ω-модуль н епрерывности и $$H^\omega = \{ f:\parallel f(x + h) - f(x)\parallel _c \leqq K_f \omega (h)\} .$$ . Мы доказываем, что есл и λ _n ?∞, то необходимым и достаточным условие м для того, чтобы для всехf∈H ^ω выполнялос ь соотношение $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _n } } \right\}^{1/\lambda _n } = 0(x \in [0;2\pi ])$$ является условие $$\omega \left( {\frac{1}{n}} \right) = o\left( {\frac{1}{{\log n}} + \frac{1}{{\lambda _n }}} \right).$$ Это же условие необхо димо и достаточно для того, чтобы выполнялось соотнош ение $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } = 0(f \in H^\omega ,x \in [0;2\pi ]).$$      

On the convergence of double Fourier series of functions from Lp,p > 1

It is proved that if a function from L_p, p > 1, satisfies the condition $$\frac{1}{{t \cdot \tau }}\int_0^t {\int_0^\tau {\left| {f(x + u,y + v) - f(x,y)} \right|} dudv = O\left( {\left[ {1n\frac{1}{{(t^2 + \tau ^2 )}}} \right]^{ - 2} } \right),}$$ then the double Fourier series of function f, under summation over a rectangle, converges almost everywhere.     

Strongly nonlinear degenerate elliptic equations with discontinuous coefficients. I

This paper is concerned with the existence and uniqueness of variational solutions of the strongly nonlinear equation $$ - \sum\limits_1^m {_i \frac{\partial }{{\partial x_i }}\left( {\sum\limits_1^m {_j a_{i,j} (x, u(x))\frac{{\partial u(x)}}{{\partial x_j }}} } \right) + g(x, u(x)) = f(x)} $$ with the coefficients a _i,j (x, s) satisfying an eHipticity degenerate condition and hypotheses weaker than the continuity with respect to the variable s. Furthermore, we establish a condition on f under which the solution is bounded in a bounded open subset Ω of R^m.     

Almansi decomposition for Dunkl operators

Let Ωbe a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ωwhich are Dunkl polyharmonic, i.e. (△h)nf =0 for some integer n. Here △h=∑j=1N Dj2 is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G, where kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x) |x|2f1(x) … |x|2(n-1)fn-1(x),(?)x∈Ω, where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.     

Pointwise estimates for Lagrange interpolation polynomials

Let f ∈ C[?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 f ∈ C ^r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$      

On the approximation by de la Vallée Poussin sums and interpolatory polynomials in Lipschitz norms

ПустьM ^m,α - множество 2π-п ериодических функци йf с конечной нормой $$||f||_{p,m,\alpha } = \sum\limits_{k = 1}^m {||f^{(k)} ||_{_p } + \mathop {\sup }\limits_{h \ne 0} |h|^{ - \alpha } ||} f^{(m)} (o + h) - f^{(m)} (o)||_{p,} $$ где1 ≦ p ≦ ∞, 0≦α≦1. Рассмотр им средние Bалле Пуссе на $$(\sigma _{n,1} f)(x) = \frac{1}{\pi }\int\limits_0^{2x} {f(u)K_{n,1} (x - u)du} $$ и $$(L_{n,1} f)(x) = \frac{2}{{2n + 1}}\sum\limits_{k = 1}^{2n} {f(x_k )K_{n,1} } (x - x_k ),$$ де0≦l≦n и x _k=2kπ/(2n+1). В работе по лучены оценки для вел ичин \(||f - \sigma _{n,1} f||_{p,r,\beta } \) и $$||f - L_{n,1} f||_{p,r,\beta } (r + \beta \leqq m + \alpha ).$$      

Asymptotically sharp error bounds for a quadrature rule for Cauchy principal value integrals based on piecewise linear interpolation

For the numerical evaluation of Cauchy principal value integrals of the form f(x)(x - λ) -1dx withλ ∈ (-1,1)and f ∈ C1[-1,1], we investigate the quadrature formula Q(n+1)Spl1[·;λ] obtained by replacing the integrand function f by its piecewise linear interpolant at an equidistant set of nodes as proposed by Rabinoivitz (Math. Comp. , 51:741 - 747,1988). We give upper bounds for the Peano - type error constantsfor s∈ {1,2}. These are the best possible constants in inequalities of the typeFurthermore, we prove that our upper bounds are asymptotically sharp.     

{W^{1,1}_0} -solutions for elliptic problems having gradient quadratic lower order terms

In this paper we deal with solutions of problems of the type $$\left\{\begin{array}{ll}-{\rm div} \Big(\frac{a(x)Du}{(1+|u|)^2} \Big)+u = \frac{b(x)|Du|^2}{(1+|u|)^3} +f \quad &{\rm in} \, \Omega,\\ u=0 &{\rm on} \partial \, \Omega, \end{array} \right.$$ where ${0 < \alpha \leq a(x) \leq \beta, |b(x)| \leq \gamma, \gamma > 0, f \in L^2 (\Omega)}$ and Ω is a bounded subset of ${\mathbb{R}^N}$ with N ≥ 3. We prove the existence of at least one solution for such a problem in the space ${W_{0}^{1, 1}(\Omega) \cap L^{2}(\Omega)}$ if the size of the lower order term satisfies a smallness condition when compared with the principal part of the operator. This kind of problems naturally appears when one looks for positive minima of a functional whose model is: $$J (v) = \frac{\alpha}{2} \int_{\Omega}\frac{|D v|^2}{(1 + |v|)^{2}} + \frac{12}{\int_{\Omega}|v|^2} - \int_{\Omega}f\,v , \quad f \in L^2(\Omega),$$ where in this case a(x) ≡ b(x) = α > 0.     

