Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

Topology and Smooth Structure for Pseudoframes

Given a closed subspace ${\mathcal{S}}Given a closed subspace S{\mathcal{S}} of a Hilbert space H{\mathcal{H}}, we study the sets F_S{\mathcal{F}_\mathcal{S}} of pseudo-frames, CF_S{\mathcal{C}\mathcal{F}_\mathcal{S}} of commutative pseudo-frames and \mathfrakX_S{\tiny{\mathfrak{X}}_{\mathcal{S}}} of dual frames for S{\mathcal{S}}, via the (well known) one to one correspondence which assigns a pair of operators (F, H) to a frame pair ({f_n}_{n ? \mathbbN},{h_n}_{n ? \mathbbN}){(\{f_n\}_{n\in\mathbb{N}},\{h_n\}_{n\in\mathbb{N}})},

F:l2? H,     F({cn}n ? \mathbbN )=?n cn fn,F:\ell^2\to\,\mathcal{H}, \quad F\left(\{c_n\}_{n\in\mathbb{N}} \right)=\sum_n c_n f_n,      3. Polynomial solutions of quasi-homogeneous partial differential equations      Luo Xuebo  ZHENG Zhujun 《中国科学A辑(英文版)》2001,44(9):1148-1155 By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation given by ( a1, …, an). Assume that either a1, …, an are positive rational numbers or for some Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝn must be infinite      4. Multilinear Singular Integrals with Rough Kernel   总被引：9，自引：0，他引：9   ShanZhenLU HuoXiongWU PuZHANG 《数学学报(英文版)》2003,19(1):51-62 For a class of multilinear singular integral operators T A , where R m (A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m − 1 in is homogeneous of degree zero, the authors prove that T A is bounded from L p (ℝ n ) to and from L 1(ℝ n ) to L n/(n−β),∞(ℝ n ) with the bound And if Ω has vanishing moments of order m − 1 and satisfies some kinds of Dini regularity otherwise, then T A is also bounded from L p (ℝ n ) to with the bound Supported by the National 973 Project (G1990751) and SEDF of China (20010027002)      5. The Bergman kernel function of some Reinhardt domains (II)      Sheng Gong  Xue&#;an Zheng 《中国科学A辑(英文版)》2000,43(5):458-469       6. Composition operators on the Lipschitz space in polydiscs      周泽华 《中国科学A辑(英文版)》2003,46(1):33-38 Let Un be the unit polydisc of Cn and φ= (φ1,...,φn? a holomorphic self-map of Un. Let 0≤α< 1. This paper shows that the composition operator Cφ, is bounded on the Lipschitz space Lipa(Un) if and only if there exists M > 0 such thatfor z∈Un. Moreover Cφ is compact on Lipa(Un) if and only if Cφ is bounded on Lipa(Un) and for every ε > 0, there exists a δ > 0 such that whenever dist(φ(z),σUn) <δ      7. Sharp Lipschitz constant of bi-Lipschitz automorphism on Cantor set      Ying Xiong  LiSha Wang  LiFeng Xi 《中国科学A辑(英文版)》2009,52(4):709-719 Suppose C r = (r C r ) ∪ (r C r + 1 − r) is a self-similar set with r ∈ (0, 1/2), and Aut(C r ) is the set of all bi-Lipschitz automorphisms on C r . This paper proves that there exists f* ∈ Aut(C r ) such that where and blip(g) = max(lip(g), lip(g −1)). This work was supported by National Natural Science Foundation of China (Grant Nos. 10671180, 10571140, 10571063, 10631040, 11071164) and Morningside Center of Mathematics      8. The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains      Yu-can Zhu  Ming-sheng Liu 《中国科学A辑(英文版)》2007,50(12):1781-1794 The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1| (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α ＜ 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn ∑nj=1|zj|pj ＜ 1} (pj ＞ 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α ＜ 1 and some suitable constants βj,γj,pj.      9. Singular Integrals and Potential Theory      Nicolas Th. Varopoulos 《Milan Journal of Mathematics》2007,75(1):1-60 In this paper I consider a class of non-standard singular integrals motivated by potential theoretic and probabilistic considerations. The probabilistic applications, which are by far the most interesting part of this circle of ideas, are only outlined in Section 