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1.
The distributionF(x +, −r) Inx+ andF(x , −s) corresponding to the functionsx + −r lnx+ andx −s respectively are defined by the equations
(1) and
(2) whereH(x) denotes the Heaviside function. In this paper, using the concept of the neutrix limit due to J G van der Corput [1], we evaluate the non-commutative neutrix product of distributionsF(x +, −r) lnx+ andF(x , −s). The formulae for the neutrix productsF(x +, −r) lnx + ox −s, x+ −r lnx+ ox −s andx −s o F(x+, −r) lnx+ are also given forr, s = 1, 2, ...  相似文献   

2.
Conditions are established when the collocation polynomials Pm(x) and PM(x), m M, constructed respectively using the system of nodes xj of multiplicities aj 1, j = O,, n, and the system of nodes x-r,,xo,,xn,,xn+r1, r O, r1 O, of multiplicities a-r,,(ao + yo),,(an + yn),,an+r1, aj + yj 1, are two sided-approximations of the function f on the intervals , xj[, j = O,...,n + 1, and on unions of any number of these intervals. In this case, the polynomials Pm (x), PM (l) (x) with l aj are two-sided approximations of the function f(1) in the neighborhood of the node xj and the integrals of the polynomials Pm(x), PM(x) over Dj are two-sided approximations of the integral of the function f (over Dj). If the multiplicities aj aj + yj of the nodes xj are even, then this is also true for integrals over the set j= µ k Dj µ 1, k n. It is shown that noncollocation polynomials (Fourier polynomials, etc.) do not have these properties.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 31–37, 1989.  相似文献   

3.
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   

4.
Let f(z) be a holomorphic Hecke eigencuspform of even weight k with respect to SL(2, Z) and let L(s, sym 2 f) = ∑ n=1 cnn−s, Re s > 1, be the symmetric square L-function associated with f. Represent the Riesz mean (ρ ≥ 0)
as the sum of the “residue function” Γ(ρ+1)−1 Ł(0, sym2f)xρ and the “error term”
. Using the Voronoi formula for Δρ(x;sym 2f), obtained earlier (see Zap. Nauchn. Semin. POMI. 314, 247–256 (2004)), the integral
is estimated. In this way, an asymptotics for 0 < ρ ≤ 1 and an upper bound for ρ = 0 are obtained. Also the existence of a limiting distribution for the function
, and, as a corollary, for the function
, is established. Bibliography: 12 titles. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 274–286.  相似文献   

5.
Let M be either the space of 2π-periodic functions Lp, where 1 ≤ p < ∞, or C; let ωr(f, h) be the continuity modulus of order r of the function f, and let
, where
, be the generalized Jackson-Vallée-Poussin integral. Denote
. The paper studies the quantity Km(f − Dn,r,l(f)). The general results obtained are applicable to other approximation methods. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 52–69.  相似文献   

6.
Given a unital C*-algebra and a right C*-module over , we consider the problem of finding short smooth curves in the sphere = {x ∈ : 〈x, x〉 = 1}. Curves in are measured considering the Finsler metric which consists of the norm of at each tangent space of . The initial value problem is solved, for the case when is a von Neumann algebra and is selfdual: for any element x 0 ∈ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ (x 0), Z ∈ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection Ix 0x 0, if the algebra f 0 f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.   相似文献   

7.
It is proved in this paper that the lowest upper bound of the number of the isolated zeros of the Abelian integral
is two for h∈(−1/12, 0), where Γh is the compact component of H(x, y)=(1/2) y2+(1/3) x3+(1/4) x4=h, and α, β, γ are arbitrary constants. Entrata in Redazione il 4 dicembre 1997. Partially supported by NSF and DPF of China.  相似文献   

8.
Summary We consider the functional equationf[x 1,x 2,, x n ] =h(x 1 + +x n ) (x 1,,x n K, x j x k forj k), (D) wheref[x 1,x 2,,x n ] denotes the (n – 1)-st divided difference off and prove Theorem. Let n be an integer, n 2, let K be a field, char(K) 2, with # K 8(n – 2) + 2. Let, furthermore, f, h: K K be functions. Then we have that f, h fulfil (D) if, and only if, there are constants aj K, 0 j n (a := an, b := an – 1) such thatf = ax n +bx n – 1 + +a 0 and h = ax + b.  相似文献   