Existence of solutions for the critical elliptic system with inverse square potentials

Let Ω ? 0 be an open bounded domain in R ^N (N ≥ 3) and $2^* (s) = \tfrac{{2(N - s)}} {{N - 2}}$ , 0 < s < 2. We consider the following elliptic system of two equations in H ₀ ¹ (Ω) × H ₀ ¹ (Ω): $$- \Delta u - t\frac{u} {{\left| x \right|^2 }} = \frac{{2\alpha }} {{\alpha + \beta }}\frac{{\left| u \right|^{\alpha - 2} u\left| v \right|^\beta }} {{\left| x \right|^s }} + \lambda u, - \Delta v - t\frac{v} {{\left| x \right|^2 }} = \frac{{2\beta }} {{\alpha + \beta }}\frac{{\left| u \right|^\alpha \left| v \right|^{\beta - 2} v}} {{\left| x \right|^s }} + \mu v,$$ where λ, µ > 0 and α, β > 1 satisfy α + β = 2*(s). Using the Moser iteration, we prove the asymptotic behavior of solutions at the origin. In addition, by exploiting the Mountain-Pass theorem, we establish the existence of solutions.     

On p dependent boundedness of singular integral operators

We study the classical Calderón Zygmund singular integral operator with homogeneous kernel. Suppose that Ω is an integrable function with mean value 0 on S ¹. We study the singular integral operator $$T_\Omega f= {\rm p.v.} \, f * \frac {\Omega (x/|x|)}{|x|^2}.$$ We show that for α > 0 the condition $$\Bigg| \int \limits _{I} \Omega (\theta) \, d\theta \Bigg| \leq C |\log|I||^{-1-\alpha} \quad\quad\quad\quad (0.1)$$ for all intervals |I| < 1 in S ¹ gives L ^p boundedness of T _Ω in the range ${|1/2-1/p| < \frac \alpha {2(\alpha+1)}}$ . This condition is weaker than the conditions from Grafakos and Stefanov (Indiana Univ Math J 47:455–469, 1998) and Fan et al. (Math Inequal Appl 2:73–81, 1999). We also construct an example of an integrable Ω which satisfies (0.1) such that T _Ω is not L ^p bounded for ${|1/2-1/p| > \frac {3\alpha +1}{6(\alpha +1)}}$ .     

On the best constants in the Khintchine inequality for Steinhaus variables

Let S _j : (Ω, P) → S ¹ ? ? be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S ¹. We determine the best constants a _p in the Khintchine-type inequality $${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$ for 0 < p < 1, verifying a conjecture of U. Haagerup that $${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$ . Both expressions are equal for p = p ₀ }~ 0.4756. For p ≥ 1 the best constants a _p have been known for some time. The result implies for a norm 1 sequence x ∈ ? ⁿ , ‖x‖₂ = 1, that $${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$ , answering a question of A. Baernstein and R. Culverhouse.     

Removable singularities of some strongly nonlinear elliptic equations

Let Ω be some open subset of ?^N containing 0 and Ω′=Ω?{0}. If g is a continuous function from ? × ? into ? satisfying some power like growth assumption, then any u∈L _loc ^∞ (Ω′) satisfying $$\begin{array}{*{20}c} { - div (Du \left| {Du} \right|^{p - 2} ) + g(.,u) = 0} & {in \mathcal{D}'(\Omega ')} \\ \end{array} $$ , remains bounded in Ω and satisfies the equation in D'(Ω). We give extensions when the singular set is some compact submanifold of Ω. When g is bounded below on ?⁺ and above on ?^?, then we prove that any subset Σ with 1-capacity zero is a removable singularity for a function u∈L _loc ^∞ (ω?Σ) satisfying $$\begin{array}{*{20}c} { - div \left( {\frac{{Du}}{{\sqrt {1 + \left| {Du} \right|^2 } }}} \right) + g(.,u) = 0} & {in \mathcal{D}'(\Omega - \Sigma )} \\ \end{array} $$ .     