1.5: They give the best approximation of the solution of the classical Dirichlet problem in a Lipschitz domain by the corresponding solution by finite differences. The potential theoretic estimate needed for this gives rise to a natural duality between the L p functions on the boundary ∂Ω and a class of functions A on Ω that was first considered by Dahlberg. The actual duality is given by ∫Ω S f(x)A(x)dx = (f, A) where S f(x) = ∫∂Ω |x − y|1−n f(y)dy is the Newtonian potential. We can identify the upper half Lipschitz space with in the obvious way and express for an appropriate kernel K. It is the boundedness properties of the above (for , ) that is the essential part of this work. This relates with more classical (but still “rough”) singular integrals that have been considered by Christ and Journé. Lecture held in the Seminario Matematico e Fisico on March 14, 2005 Received: April 2007      10. On the strong law of large numbers and additive functions      István Berkes  Wolfgang Müller  Michel Weber 《Periodica Mathematica Hungarica》2011,62(1):1-12 Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f, we prove the following weighted strong law of large numbers: if X,X 1,X 2, … is any sequence of integrable i.i.d. random variables, then $ \mathop {\lim }\limits_{N \to \infty } \frac{{\sum\nolimits_{n = 1}^N {f(n)X_n } }} {{\sum\nolimits_{n = 1}^N {f(n)} }} = \mathbb{E}Xa.s. $ \mathop {\lim }\limits_{N \to \infty } \frac{{\sum\nolimits_{n = 1}^N {f(n)X_n } }} {{\sum\nolimits_{n = 1}^N {f(n)} }} = \mathbb{E}Xa.s.       11. From Chaos to Global Convergence      Giovanni Di Lena  Davide Franco  Mario Martelli  Basilio Messano 《Mediterranean Journal of Mathematics》2011,8(4):473-489 The main purpose of this paper is to investigate dynamical systems F : \mathbbR2 ? \mathbbR2{F : \mathbb{R}^2 \rightarrow \mathbb{R}^2} of the form F(x, y) = (f(x, y), x). We assume that f : \mathbbR2 ? \mathbbR{f : \mathbb{R}^2 \rightarrow \mathbb{R}} is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x0 = (x 0, y 0), such that the orbit O(x0) = {x0, x1 = F(x0), x2 = F(x1), . . . }, O({\rm{x}}_0) = \{{\rm{x}}_0, {\rm{x}}_1 = F({\rm{x}}_0), {\rm{x}}_2 = F({\rm{x}}_1), . . . \},      12. The unitary connections on the complex Grassmann manifold      Qikeng Lu 《中国科学A辑(英文版)》1998,41(12):1248-1254 In the complex Grassmann manifold ℱ(m,n), the space of complexn-planes passes through the origin of Cm+n; the local coordinate of the space can be arranged into anm ×n matrixZ. It is proved that is a U(m)-connection of ℱ(m,n) and its curvature form satisfies the Yang-Mills equation. Moreover, is an (Sum)-connection and its curvature form satisfies the Yang-Mills equation. Project partially supported by the National Natural Science Foundation of China (Grant No. 19631010) and Fundamental Research Bureau of CAS.      13. Invariant Mean—Value Property and M—Harmonicity in the Unit Ball of R^n      CongWenLIU JiHuaiSHI 《数学学报(英文版)》2003,19(1):187-200 In 1993,Ahern,Flores and Rudin showed that,if f is integrable over the unit ball BC^n of C^n and satisfies∫BC^nfoφdv=f（φ（0)) for every φ∈Aut(BC^n),then f is M-harmonic if and only if n≤11.The present paper is about an analogous question in the context of the unit ball Bn of R^n as well as in the weighted setting.      14. The Beckman-Quarles theorem for mappings from ℝ2 to $${\mathbb{F}}^2 $$ , where F is a subfield of a commutative field extending ℝ, where F is a subfield of a commutative field extending ℝ      A.?TyszkaEmail author 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2004,74(1):77-87 Let F be a subfield of a commutative field extending ℝ. Let We say thatf : preserves distanced ≥ 0 if for eachx,y ∈ ℝ ∣x- y∣= d implies ϕ2(f(x),f(y)) = d2 . We prove that each unit-distance preserving mappingf : has a formI o (ρ,ρ), where is a field homomorphism and is an affine mapping with orthogonal linear part.      15. Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum      Onur Yavuz 《Integral Equations and Operator Theory》2007,58(3):433-446 We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ1, λ2, . . . , λ n , and the operators T and r j (T − λ j I)−1 are polynomially bounded, then there exists a nontrivial common invariant subspace for T * and (T − λ j I)*-1.      16. Toeplitz operators related to strongly singular Calderón-Zygmund operators      Lin Yan  Lu Shanzhen 《中国科学A辑(英文版)》2006,49(8):1048-1064 In this paper, the boundedness of Toeplitz operator T b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝn) is discussed from L p(ℝn) to L q(ℝn), , and from L p(ℝn) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ α0 b from L p(ℝn) to L q(ℝn), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L p(ℝn), 1 < p < ∞.      17. Isotropic bodies and Bourgain''''s problem   总被引：1，自引：0，他引：1   HE Binwu & LENG Gangsong Department of Mathematics  Shanghai University  Shanghai  China 《中国科学A辑(英文版)》2005,48(5):666-679 Let K (?) Rn be a convex body of volume 1 whose barycenter is at the origin, LK be the isotropic constant of K. Finding the least upper bound of LK , being called Bourgain's problem, is a well known open problem in the local theory of Banach space. The best estimate known today is LK < cn1/4 log n, recently shown by Bourgain, for an arbitrary convex body in any finite dimension. Utilizing the method of spherical section function, it is proven that if K is a convex body with volume 1 and r1Bn2 (?) K (?) r2Bn2,(r1≥1/2, r2≤(?)/2), then (?) ≤(?) and find the conditions with equality. Further, the geometric characteristic of isotropic bodies is shown.      18. Exponential generating functions and complexity of Lie varieties      V. M. Petrogradsky 《Israel Journal of Mathematics》1999,113(1):323-339 Suppose that % MathType!End!2!1! is a variety of Lie algebras, and letc n( % MathType!End!2!1!) be the dimension of the linear span of all multilinear words onn distinct letters in the free algebraF( % MathType!End!2!1!,X) of the variety % MathType!End!2!1!. We consider an exponential generating function % MathType!End!2!1!, called the complexity function. The complexity function is an entire function of a complex variable provided the variety of Lie algebras is nontrivial. In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the complexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the complexities of multiplicands. The main observation here is thatC( % MathType!End!2!1!),z) behaves like a composition of three functionsC( % MathType!End!2!1!),z), exp(z), andC( % MathType!End!2!1!),z). Partially supported by grant RFFI 96-01-00146; the author is grateful to the University of Bielefeld for hospitality, where he was DAAD-fellow.      19. Strong Approximation by Cesàro Means with Critical Index in the Hardy Spaces H p (\mathbb{S}^{{d - 1}} ) (0 < p ≤ 1)      Feng Dai  Kun Yang Wang 《数学学报(英文版)》2005,21(2):439-448 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      20. Congruence Properties of Binary Partition Functions      Katherine Anders  Melissa Dennison  Jennifer Weber Lansing  Bruce Reznick 《Annals of Combinatorics》2013,17(1):15-26 Let ${\mathcal{A}}$ be a finite subset of ${\mathbb{N}}$ containing 0, and let f (n) denote the number of ways to write n in the form ${\sum \varepsilon _{j}2^{j}}$ , where ${\varepsilon _{j} \epsilon \mathcal{A}}$ . We show that there exists a computable ${T = T (\mathcal{A})}$ so that the sequence (f (n) mod 2) is periodic with period T. Variations and generalizations of this problem are also discussed.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Polynomial solutions of quasi-homogeneous partial differential equations

