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1.
In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting vertically the zeros of (1-x)2=P (a,) n(x),a>0,>0,(1-x)P(a,) n(x),a>0,>-1,(1+x)P P(a,) n(x),a>-1,0 and P(a,) n(x),a>-1,>-1, respectively, onto the unit circle, where P(a,) n(x),a>-1,>-1, stands for the n-th Jacobi polynomial. Moreover, a result of Saff and Walsh is also extended.  相似文献   

2.
LetW(x):= exp(-{tiQ(x})), where, for example, Q(x) is even and convex onR, and Q(x)/logx → ∞ asx → ∞. A result of Mhaskar and Saff asserts that ifa n =a n (W) is the positive root of the equation $$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {{{a_n xQ'(a_n x)} \mathord{\left/ {\vphantom {{a_n xQ'(a_n x)} {\sqrt {1 - x^2 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - x^2 } }}dx,}$$ then, given any polynomialP n(x) of degree at mostn, the sup norm ofP n(x)W(a n x) overR is attained on [-1, 1]. In addition, any sequence of weighted polynomials {p n (x)W(a n x)} 1 that is uniformly bounded onR will converge to 0, for ¦x¦>1. In this paper we show that under certain conditions onW, a function g(x) continuous inR can be approximated in the uniform norm by such a sequence {p n (x)W(a n x)} 1 if and only if g(x)=0 for ¦x¦? 1. We also prove anL p analogue for 0W(x)=exp(?|x| α ), when α >1. Further applications of our results are upper bounds for Christoffel functions, and asymptotic behavior of the largest zeros of orthogonal polynomials. A final application is an approximation theorem that will be used in a forthcoming proof of Freud's conjecture for |x| p exp(?|x| α ),α > 0,p > ?1.  相似文献   

3.
An even-order three-point boundary value problem on time scales   总被引:1,自引:0,他引:1  
We study the even-order dynamic equation (−1)nx(Δ∇)n(t)=λh(t)f(x(t)), t∈[a,c] satisfying the boundary conditions x(Δ∇)i(a)=0 and x(Δ∇)i(c)=βx(Δ∇)i(b) for 0?i?n−1. The three points a,b,c are from a time scale , where 0<β(ba)<ca for b∈(a,c), β>0, f is a positive function, and h is a nonnegative function that is allowed to vanish on some subintervals of [a,c] of the time scale.  相似文献   

4.
Let {Q n (α,β) (x)} n=0 denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product
$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$
where λ>0 and d μ α,β(x)=(x?a)(1?x)α?1(1+x)β?1 dx, d ν α,β(x)=(1?x) α (1+x) β dx with aα,β>0. Their inner strong asymptotics on (?1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q n (α,β) are obtained.
  相似文献   

5.
In the present paper, we consider a preconditioning strategy for Finite Element (FE) matrix sequences {A n (a)} n discretizing the elliptic problem $$\left\{ \begin{gathered} A_a u \equiv ( - )^k \nabla ^k [a(x,y)\nabla ^k u(x,y)] = f(x,y),{ }(x,y) \in \Omega = (0,1)^2 , \hfill \\ \left. {\left( {\frac{{\partial ^s }}{{\partial v^s }}u(x,y)} \right)} \right|_{\partial \Omega } \equiv 0,{ }s = 0,...,k - 1,{ }^{^{^{^{^{^{(1)} } } } } } \hfill \\ \end{gathered} \right.$$ with a(x,y) being a uniformly positive function and ν denoting the unit outward normal direction. More precisely, in connection with preconditioned conjugate gradient (PCG) like methods, we define the preconditioning sequence: {P n (a)} n , P n (a):= $$\widetilde D$$ n 1/2(a)A n (1) $$\widetilde D$$ n 1/2(a), where $$\widetilde D$$ n (a) is the suitable scaled main diagonal of A n (a). In fact, under the mild assumption of Lebesgue integrability of a(x), the weak clustering at the unity of the corresponding preconditioned sequence is proved. Moreover, if a(x,y) is regular enough and if a uniform triangulation is considered, then the preconditioned sequence shows a strong clustering at the unity so that the sequence {P n (a)} n turns out to be a superlinear preconditioning sequence for {A n (a)} n . The computational interest is due to the fact that the computation with A n (a) is reduced to computations involving diagonals and two-level Toeplitz structures {A n (1)} n with banded pattern. Some numerical experimentations confirm the efficiency of the discussed proposal.  相似文献   

