On L 1-convergence of Fourier series of complex valued functions under the GM7 condition

Let f??L _2?? be a real-valued even function with its Fourier series $\frac {a_{0}}{2}+\sum\limits_{n=1}^{\infty}{a_{n}}\cos nx$ , and let S _n (f,x) be the nth partial sum of the Fourier series, n?R1. The classical result says that if the nonnegative sequence {a _n } is decreasing and $\lim\limits_{n\to\infty} a_{n} =0$ , then $\lim\limits_{n\to\infty} \|{f-S_{n}(f)}\|_{L}=0$ if and only if $\lim\limits_{n\to\infty} a_{n}\log n=0$ . Later, the monotonicity condition set on {a _n } is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Kórus further generalized the condition in the classical result to the so-called GM₇ condition in real space. In this paper, we give a complete generalization to the complex space.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

Approximation operators and tauberian constants

The explicit expression of the smallest constantC satisfying $$\mathop {lim}\limits_{\lambda \to \infty } \left| {t_{n(\lambda )}^{(1)} - t_{m(\lambda )}^{(2)} } \right| \leqq C. \mathop {lim sup}\limits_{n \to \infty } \left| {d_n } \right|$$ for all sequences {s _n} satisfying lim sup _n→∞ |d _n| <∞, where {t _n ⁽¹⁾ }, {t _n ⁽²⁾ } are two generalised Hausdorff transforms of {s _n }, {d _n} is the generalised (C, α)-transform (0≦α≦1) of {λ _n a _n} andn(λ, m(λ) are suitably related, is obtained. These results are obtained by using new properties of positive approximation operators and generalised Bernstein approximation operators.     

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]).     

Moment multipliers for power series

По определению после довательность {μ _n пр инадлежит классуG _s , если звезда М иттагЛеффлера произвольного степе нного ряда (1) $$\mathop \sum \limits_0^\infty a_n z^n , \mathop {lim sup}\limits_{n \to \infty } \left| {a_n } \right|^{1/n}< \infty $$ , совпадает со звёздам и Миттаг-Леффлера сте пенных рядов $$\mathop \sum \limits_0^\infty \mu _n a_n z^n ,\mathop \sum \limits_0^\infty \mu _n^{ - 1} a_n z^n $$ . В работе установлены следующие утвержден ия Теорема 1.Для произво льной последователь ности ? _n с условиями $$0< \varphi _n< 1,\mathop {lim}\limits_{n \to \infty } \varphi _n = 0,\mathop {lim}\limits_{n \to \infty } \varphi _n^{1/n} = 1$$ существует неубываю щая функция χ(t) такая, ч то моменты \(\mu _n = \int\limits_0^1 {t^n d\chi (t)} \) удовлетворяют условию 0<μ_nn звезда М иттаг-Леффлера любог о ряда (1) совпадает со звездой МиттагЛеффлера степенных рядов . Теорема 2. Для произвол ьной неотрицательно й последовательности {а_n} с условием {a _n } и для любой последов ательности {?_n} для к оторой 0n<1, \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n = 0\) сущест вуютπ={π _n }∈G _s и последовательнос ть {п_i} такие, что a_nμ_n≦1 (n≧n₀), \(a_{n_i } \mu _{\mu _i } \geqq exp( - \varepsilon _{n_i } )\) (i=1, 2, ...) и при эmom звезда Миттаг-Леффлера ряда (1) совпа дает со звездой Миттаг- Леффлера степенных р ядов .     

Certain numerical characteristics of KN-lineals

We study the properties of certain numerical characteristics in a normed lattice that characterize its conjugate space. A typical result is as follows: let X be a K_σN-space or a KB-lineal. If for every sequence {x_n} ? X of pairwise disjoint positive elements with norms not exceeding 1 we have $$\mathop {\underline {\lim } }\limits_{n \to \infty } \frac{1}{n} x_1 \vee x_2 \vee \ldots \vee x_n \parallel = 0,$$ then all the odd conjugate spaces X^*, X^***, ... are KB-spaces.     

Freud's conjecture and approximation of reciprocals of weights by polynomials

LetW(x) be a function that is 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, and let {α _n } ₁ ^∞ and {β _n } ₁ ^∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ^?1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec _{n 1} ^∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR.     

