Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

Universally L 1 good sequences with gaps tending to infinity

We construct a sequence (n _k ) such that n _{k + 1} − n _k → ∞ and for any ergodic dynamical system (X, Σ, μ, T) and f ε L ¹(μ) the averages converge to ∝ _X f dμ for μ almost every x. Since the above sequence is of zero Banach density this disproves a conjecture of J. Rosenblatt and M. Wierdl about the nonexistence of such sequences. Research supported by the Hungarian National Foundation for Scientific research T049727.     

Metric diophantine approximation in Julia sets of expanding rational maps

Let T : J → J be an expanding rational map of the Riemann sphere acting on its Julia set J andf : J →R denote a Hölder continuous function satisfyingf(x)?log | T′(x vb for allx in J. Then for any pointz ₀ in J define the set D_{z 0}(f) of “well-approximable” points to be the set of points in J which lie in the Euclidean ball $B(\gamma ,{\text{ exp(}} - \sum {_{i - 0}^{\mathfrak{n} - 1} } f(T^\ell x)))$ for infinitely many pairs (y, n) satisfying T ⁿ (y)=z₀. We prove that the Hausdorff dimension of D_{z 0}(f) is the unique positive numbers(f) satisfying the equation P(T,?s(f).f)=0, where P is the pressure on the Julia set. This result is then shown to have consequences for the limsups of ergodic averages of Hölder continuous functions. We also obtain local counting results which are analogous to the orbital counting results in the theory of Kleinian groups.     

A note on strong summability of two-dimensional Walsh-Fourier series

The paper deals with the strong summability of Marcinkiewicz means with a variable power. Let $$H_n \left( {f,x,y,A_n } \right): = \tfrac{1} {n}\sum\nolimits_{l = 1}^n {\left( {e^{\left. {A_n } \right|\left. {S_{ll} \left( {f,x,y} \right) - f\left( {x,y} \right)} \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } - 1} \right)} .$$ It is shown that if A _n ↑ ∞ arbitrary slowly, there exists f ∈ C(I ²) such that lim _n→∞ H _n (f, 0, 0, A _n ) = +∞. At the same time, for every f ∈ C (I ²) there exists A _n (f) ↑ ∞ such that lim _n→∞ H _n (f, x, y, A _n ) = 0 uniformly on I ².     

On a conjecture of Chowla and Walum

We show the asymptotic behaviour of the mean square of the sum n c√x naPk(x/n),where Pk(x) = Bk({x}) and Bk(x) denotes the Bernoulli polynomial of degree k and c 0 is a real number such that c2 is rational.Our result implies that a conjecture of Chowla and Walum is true on average.     

Arithmetic of semigroups of series in multiplicative systems

We study the arithmetic of a semigroup M_P\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?_{n = 0}^￥ a_nc_n(x)    ( a_n 3 0,?_{n = 0}^￥ a_n = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { c_n }_{n = 0}^￥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup M_P\mathcal{M}_{\mathcal{P}}, analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R ⁿ are true. We describe the class I₀(M_P)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in M_P\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence.     

Uniform estimates for sums of Fourier coefficients of cusp forms

Let k be an even positive integer and f a holomorphic Hecke eigenform of weight k with respect to the full modular group SL(2, ?). Let c _n be the nth coefficient of the symmetric square L-function associated to f. We study the uniform bound for the sum C(x) = Σ _n≦x c _n with respect to the weight k and establish that $$ C(x) = \sum\limits_{n \leqq x} {c_n } \ll x^{\tfrac{3} {5}} (\log x)^{\tfrac{{22}} {5}} + k^{\tfrac{3} {2}} (\log x)^5 $$ . Other similar results are also established.     

A note on uniqueness problem for meromorphic mappings with 2N + 3 hyperplanes

In this paper,a uniqueness theorem for meromorphic mappings partially sharing 2N+3 hyperplanes is proved.For a meromorphic mapping f and a hyperplane H,set E(H,f) = {z|ν(f,H)(z) 0}.Let f and g be two linearly non-degenerate meromorphic mappings and {Hj}j2=N1+ 3be 2N + 3 hyperplanes in general position such that dim f-1(Hi) ∩ f-1(Hj) n-2 for i = j.Assume that E(Hj,f) E(Hj,g) for each j with 1 j 2N +3 and f = g on j2=N1+ 3f-1(Hj).If liminfr→+∞ 2j=N1+ 3N(1f,Hj)(r) j2=N1+ 3N(1g,Hj)(r) NN+1,then f ≡ g.     

Rate of decay of weakly lacunary series

Let function f(z) ≠ 0 be analytic in the unit disk and have sparse nonzero Taylor coefficients. Then the rate of decay of the function f as x → 1 − 0 depends on the rate of sparseness of its nonzero Taylor coefficients. In this paper, we consider the case f(z) = $ \sum\nolimits_{k = 0}^\infty {a_k z^{n_k } } $ \sum\nolimits_{k = 0}^\infty {a_k z^{n_k } } , where n _k ≥ A ₀(k + 2) ^p logb(k + 2).     

Copositive pointwise approximation

We prove that if a functionf ∈C ⁽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 iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k ? 1, there exists a polynomialP _n =P _n (x) of degree ≤n such thatP _n (x) ≥ 0,x ∈I, and |f(x) ?P _n (x)| ≤c(k)ω _k (f;n ^?2 +n ^?1 √1 ?x ²),x ∈I.     