6.
We propose a method of constructing orthogonal polynomials Pn(x) (Krall's polynomials) that are eigenfunctions of higher-order differential operators. Using this method we show that recurrence coefficients of Krall's polynomials Pn(x) are rational functions of n. Let Pn(a,b;M)(x) be polynomials obtained from the Jacobi polynomials Pn(a,b)(x) by the following procedure. We add an arbitrary concentrated mass M at the endpoint of the orthogonality interval with respect to the weight function of the ordinary Jacobi polynomials. We find necessary conditions for the parameters a,b in order for the polynomials Pn(a,b;M)(x) to obey a higher-order differential equation. The main result of the paper is the following. Let a be a positive integer and b⩾−1/2 an arbitrary real parameter. Then the polynomials Pn(a,b;M)(x) are Krall's polynomials satisfying a differential equation of order 2a+4.  相似文献   

7.
For the singular operator $$Su = \int_a^b {\frac{{K(x, s) u (s)}}{{s - x}}} ds$$ invariant weight spacesλ α β , p (u(x)∈λ α β , p if 10,u (x) ρ (x)∈ H β 0 , 20.‖uL p0)<∞, ρ (x) = (x?a) (b ?x)1+β, ρ0(x)=(b?x)α(p?1), 0<α, β<1,p>1H 0 β is a Hölder space. Multiplicative inequalities of the type of Kh. Sh. Mukhtarov are also obtained.  相似文献   

8.
Explicit upper and lower estimates are given for the norms of the operators of embedding of , n ∈ ?, in L q (dµ), 0 < q < ∞. Conditions on the measure µ are obtained under which the ratio of the above estimates tends to 1 as n → ∞, and asymptotic formulas are presented for these norms in regular cases. As a corollary, an asymptotic formula (as n → ∞) is established for the minimum eigenvalues λ1, n, β , β > 0, of the boundary value problems (?d 2/dx 2) n u(x) = λ|x| β?1, x ∈ (?1, 1), u (k)(±1) = 0, k ∈ {0, 1, ..., n ? 1}.  相似文献   

9.
Let P(x) = Σi=0naixi be a nonnegative integral polynomial. The polynomial P(x) is m-graphical, and a multi-graph G a realization of P(x), provided there exists a multi-graph G containing exactly P(1) points where ai of these points have degree i for 0≤in. For multigraphs G, H having polynomials P(x), Q(x) and number-theoretic partitions (degree sequences) π, ?, the usual product P(x)Q(x) is shown to be the polynomial of the Cartesian product G × H, thus inducing a natural product π? which extends that of juxtaposing integral multiple copies of ?. Skeletal results are given on synthesizing a multi-graph G via a natural Cartesian product G1 × … × Gk having the same polynomial (partition) as G. Other results include an elementary sufficient condition for arbitrary nonnegative integral polynomials to be graphical.  相似文献   

