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1.
Let D be an increasing sequence of positive integers, and consider the divisor functions: d(n, D) =∑d|n,d∈D,d≤√n1, d2(n,D)=∑[d,δ]|n,d,δ∈D,[d,δ]≤√n1, where [d,δ]=1.c.m.(d,δ). A probabilistic argument is introduced to evaluate the series ∑n=1^∞and(n,D) and ∑n=1^∞and2(n,D).  相似文献   

2.
We establish the existence of fundamental solutions for the anisotropic porous medium equation, ut = ∑n i=1(u^mi)xixi in R^n × (O,∞), where m1,m2,..., and mn, are positive constants satisfying min1≤i≤n{mi}≤ 1, ∑i^n=1 mi 〉 n - 2, and max1≤i≤n{mi} ≤1/n(2 + ∑i^n=1 mi).  相似文献   

3.
Let , be a family of compatible couples of Lp-spaces. We show that, given a countably incomplete ultrafilter in , the ultraproduct of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type , an intermediate K?the space between and being a purely atomic measure space, and a K?the function space K3) defined on some purely non atomic measure space (Ω3, ν3) in such a way that Ω2 ∪ Ω3 ≠∅. The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.  相似文献   

4.
This paper concerns the problem of average σ-width of Sobolev-Wiener classes W^rpq(R^d),W^rpq(M,R^d),and Besov-Wiener classes S^rpqθb(R^d).S^rpqθB(R^d),S^rpqθb(M,R^d),S^rpqθB(R^d)in the metric Lq(R^d) for 1≤q≤p≤∞.The weak asymptotic results concerning the average linear widths,the average Bernstein widths and the infinite-dimensional Gel‘fand widths are obtained,respectively.  相似文献   

5.
Jackson's Theorem on Bounded Symmetric Domains   总被引:1,自引:0,他引:1  
Polynomial approximation is studied on bounded symmetric domain Ω in C^n for holomorphic function spaces X such as Bloch-type spaces, Bergman-type spaces, Hardy spaces, Ω algebra and Lipschitz space. We extend the classical Jackson's theorem to several complex variables:Eκ(f,X)≤ω(1/k,f,X), where Eκ(f,X) is the deviation of the best approximation of f ∈X by polynomials of degree at most k with respect to the X-metric and ω(1/k,f,X) is the corresponding modulus of continuity.  相似文献   

6.
Abstract With Littlewood–Paley analysis, Peetre and Triebel classified, systematically, almost all the usual function spaces into two classes of spaces: Besov spaces and Triebel–Lizorkin spaces ; but the structure of dual spaces of is very different from that of Besov spaces or that of Triebel–Lizorkin spaces, and their structure cannot be analysed easily in the Littlewood–Paley analysis. Our main goal is to characterize in tent spaces with wavelets. By the way, some applications are given: (i) Triebel–Lizorkin spaces for p = ∞ defined by Littlewood–Paley analysis cannot serve as the dual spaces of Triebel–Lizorkin spaces for p = 1; (ii) Some inclusion relations among these above spaces and some relations among and L 1 are studied. Supported by NNSF of China (Grant No. 10001027)  相似文献   

7.
When A ∈ B(H) and B ∈ B(K) are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = ( A0 CB). In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm (lower semi-Fredholm, or Fredholm) operator for some C ∈B(K,H). In addition, let σSF+(A) = {λ ∈ C : A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B(H) and let σrsF- (A) = {λ∈ C : A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±(A) U σSF±(B) to σSF±(Mc) is accomplished by removing certain open subsets of σSF-(A) ∩σSF+ (B) from the former, that is, there is an equality σSF±(A) ∪σSF± (B) = σSF± (Mc) ∪& where L is the union of certain of the holes in σSF±(Mc) which ilappen to be subsets of σSF- (A) A σSF+ (B). Weyl's theorem and Browder's theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl's theorem, Browder's theorem, a-Weyl's theorem and a-Browder's theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.  相似文献   

8.
For an integer m ≥ 4, we define a set of 2[m/2] × 2[m/2] matrices γj (m), (j = 0, 1,..., m - 1) which satisfy γj (m)γk (m) +γk (m)γj (m) = 2ηjk (m)I[m/2], where (ηjk (m)) 0≤j,k≤m-1 is a diagonal matrix, the first diagonal element of which is 1 and the others are -1, I[m/2] is a 2[m/1] × 2[m/2] identity matrix with [m/2] being the integer part of m/2. For m = 4 and 5, the representation (m) of the Lorentz Spin group is known. For m≥ 6, we prove that (i) when m = 2n, (n ≥ 3), (m) is the group generated by the set of matrices {T|T=1/√ξ((I+k) 0 + 0 I-K) ( U 0 0 U), (ii) when m = 2n + 1 (n≥ 3), (m) is generated by the set of matrices {T|T=1/√ξ(I -k^- k I)U,U∈ (m-1),ξ=1-m-2 ∑k,j=0 ηkja^k a^j〉0, K=i[m-3 ∑j=0 a^j γj(m-2)+a^(m-2) In],K^-=i[m-3∑j=0 a^j γj(m-2)-a^(m-2) In]}  相似文献   

9.
In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C*-ternary ring homomorphisms associated to the Trif functional equation
  相似文献   

10.
In this paper we consider the effective reducibility of the following linear differentialequation: x = (A ∈Q(t,∈))x, |∈| ≤ ∈0, where A is a constant matrix, Q(t,e) is quasiperiodic in t, and e is a small perturbation parameter. We prove that if the eigenvalues of A and the basic frequencies of Q satisfy some non-resonant conditions, the linear differential equation can be reduced to y = (A^*(∈) R^*(t, ∈))y, |∈| ≤ ∈o, where R^* is exponentially small in ∈.  相似文献   