Approximation of the function sign x in the uniform and integral metrics by means of rational functions

Estimates are obtained for the nonsymmetric deviations R_n [sign x] and R_n [sign x]_L of the function sign x from rational functions of degree ≤n, respectively, in the metric $$c([ - 1, - \delta ] \cup [\delta ,1]), 0< \delta< exp( - \alpha \surd \overline n ), \alpha > 0,$$ and in the metric L[?1, 1]: $$\begin{gathered} R_n [sign x] _{\frown }^\smile exp \{ - \pi ^2 n/(2 ln 1/\delta )\} , n \to \infty , \hfill \\ 10^{ - 3} n^{ - 2} \exp ( - 2\pi \surd \overline n )< R_n [sign x_{|L}< \exp ( - \pi \surd \overline {n/2} + 150). \hfill \\ \end{gathered} $$ Let 0 < δ < 1, Δ (δ)=[?1, ? δ] ∪ [δ, 1]; $$\begin{gathered} R_n [f;\Delta (\delta )] = R_n [f] = inf max |f(x) - R(x)|, \hfill \\ R_n [f;[ - 1,1] ]_L = R_n [f]_L = \mathop {inf}\limits_{R(x)} \smallint _{ - 1}^1 |f(x) - R(x)|dx, \hfill \\ \end{gathered} $$ where R(x) is a rational function of order at most n. Bulanov [1] proved that for δ ε [e^?n, e^?1] the inequality $$\exp \left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta }}} \right) \leqslant R_n [sign x] \leqslant 30 exp\left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta + 4 ln ln (e/\delta ) + 4}}} \right)$$ is valid. The lower estimate in this inequality was previously obtained by Gonchar ([2], cf. also [1]).     

Simultaneouse approximation to a differentiable function and its derivatives by Lagrange interpolating polynomials

This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima-tion:then|f~(k)(x)-P_n~(k)(f,x)|=O(1)△_n~(q-k)(x)ωwhere P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodesX_nUY_n(see the definition of the next).     

A Mean-Value Theorem for Multiplicative Functions on the Set of Shifted Primes

Let f be a complex-valued multiplicative function, letp denote a prime and let π(x) be the number of primes not exceeding x. Further put $$m_p (f): = \mathop {\lim }\limits_{x \to \infty } \frac{1}{{\pi (x)}}\sum\limits_{p \leqslant x} {f(p + 1)} {\text{, }}M(f): = \mathop {\lim }\limits_{x \to \infty } \frac{1}{x}\sum\limits_{n \leqslant x} {f(n)}$$ and suppose that $$\mathop {\lim \sup }\limits_{x \to \infty } \frac{1}{x}\sum\limits_{n \leqslant x} {\left| {f\left( n \right)} \right|^2 } < \infty ,\sum\limits_{p \leqslant x} {\left| {f\left( n \right)} \right|^2 } \ll x\left( {\ln x} \right)^{ - \varrho } ,$$ with some \varrho > 0. For such functions we prove: If there is a Dirichlet character χ_d such that the mean-value M(f χ_d) exists and is different from zero,then the mean-value m_p(f) exists. If the mean-value M(f) exists, then the same is true for the mean-valuem_p(f) .     

Divided differences for functions of two variables for irregularly spaced arguments

Divided differences forf (x, y) for completely irregular spacing of points (x _i ,y _i ) are developed here by a natural generalization of Newton's scheme. Existing bivariate schemes either iterate the one-dimensional scheme, thus constraining (x _i ,y _i ) to be at corners of rectangles, or give polynomials Σa _jk x ^j y ^k having more coefficients than interpolation conditions. Here the generalizedn ^th divided difference is defined by (1)\(\left[ {01... n} \right] = \sum\limits_{i = 0}^n {A_i f\left( {x_i , y_i } \right)} \) where (2)\(\sum\limits_{i = 0}^n {A_i x_i^j , y_i^k = 0} \), and 1 for the last or (n+1)^th equation, for every (j, k) wherej+k=0, 1, 2,... in the usual ascending order. The gen. div. diff. [01...n] is symmetric in (x _i ,y _i ), unchanged under translation, 0 forf (x, y) an, ascending binary polynomial as far asn terms, degree-lowering with respect to (X, Y) whenf(x, y) is any polynomialP(X+x, Y+y), and satisfies the 3-term recurrence relation (3) [01...n]=λ{[1...n]?[0...n?1]}, where (4) λ= |1...n|·|01...n?1|/|01...n|·|1...n?1|, the |...i...| denoting determinants inx _i ^j y _i ^k . The generalization of Newton's div. diff. formula is (5)

$$\begin{gathered} f\left( {x, y} \right) = f\left( {x_0 , y_0 } \right) - \frac{{\left| {\alpha 0} \right|}}{{\left| 0 \right|}}\left[ {01} \right] + \frac{{\left| {\alpha 01} \right|}}{{\left| {01} \right|}}\left[ {012} \right] - \frac{{\left| {\alpha 012} \right|}}{{\left| {012} \right|}}\left[ {0123} \right] + \cdots + \hfill \\ + \left( { - 1} \right)^n \frac{{\left| {\alpha 01 \ldots n - 1} \right|}}{{\left| {01 \ldots n - 1} \right|}}\left[ {01 \ldots n} \right] + \left( { - 1} \right)^{n + 1} \frac{{\left| {\alpha 01 \ldots n} \right|}}{{\left| {01 \ldots n} \right|}}\left[ {01 \ldots n} \right], \hfill \\ \end{gathered} $$