9.
Summary While looking for solutions of some functional equations and systems of functional equations introduced by S. Midura and their generalizations, we came across the problem of solving the equationg(ax + by) = Ag(x) + Bg(y) + L(x, y) (1) in the class of functions mapping a non-empty subsetP of a linear spaceX over a commutative fieldK, satisfying the conditionaP + bP P, into a linear spaceY over a commutative fieldF, whereL: X × X Y is biadditive,a, b K\{0}, andA, B F\{0}. Theorem.Suppose that K is either R or C, F is of characteristic zero, there exist A 1,A 2,B 1,B 2, F\ {0}with L(ax, y) = A 1 L(x, y), L(x, ay) = A 2 L(x, y), L(bx, y) = B 1 L(x, y), and L(x, by) = B 2 L(x, y) for x, y X, and P has a non-empty convex and algebraically open subset. Then the functional equation (1)has a solution in the class of functions g: P Y iff the following two conditions hold: L(x, y) = L(y, x) for x, y X, (2)if L 0, then A 1 =A 2,B 1 =B 2,A = A 1 2 ,and B = B 1 2 . (3) Furthermore, if conditions (2)and (3)are valid, then a function g: P Y satisfies the equation (1)iff there exist a y 0 Y and an additive function h: X Y such that if A + B 1, then y 0 = 0;h(ax) = Ah(x), h(bx) =Bh(x) for x X; g(x) = h(x) + y 0 + 1/2A 1 -1 B 1 -1 L(x, x)for x P.  相似文献   

10.
We consider the ordinary differential operator L generated on [0, 1] by the differential expression
and n linearly independent, homogeneous boundary conditions at the endpoints. We assume that the coefficients p k (x) are Lebesgue-integrable complex functions. If the boundary conditions are Birkhoff regular, then the Green function G(λ), being the kernel of the operator (Lλ)−1, admits the asymptotic estimate (for sufficiently large |λ| > c 0)
, where M = M(c 0) is a certain constant. In the present paper, we prove the converse assertion: the fulfillment of this estimate on some rays implies the regularity of the operator L. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 231–239, 2006.  相似文献   

11.
The asymptotic expansions are studied for the vorticity to 2D incompressible Euler equations with-initial vorticity , where ϕ0(x) satisfies |d ϕ0(x)|≠0 on the support of and is sufficiently smooth and with compact support in ℝ2 (resp. ℝ2×T) The limit,v(t,x), of the corresponding velocity fields {v ɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω0(x). Moreover, (ℤ2)) for all 1≽p∞, where and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively.  相似文献   

12.
Let X be a complex Banach space and H1 the usual Hardy space. Various properties of operators L1/H 0 1 X and, mainly, H1X are considered, e.g. being weakly compact, Riesz representable, Dunford-Pettis. Connections with RNP resp. aRNP and with the validity of the equation are also studied, the latter space being an X-valued Hardy space. Whereas results for operators L1/H 0 1 X closely resemble well-known theorems about operators L1 X, this is not the case for operators H1X. E.g., for most classical Banach spaces X it isnot true that (canonically).  相似文献   

13.
For γ≥1 we consider the solution u=u(x) of the Dirichlet boundary value problem Δu + u^-γ=0 in Ω, u=0 on δΩ. For γ= 1 we find the estimate u(x)=p(δ(x))[1+A(x)(log 1/δ(x)^-6], where p(r) ≈ r r√2 log(1/r) near r = 0,δ(x) denotes the distance from x to δΩ, 0 〈ε 〈 1/2, and A(x) is a bounded function. For 1 〈 γ 〈 3 we find u(x)=(γ+1/√2(γ-1)δ(x))^2/γ+[1+A(x)(δ(x))2γ-1/γ+1] For γ3= we prove that u(x)=(2δ(x))^1/2[1+A(x)δ(x)log 1/δ(x)]  相似文献   

14.
LetW (x) be a function nonnegative inR, positive on a set of positive measure, and such that all power moments ofW 2(x) are finite. Let {p n (W 2;x)} 0 denote the sequence of orthonormal polynomials with respect to the weightW 2(x), and let {A n } 1 and {B n } 1 denote the coefficients in the recurrence relation
  相似文献   