10.
п. л. ЧЕБышЕВыМ БылА пОс тАВлЕНА И РЕшЕНА жАДА ЧА: пРИ пРОИжВОльНО жАДАННО М НА [?1,1] пОлОжИтЕльНОМ МНОг ОЧлЕНЕP l (x) жАДАННОИ ст ЕпЕНИl НАИтИ пРИ кАжДОМn≧1 МНОгОЧлЕНР n * (x) стЕпЕН Ип с кОЁФФИцИЕНтОМ п РИx n , РАВНыМ 1, кОтОРыИ ьВл ьлсь Бы МНОгО-ЧлЕНОМ, НАИМЕНЕЕ УклОНьУЩИМ сь От НУль с ВЕсОМP l ?1 (x) В МЕтРИкЕC[? 1,1]. А. А. МАРкОВ ОБОБЩИл ЁтО т РЕжУльтАт И пРИ тЕх ж Е УслОВИьх НАP l (x) пОстРО Ил Дль1/2 МНОгОЧлЕНыP n * (x) стЕп ЕНИ п, НАИМЕНЕЕ УклОНьУЩИЕсь От НУль с ВЕсОМP l -1/2 (x). В ДАННОИ стАтьЕ УкАжы ВАЕтсь БОлЕЕ пРОстОИ, ЧЕМ В [3], спОсОБ пОстРОЕНИь МН ОгОЧлЕНОВP n * (x), ДАУЩИх РЕшЕНИЕ жАД АЧИ МАРкОВА, И пРИ ЁтОМ, ВО-пЕРВых, УстАНОВлЕН О, ЧтО ВськИИ тАкОИ МНОгОЧлЕН МОжН О пРЕДстАВИть В ВИДЕ л ИНЕИНОИ кОМБИНАцИИ НЕ БОлЕЕ Ч ЕМ Ижl+1 МНОгОЧлЕНОВ ЧЕБышЕВ АT j (x)=cos (jarc cosx): $$P_n^* (x) = \mathop \Sigma \limits_{k = 0}^l \gamma _k T_{|n - l + k|} (x)$$ , В кОтОРОИ кОЁФФИцИЕН тыγ k НЕ жАВИсьт Отп И, ВО-ВтОРых, УкАжАН спОс ОБ ЁФФЕктИВНОгО РАжы с-кАНИь кОЁФФИцИЕНтОВγ k пО М НОгОЧлЕНУР l (х).  相似文献   

11.
We prove that if a functionfC (1) (I),I: = [?1, 1], changes its signs times (s ∈ ?) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ?, there exists an algebraic polynomialP n =P n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iffC (I) andf(x) ≥ 0,xI then, for anynk ? 1, there exists a polynomialP n =P n (x) of degree ≤n such thatP n (x) ≥ 0,xI, and |f(x) ?P n (x)| ≤c(k k (f;n ?2 +n ?1 √1 ?x 2),xI.  相似文献   

12.
Let f∈C [−1,1] (r≥1) and Rn(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions Rn(f,α,β,xk)=f(xk),Rn (f,α,β,xk)=f′(xk)(k=1,2,…,n), where {xk} are the roots of n-th Jacobi polynomial Pn(α,β,x),α,β>−1 and {x k } are the roots of (1−x2)Pn″(α,β,x). In this paper, we prove that holds uniformly on [0,1]. In Memory of Professor M. T. Cheng Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang.  相似文献   

13.
In this note, we show that, if the Druzkowski mappings F(X)=X+(AX)∗3, i.e. F(X)=(x1+(a11x1+?+a1nxn)3,…,xn+(an1x1+?+annxn)3), satisfies TrJ((AX)∗3)=0, then where δ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension ?9 in the case .  相似文献   

14.
Summary. We study the role of preconditioning strategies recently developed for coercive problems in connection with a two-step iterative method, based on the Hermitian skew-Hermitian splitting (HSS) of the coefficient matrix, proposed by Bai, Golub and Ng for the solution of nonsymmetric linear systems whose real part is coercive. As a model problem we consider Finite Differences (FD) matrix sequences {An(a,p)}n discretizing the elliptic (convection-diffusion) problem with being a plurirectangle of Rd with a(x) being a uniformly positive function and p(x) denoting the Reynolds function: here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes. More precisely, in connection with preconditioned HSS/GMRES like methods, we consider the preconditioning sequence {Pn(a)}n, Pn(a):= Dn1/2(a)An(1,0) Dn1/2(a) where Dn(a) is the suitably scaled main diagonal of An(a,0). If a(x) is positive and regular enough, then the preconditioned sequence shows a strong clustering at unity so that the sequence {Pn(a)}n turns out to be a superlinear preconditioning sequence for {An(a,0)}n where An(a,0) represents a good approximation of Re(An(a,p)) namely the real part of An(a,p). The computational interest is due to the fact that the preconditioned HSS method has a convergence behavior depending on the spectral properties of {Pn-1(a)Re(An(a,p))}n {Pn-1(a)An(a,0)}n: therefore the solution of a linear system with coefficient matrix An(a,p) is reduced to computations involving diagonals and to the use of fast Poisson solvers for {An(1,0)}n.Some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques.Mathematics Subject Classification (1991): 65F10, 65N22, 15A18, 15A12, 47B65  相似文献   