By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation given by ( a₁, …, a_n). Assume that either a₁, …, a_n are positive rational numbers or for some Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝⁿ must be infinite     

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

Composition operators on the Lipschitz space in polydiscs

Let Un be the unit polydisc of Cn and φ= (φ1,...,φn? a holomorphic self-map of Un. Let 0≤α< 1. This paper shows that the composition operator Cφ, is bounded on the Lipschitz space Lipa(Un) if and only if there exists M > 0 such thatfor z∈Un. Moreover Cφ is compact on Lipa(Un) if and only if Cφ is bounded on Lipa(Un) and for every ε > 0, there exists a δ > 0 such that whenever dist(φ(z),σUn) <δ     

Sharp Lipschitz constant of bi-Lipschitz automorphism on Cantor set

Suppose C _r = (r C _r ) ∪ (r C _r + 1 − r) is a self-similar set with r ∈ (0, 1/2), and Aut(C _r ) is the set of all bi-Lipschitz automorphisms on C _r . This paper proves that there exists f* ∈ Aut(C _r ) such that

The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains

The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1| (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α ＜ 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn ∑nj=1|zj|pj ＜ 1} (pj ＞ 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α ＜ 1 and some suitable constants βj,γj,pj.     

Singular Integrals and Potential Theory

In this paper I consider a class of non-standard singular integrals motivated by potential theoretic and probabilistic considerations. The probabilistic applications, which are by far the most interesting part of this circle of ideas, are only outlined in Section 1.5: They give the best approximation of the solution of the classical Dirichlet problem in a Lipschitz domain by the corresponding solution by finite differences. The potential theoretic estimate needed for this gives rise to a natural duality between the L _p functions on the boundary ∂Ω and a class of functions A on Ω that was first considered by Dahlberg. The actual duality is given by ∫_Ω S f(x)A(x)dx = (f, A) where S f(x) = ∫_∂Ω |x − y|¹⁻ⁿ f(y)dy is the Newtonian potential. We can identify the upper half Lipschitz space with in the obvious way and express for an appropriate kernel K. It is the boundedness properties of the above (for , ) that is the essential part of this work. This relates with more classical (but still “rough”) singular integrals that have been considered by Christ and Journé. Lecture held in the Seminario Matematico e Fisico on March 14, 2005 Received: April 2007     

On the strong law of large numbers and additive functions

Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f, we prove the following weighted strong law of large numbers: if X,X ₁,X ₂, … is any sequence of integrable i.i.d. random variables, then

From Chaos to Global Convergence

The main purpose of this paper is to investigate dynamical systems F : \mathbbR² ? \mathbbR²{F : \mathbb{R}^2 \rightarrow \mathbb{R}^2} of the form F(x, y) = (f(x, y), x). We assume that f : \mathbbR² ? \mathbbR{f : \mathbb{R}^2 \rightarrow \mathbb{R}} is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x₀ = (x ₀, y ₀), such that the orbit

The unitary connections on the complex Grassmann manifold

In the complex Grassmann manifold ℱ(m,n), the space of complexn-planes passes through the origin of C^m+n; the local coordinate of the space can be arranged into anm ×n matrixZ. It is proved that

Invariant Mean—Value Property and M—Harmonicity in the Unit Ball of R^n

In 1993,Ahern,Flores and Rudin showed that,if f is integrable over the unit ball BC^n of C^n and satisfies∫BC^nfoφdv=f（φ（0)) for every φ∈Aut(BC^n),then f is M-harmonic if and only if n≤11.The present paper is about an analogous question in the context of the unit ball Bn of R^n as well as in the weighted setting.     

The Beckman-Quarles theorem for mappings from ℝ2 to $${\mathbb{F}}^2 $$ , where F is a subfield of a commutative field extending ℝ, where F is a subfield of a commutative field extending ℝ

Let F be a subfield of a commutative field extending ℝ. Let We say thatf : preserves distanced ≥ 0 if for eachx,y ∈ ℝ ∣x- y∣= d implies ϕ₂(f(x),f(y)) = d² . We prove that each unit-distance preserving mappingf : has a formI o (ρ,ρ), where is a field homomorphism and is an affine mapping with orthogonal linear part.     

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r _j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded, then there exists a nontrivial common invariant subspace for T ^* and (T − λ _j I)^*-1.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

Isotropic bodies and Bourgain''''s problem

Let K (?) Rn be a convex body of volume 1 whose barycenter is at the origin, LK be the isotropic constant of K. Finding the least upper bound of LK , being called Bourgain's problem, is a well known open problem in the local theory of Banach space. The best estimate known today is LK < cn1/4 log n, recently shown by Bourgain, for an arbitrary convex body in any finite dimension. Utilizing the method of spherical section function, it is proven that if K is a convex body with volume 1 and r1Bn2 (?) K (?) r2Bn2,(r1≥1/2, r2≤(?)/2), then (?) ≤(?) and find the conditions with equality. Further, the geometric characteristic of isotropic bodies is shown.     

Exponential generating functions and complexity of Lie varieties

Suppose that % MathType!End!2!1! is a variety of Lie algebras, and letc _n( % MathType!End!2!1!) be the dimension of the linear span of all multilinear words onn distinct letters in the free algebraF( % MathType!End!2!1!,X) of the variety % MathType!End!2!1!. We consider an exponential generating function % MathType!End!2!1!, called the complexity function. The complexity function is an entire function of a complex variable provided the variety of Lie algebras is nontrivial. In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the complexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the complexities of multiplicands. The main observation here is thatC( % MathType!End!2!1!),z) behaves like a composition of three functionsC( % MathType!End!2!1!),z), exp(z), andC( % MathType!End!2!1!),z). Partially supported by grant RFFI 96-01-00146; the author is grateful to the University of Bielefeld for hospitality, where he was DAAD-fellow.     

Strong Approximation by Cesàro Means with Critical Index in the Hardy Spaces H p (\mathbb{S}^{{d - 1}} ) (0 < p ≤ 1)

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

Congruence Properties of Binary Partition Functions

Let ${\mathcal{A}}$ be a finite subset of ${\mathbb{N}}$ containing 0, and let f (n) denote the number of ways to write n in the form ${\sum \varepsilon _{j}2^{j}}$ , where ${\varepsilon _{j} \epsilon \mathcal{A}}$ . We show that there exists a computable ${T = T (\mathcal{A})}$ so that the sequence (f (n) mod 2) is periodic with period T. Variations and generalizations of this problem are also discussed.