11.
In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai+kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞) is an absolutely solutely summable sequence of real numbers.  相似文献   

12.
Let{ψμ} be an orthonormal wavelet of L^2(R^d) and the support of a whole of its Fourier transform be Uμsupp{ψμ}=Пi=1^d[Ai, Di]-Пi=1^d(Bi, Ci), Ai≤Bi≤Ci≤Di. Under the weakest condition that each │ψμ│, is continuous for ω ∈ δ(Пi=1^d[Ai, Di]), a characterization of the above support of a whole is given.  相似文献   

13.
The aim of this paper is to study the stability problem of the generalized sine functional equations as follows:
g(x)f(y)=f(x+y/2)^2-f(x-y/2)^2 f(x)g(y)=f(x+y/2)^2-f(x-y/2)^2,g(x)g(y)=f(x+y/2)^-f(x-y/2)^2
Namely, we have generalized the Hyers Ulam stability of the (pexiderized) sine functional equation.  相似文献   

14.
In this paper, we prove that the weak solutions u∈Wloc^1, p (Ω) (1 〈p〈∞) of the following equation with vanishing mean oscillation coefficients A(x): -div[(A(x)△↓u·△↓u)p-2/2 A(x)△↓u+│F(x)│^p-2 F(x)]=B(x, u, △↓u), belong to Wloc^1, q (Ω)(A↓q∈(p, ∞), provided F ∈ Lloc^q(Ω) and B(x, u, h) satisfies proper growth conditions where Ω ∪→R^N(N≥2) is a bounded open set, A(x)=(A^ij(x)) N×N is a symmetric matrix function.  相似文献   

15.
A Note on Certain Block Spaces on the Unit Sphere   总被引:1,自引:0,他引:1  
In this note, we clarify a relation between block spaces and the Hardy space. We obtain Bq^0.v belong to H^1(S^n-1)+L(ln+L)^1+v(s^n-1),v〉-1,q〉1,Furthermore,if v≥ 0, q 〉 1. we verify that block spaces Rq^0.v(S^n-1)are proper subspaces of H1 (S^n- 1),  相似文献   

16.
Let be a unit sphere of the d–dimensional Euclidean space ℝ d and let (0 < p ≤ 1) denote the real Hardy space on For 0 < p ≤ 1 and let E j (f,H p ) (j = 0, 1, ...) be the best approximation of f by spherical polynomials of degree less than or equal to j, in the space Given a distribution f on its Cesàro mean of order δ > –1 is denoted by For 0 < p ≤ 1, it is known that is the critical index for the uniform summability of in the metric H p . In this paper, the following result is proved: Theorem Let 0<p<1 and Then for
where A N (f)≈B N (f) means that there’s a positive constant C, independent of N and f, such that
In the case d = 2, this result was proved by Belinskii in 1996. The authors are partially supported by NNSF of China under the grant # 10071007  相似文献   

17.
In this paper, we give a characterization of tori S^1 ( √ nr+2-n/nr)×S^n-1(√ n-2/nr) and S^m ( √n/m ) ×S^n-m (√n-m/n). Our result extends the result due to Li (1996)on the condition that M is an n-dimensional complete hypersurface in Sn+1 with two distinct principal curvatures. Keywords principal curvature, Clifford torus, Gauss equations  相似文献   

18.
Let{(t);t∈R_ ~N}be a d-dimensional N-parameter generalized Brownian sheet.Necessaryand sufficient conditions for a compact set E×F to be a polar set for(t,(t))are proved.It is also provedthat if 2N≤αd,then for any compact set ER_>~N,d-2/2 Dim E≤inf{dimF:F ∈ B(R~d),P{(E)∩F≠φ}>0}≤d-2/β DimE,and if 2N>αd,then for any compact set FR~d\{0},α/2(d-DimF)≤inf{dimE:E∈B(R_>~N),P{(E)∩F≠φ}>0}≤β/2(d-DimF),where B(R~d)and B(R_>~N)denote the Borel σ-algebra in R~d and in R_>~N respectively,dim and Dim are Hausdorffdimension and Packing dimension respectively.  相似文献   

19.
We study the scaling limit for a catalytic branching particle system whose particles perform random walks on Z and can branch at 0 only. Varying the initial (finite) number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are n^β particles and consider the scaled process Zt^n(·) = Znt(√n·), where Zt is the measure-valued process 1 and to a representing the original particle system. We prove that Ztn converges to 0 when β 〈1/4 and to a nondegenerate discrete distribution when β=1/4.In addition,if 1/4〈β〈1/2 then n-^(2β-1/2)Zt^n converges to a random limit,while if β 〉21then n^-βZtn converges to a deterministic limit.  相似文献   

20.
This paper, by using of windowed Fourier transform (WFT), gives a family of embedding operators Tn:L^2(R)→L^2(C,e^-|z|^2/2dzd-↑z/4πi), s.t. TnL^2(R) lontain in L^2(C,e^-|z|^2/2dzd-↑z/4πi) are reproducing subspaces (n=0, Bargmann Space); and gives a reproducing kernel and an orthonormal basis (ONB) of TnL^2(R), Furthermore, it shows the orthogonal spaces decomposition of L^2(C,e^-|z|^2/2dzd-↑z/4πi). Finally, by using the preceding results, it shows the eigenvalues and eigenfunctions of a class of localization operators associated with WFT, which extends the result of Daubechies in [1] and [6].  相似文献   

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