15.
Let K be a field of characteristic 0 and let (K*)n denote the n-fold Cartesian product of K*, endowed with coordinatewise multiplication. Let Γ be a subgroup of (K*)n of finite rank. We consider equations (*) a1x1 + … + anxn = 1 in x = (x1xn)Γ, where a = (a1,an)(K*)n. Two tuples a, b(K*)n are called Γ-equivalent if there is a uΓ such that b = u · a. Gy?ry and the author [Compositio Math. 66 (1988) 329-354] showed that for all but finitely many Γ-equivalence classes of tuples a(K*)n, the set of solutions of (*) is contained in the union of not more than 2(n+1! proper linear subspaces of Kn. Later, this was improved by the author [J. reine angew. Math. 432 (1992) 177-217] to (n!)2n+2. In the present paper we will show that for all but finitely many Γ-equivalence classes of tuples of coefficients, the set of non-degenerate solutions of (*) (i.e., with non-vanishing subsums) is contained in the union of not more than 2n proper linear subspaces of Kn. Further we give an example showing that 2n cannot be replaced by a quantity smaller than n.  相似文献   

16.
A linear differential operator P(x, D) = P(x1,... x n , D1,..., D n ) = ∑αγα(x)Dα with coefficients γα(x) defined in E n is called formally almost hypoelliptic in E n if all the derivatives DνξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in En. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.  相似文献   

17.
Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx 1,x 2,…,x n andx*1,x*2,…,x* n?1 denote the roots ofL n (α) (x) andL n (α)′ (x) respectively and putx*0=0. In this paper we prove the following theorem: Ify 0,y 1,…,y n ?1 andy 1 ,…,y n are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ .  相似文献   

18.
19.
The sequence of Jacobi polynomials \(\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }\) is orthogonal on (??1,1) with respect to the weight function (1 ? x)α(1 + x)β provided α > ??1,β > ??1. When the parameters α and β lie in the narrow range ??2 < α, β < ??1, the sequence of Jacobi polynomials \(\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }\) is quasi-orthogonal of order 2 with respect to the weight function (1 ? x)α+?1(1 + x)β+?1 and each polynomial of degree n,n ≥?2, in such a Jacobi sequence has n real zeros. We show that any sequence of Jacobi polynomials \(\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }\) with ??2 < α, β < ??1, cannot be orthogonal with respect to any positive measure by proving that the zeros of Pn??1(α,β) do not interlace with the zeros of Pn(α,β) for any \(n \in \mathbb {N},\)n ≥?2, and any α,β lying in the range ??2 < α, β < ??1. We also investigate interlacing properties satisfied by the zeros of equal degree Jacobi polynomials Pn(α,β),Pn(α,β+?1) and Pn(α+?1,β+?1) where ??2 < α, β < ??1. Upper and lower bounds for the extreme zeros of quasi-orthogonal order 2 Jacobi polynomials Pn(α,β) with ??2 < α, β < ??1 are derived.  相似文献   

20.
In the space A (θ) of all one-valued functions f(z) analytic in an arbitrary region G ? ? (0 ∈ G) with the topology of compact convergence, we establish necessary and sufficient conditions for the equivalence of the operators L 1 n z n Δ n + ... + α1 zΔ+α0 E and L 2= z n a n (z n + ... + za 1(z)Δ+a 0(z)E, where δ: (Δ?)(z)=(f(z)-?(0))/z is the Pommier operator in A(G), n ∈ ?, α n ∈ ?, a k (z) ∈ A(G), 0≤kn, and the following condition is satisfied: Σ j=s n?1 α j+1 ∈ 0, s=0,1,...,n?1. We also prove that the operators z s+1Δ+β(z)E, β(z) ∈ A R , s ∈ ?, and z s+1 are equivalent in the spaces A R, 0?R?-∞, if and only if β(z) = 0.  相似文献   

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