Congruences involving generalized central trinomial coefficients

For integers b and c the generalized central trinomial coefficient Tn(b,c)denotes the coefficient of xnin the expansion of(x2+bx+c)n.Those Tn=Tn(1,1)(n=0,1,2,...)are the usual central trinomial coefficients,and Tn(3,2)coincides with the Delannoy number Dn=n k=0n k n+k k in combinatorics.We investigate congruences involving generalized central trinomial coefficients systematically.Here are some typical results:For each n=1,2,3,...,we have n-1k=0(2k+1)Tk(b,c)2(b2-4c)n-1-k≡0(mod n2)and in particular n2|n-1k=0(2k+1)D2k;if p is an odd prime then p-1k=0T2k≡-1p(mod p)and p-1k=0D2k≡2p(mod p),where(-)denotes the Legendre symbol.We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.     

Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.     

Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Let ? = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d _W (·, ·) associated to the generating set {a, b, a ^?1, b ^?1}. Letting B _n = {x ∈ ?: d _W (x, e _?) ? n} denote the corresponding closed ball of radius n ∈ ?, and writing c = [a, b] = aba ^?1 b ^?1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ? → X satisfies $$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)} $$ . It follows that for every n ∈ ? the bi-Lipschitz distortion of every f: B _n → X is at least a constant multiple of (log n)^1/q , an asymptotically optimal estimate as n → ∞.     

Estimate of a sum of Legendre symbols of polynomials of even degree

Let n≥4 be even, p > (n²?2n)/2 be simple odd, andf(x)=a ₀+a ₁+...+a _nxⁿ be a polynomial with integral coefficients that are not quadratic over the residue field modulo p, (a _n, p)=1. The following inequality is proved: $$\left| {\sum\nolimits_{x = 1}^p {\left( {\frac{{f(x)}}{p}} \right)} } \right| \leqslant (n - 2)\sqrt {p + 1 - \frac{{n(n - 4)}}{4}} + 1.$$      

On Laplace derivative

If f: ? → ? is integrable in a right neighbourhood of x ∈ ? and if there are real numbers α ₀, α ₁, ..., α _n?1 such that the limit lim $$ \mathop {\lim }\limits_{s \to \infty } s^{n + 1} \int_0^\delta {e^{ - st} } \left[ {f(x + t) - \sum\limits_{i = 0}^{n - 1} {\frac{{t^i }} {{i!}}\alpha _i } } \right]dt $$ exists, then this limit is called the right-hand Laplace derivative of f at x of order n and is denoted by LD _n ⁺ f(x). There is a corresponding definition for the left-hand derivative and if they are equal the common value is the Laplace derivative LD _n f(x). In this paper, it is shown that the basic properties of the Peano derivatives are also possessed by this derivative (cf. [5]).     

Integration and Lipschitz functions

The aim of the paper is to prove that every f ∈ L ¹([0,1]) is of the form f = , where j _n,k is the characteristic function of the interval [k- 1 / 2 ⁿ , k / 2 ⁿ ) and Σ _n=0^∞Σ _k=1²ⁿ |a _n,k | is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b _n,k ) _n≧0 ^k=1,...,2n of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).      

Estimates for sums of moduli of blocks in trigonometric Fourier series

We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets A _k ? ? and a sequence of functions λ _k : A _k → ? provide the existence of a number C such that any function f ∈ L ₁ satisfies the inequality ‖U _A,Λ(f)‖ _p ≤ C‖f‖₁ and what is the exact constant in this inequality? Here, \(U_{\mathcal{A},\Lambda } \left( f \right)\left( x \right) = \sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {\lambda _k \left( m \right)c_m \left( f \right)e^{imx} } } \right|}\) and c _m (f) are Fourier coefficients of the function f ∈ L ₁. Problem 2: what conditions on a sequence of finite subsets A _k ? ? guarantee that the function \(\sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {c_m \left( h \right)e^{imx} } } \right|}\) belongs to L _p for every function h of bounded variation?     

On the error term for the counting functions of finite Abelian groups

Leta(n) denote the number of non-isomorphic Abelian groups withn elements, and Δ(x) (resp. Δ _x ) appropriate error terms in the asymptotic formulas for the counting function \(\sum\nolimits_{n \leqslant x} {a(n)} (resp. \sum\nolimits_{m n \leqslant x} {a(m)} a(n))\) . Sharp bounds for $$\int\limits_1^X {\Delta (x) dx} , \int\limits_1^X {\Delta _{ 1} (x) dx} ,\int\limits_1^X {\Delta _1^2 (x) dx} $$ are given by using results on power moments of the Riemann zeta-function.     

Convergence of weighted averages of noncommutative martingales

Let x = (x _n ) _n?1 be a martingale on a noncommutative probability space ( $\mathcal{M}$ , τ) and (w _n ) _n?1 a sequence of positive numbers such that $W_n = \sum\nolimits_{k = 1}^n {w_k \to \infty } $ as n → ∞. We prove that x = (x _n ) _n?1 converges bilaterally almost uniformly (b.a.u.) if and only if the weighted average (σ _n (x)) _n?1 of x converges b.a.u. to the same limit under some condition, where σ _n (x) is given by $\sigma _n (x) = \frac{1} {{W_n }}\sum\limits_{k = 1}^n {w_k x_k } ,n = 1,2,... $ Furthermore, we prove that x = (x _n ) _n?1 converges in L _p ( $\mathcal{M}$ ) if and only if (σ _n (x)) _n?1 converges in L _p ( $\mathcal{M}$ ), where 1 ? p < ∞. We also get a criterion of uniform integrability for a family in L ₁( $\mathcal{M}$ ).