Statistical extension of classical Tauberian theorems in the case of logarithmic summability

Let s: [1,∞) → ? be a locally integrable function in Lebesgue’s sense. The logarithmic (also called harmonic) mean of the function s is defined by $$\tau (t): = \frac{1} {{\log t}}\int_1^t {\frac{{s(x)}} {x}dx, t > 1,}$$ where the logarithm is to the natural base e. Besides the ordinary limit lim _x→∞ s(x), we use the notion of the so-called statistical limit of s at ∞, in notation: st-lim _x→∞ s(x) = l, by which we mean that for every ? > 0, $$\mathop {\lim }\limits_{b \to \infty } \frac{1} {b}\left| {\left\{ {x \in (1,b):\left| {s(x) - \ell } \right| > \varepsilon } \right\}} \right| = 0.$$ We also use the ordinary limit limt→∞ τ (t) as well as the statistical limit st-limt→∞ τ (t). We will prove the following Tauberian theorem: Suppose that the real-valued function s is slowly decreasing or the complex-valued s is slowly oscillating. If the statistical limit st-limtt→∞ τ (t) = l exists, then the ordinary limit limx→∞ s (x) = l also exists.     

Strong convergence of a regularization method for Rockafellar’s proximal point algorithm

In this paper, for a monotone operator T, we shall show strong convergence of the regularization method for Rockafellar’s proximal point algorithm under more relaxed conditions on the sequences {r _k } and {t _k }, $$\lim\limits_{k\to\infty}t_k = 0;\quad \sum\limits_{k=0}^{+\infty}t_k = \infty;\quad\ \liminf\limits_{k\to\infty}r_k > 0.$$ Our results unify and improve some existing results.     

Distribution of points for convergent interpolatory integration rules on (−∞, ∞)

Letw be a “nice” positive weight function on (?∞, ∞), such asw(x)=exp(??x?^α) α>1. Suppose that, forn≥1, $$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} } f(x_{jn} )$$ is aninterpolatory integration rule for the weightw: that is for polynomialsP of degree ≤n-1, $$I_n [P]: = \int\limits_{ - \infty }^\infty {P(x)w(x)dx.} $$ Moreover, suppose that the sequence of rules {I _n} _n=1 ^t8 isconvergent: $$\mathop {\lim }\limits_{n \to \infty } I_n [f] = \int\limits_{ - \infty }^\infty {f(x)w(x)dx} $$ for all continuousf:R→R satisfying suitable integrability conditions. What then can we say about thedistribution of the points {x _jn} _j=1 ⁿ ,n≥1? Roughly speaking, the conclusion of this paper is thathalf the points are distributed like zeros of orthogonal polynomials forw, and half may bearbitrarily distributed. Thus half the points haveNevai-Ullmann distribution of order α, and the rest are arbitrarily distributed. We also describe the possible distributions of the integration points, when the ruleI _n has precision other thann-1.     

Some results on non-linear recurrence

We study the behavior of measure-preserving systems with continuous time along sequences of the form {n ^α}n∈#x2115;} where α is a positive real number¹. Let {S ^t } _t∈? be an ergodic continuous measure preserving flow on a probability Lebesgue space (X, β, μ). Among other results we show that:

1. For all but countably many α (in particular, for all α∈???) one can find anL ^∞-functionf for which the averagesA _N (f)(1/N)=Σ _n=1 ^N f(S ^nα x) fail to converge almost everywhere (the convergence in norm holds for any α!).
2. For any non-integer and pairwise distinct numbers α₁, α₂,..., α _k ∈(0, 1) and anyL ^∞-functionsf ₁,f ₂, ...,f _k , one has $$\mathop {lim}\limits_{N \to \infty } \left\| {\frac{1}{N}\sum\limits_{n - 1}^N {\prod\limits_{i - 1}^k {f_i (S^{n^{\alpha _i } } x) - \prod\limits_{i - 1}^k {\int_X {f_i d\mu } } } } } \right\|_{L^2 } = 0$$

We also show that Furstenberg’s correspondence principle fails for ?-actions by demonstrating that for all but a countably many α>0 there exists a setE?? having densityd(E)=1/2 such that, for alln∈?, $$d(E \cap (E - n^\alpha )) = 0$$ .     