15.
Let f(x, y) be a periodic function defined on the region D
with period 2π for each variable. If f(x, y) ∈ C p (D), i.e., f(x, y) has continuous partial derivatives of order p on D, then we denote by ω α,β(ρ) the modulus of continuity of the function
and write
For p = 0, we write simply C(D) and ω(ρ) instead of C 0(D) and ω 0(ρ). Let T(x,y) be a trigonometrical polynomial written in the complex form
We consider R = max(m 2 + n 2)1/2 as the degree of T(x, y), and write T R(x, y) for the trigonometrical polynomial of degree ⩾ R. Our main purpose is to find the trigonometrical polynomial T R(x, y) for a given f(x, y) of a certain class of functions such that
attains the same order of accuracy as the best approximation of f(x, y). Let the Fourier series of f(x, y) ∈ C(D) be
and let
Our results are as follows Theorem 1 Let f(x, y) ∈ C p(D (p = 0, 1) and
Then
holds uniformly on D. If we consider the circular mean of the Riesz sum S R δ (x, y) ≡ S R δ (x, y; f):
then we have the following Theorem 2 If f(x, y) ∈ C p (D) and ω p(ρ) = O(ρ α (0 < α ⩾ 1; p = 0, 1), then
holds uniformly on D, where λ 0 is a positive root of the Bessel function J 0(x) It should be noted that either
or
implies that f(x, y) ≡ const. Now we consider the following trigonometrical polynomial
Then we have Theorem 3 If f(x, y) ∈ C p(D), then uniformly on D,
Theorems 1 and 2 include the results of Chandrasekharan and Minakshisundarm, and Theorem 3 is a generalization of a theorem of Zygmund, which can be extended to the multiple case as follows Theorem 3′ Let f(x 1, ..., x n) ≡ f(P) ∈ C p and let
where
and
being the Fourier coefficients of f(P). Then
holds uniformly. __________ Translated from Acta Scientiarum Naturalium Universitatis Pekinensis, 1956, (4): 411–428 by PENG Lizhong.  相似文献   

16.
Let ξn −1 < ξn −2 < ξn − 2 < ... < ξ1 be the zeros of the the (n−1)-th Legendre polynomial Pn−1(x) and −1=xn<xn−1<...<x1=1, the zeros of the polynomial . By the theory of the inverse Pal-Type interpolation, for a function f(x)∈C [−1,1] 1 , there exists a unique polynomial Rn(x) of degree 2n−2 (if n is even) satisfying conditions Rn(f, ξk) = f (εk) (1 ⩽ k ⩽ n −1); R1 n(f,xk)=f1(xk)(1≤k≤n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f, x)} (n is even) and the main result of this paper is that if f∈C [1,1] r , r≥2, n≥r+2, and n is even then |R1 n(f,x)−f1(x)|=0(1)|Wn(x)|h(x)·n3−r·E2n−r−3(f(r)) holds uniformly for all x∈[−1,1], where .  相似文献   

17.
We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ N . Our attention is focused on two cases when , where m(x) = max{p 1(x), p 2(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(Nm(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.  相似文献   

18.
The paper is concerned with bounds for integrals of the type
, involving Jacobi polynomials p n (α,β) and Jacobi weights w (a,b) depending on α,β, a, b > −1, where the subsets U k (x) ⊂ [−1, 1] located around x and are given by with . The functions to be integrated will also be of the type on the domain [−1,1] t/ U k (x). This approach uses estimates of Jacobi polynomials modified Jacobi weights initiated by Totik and Lubinsky in [1]. Various bounds for integrals involving Jacobi weights will be derived. The results of the present paper form the basis of the proof of the uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms in [3].   相似文献   

19.
For an entire Dirichlet series , sufficient conditions on the exponents are established such that the following relations hold outside a set of finite measure asx→+∞:
, where ψ(x) is a function increasing to +∞ and such thatx≤ψ(x)≤e x (x≥0). Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 282–292, August, 1999  相似文献   

20.
Consider the discrete cube Ω={0,1} N , provided with the uniform probabilityP. We denote byd(x, A) the Hamming distance of a pointx of Ω and a subsetA of Ω. We define the influenceI(A) of theith coordinate onA as follows. Forx in Ω, consider the pointT i (x) obtained by changing the value of theith coordinate. Then We prove that we always have Since it is easy to see that , this recovers the well known fact that ∫Ω d(x, A)dP(x) is at most of order whenP(A)≥1/2. The new information is that ∫Ω d(x, A)dP(x) can be of order only ifA reassembles the Hamming ball {x; ∑1≤N x i N/2}.  相似文献   

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