The convergence of moments in the central limit theorem for stationary ϕ-mixing processes

Пусть {X_j} - строго стац ионарная последоват ельностьс ?перемешиванием, EXj-Q,E¦-X _j¦^r<∞ для некоторогоr>2. Положим \(S_n = \mathop \sum \limits_{j = 1}^n X_j \) . Ибрагимов (1962) доказал, что если приn →∞, то 1 $$\mathop {\lim }\limits_{n \to \infty } P\{ S_n /\sigma _n< x\} = (2\pi )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \mathop \smallint \limits_{ - \infty }^x e^{{{ - u^2 } \mathord{\left/ {\vphantom {{ - u^2 } 2}} \right. \kern-\nulldelimiterspace} 2}} du.$$ В работе установлено, что при указанных выш е условиях в этой центральной пр едельной теореме имеет место т акже и сходимостьr-ых абсолютных моментов, т.е. если σ _n ² →∞ приn→ ∞, то $$\mathop {\lim }\limits_{n \to \infty } E|S_n /\sigma _n |^r = (2\pi )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \mathop \smallint \limits_{ - \infty }^{ + \infty } |u|^r e^{ - u^2 /2} du.$$ Этот результат обобщ ает один более ранний результат автора (1980 г.).     

Über arithmetische Eigenschaften von Lückenreihen mit algebraischen Koeffizienten und algebraischem Argument

Let \(f(z) = \sum\limits_{h = 0}^\infty {f_h z^h } \) be a power series with positive radius of convergenceR _f ≤1,f _h algebraic and lacunary in the following sense: Let {r _n }, {s _n } be two infinite sequences of integers, satisfying $$0 = s_0 \leqslant r_1< s_1 \leqslant r_2< s_2 \leqslant r_3< s_3 \leqslant ..., \mathop {lim}\limits_{n \to \infty } (s_n /F(n)) = \infty $$ such that $$f_h = 0 if r_n< h< s_n ,f_{r_n } \ne 0,f_{s_n } \ne 0 for n = 1,2,3,...;$$ F(n) denotes a certain function ofn, dependent onr _n and \(f_0 ,f_1 ,f_2 , \ldots f_{r_n } \) . Using ideas from a note ofK. Mahler, among other results the following main theorem is proved: The function valuef(α) (with α algebraic, 0<|α|<R _f ) is algebraic if and only if there exists a positive integerN=N(α) such that $$P_n (\alpha ): = \sum\limits_{h = s_n }^{r_{n + 1} } {f_h \alpha ^h = 0 for all n \geqslant N.} $$      

Rational approximations of real numbers

For anyx ∈ r put $$c(x) = \overline {\mathop {\lim }\limits_{t \to \infty } } \mathop {\min }\limits_{(p,q\mathop {) \in Z}\limits_{q \leqslant t} \times N} t\left| {qx - p} \right|.$$ . Let [x₀; x₁,..., x_n, ...] be an expansion of x into a continued fraction and let \(M = \{ x \in J,\overline {\mathop {\lim }\limits_{n \to \infty } } x_n< \infty \}\) .Forx∈M put D(x)=c(x)/(1?c(x)). The structure of the set \(\mathfrak{D} = \{ D(x),x \in M\}\) is studied. It is shown that $$\mathfrak{D} \cap (3 + \sqrt 3 ,(5 + 3\sqrt 3 )/2) = \{ D(x^{(n,3} )\} _{n = 0}^\infty \nearrow (5 + 3\sqrt 3 )/2,$$ where \(x^{(n,3)} = [\overline {3;(1,2)_n ,1} ].\) This yields for \(\mu = \inf \{ z,\mathfrak{D} \supset (z, + \infty )\}\) (“origin of the ray”) the following lower bound: μ?(5+3√3)/2=5.0n>(5 + 3/3)/2=5.098.... Suppose a∈n. Put \(M(a) = \{ x \in M,\overline {\mathop {\lim }\limits_{n \to \infty } } x_n = a\}\) , \(\mathfrak{D}(a) = \{ D(x),x \in M(a)\}\) . The smallest limit point of \(\mathfrak{D}(a)(a \geqslant 2)\) is found. The structure of (a) is studied completely up to the smallest limit point and elucidated to the right of it.     

On a certain class of infinitely divisible distributions

We characterize the class of distribution functions Φ(x), which are limits in the following sense: there exist a sequence of independent and equally distributed random variables {ξ _n }, numerical sequences {a _k }, {b _k } and natural numbers {n _k } such that $$\mathop {lim}\limits_{k \to \infty } Prob\left\{ {\frac{1}{{a_k }}\mathop {\Sigma }\limits_{k = 1}^{n_k } \xi _k - b_k< x} \right\} = \Phi (x)$$ and $$\mathop {\lim \inf }\limits_{k \to \infty } (n_k /n_{k + 1} ) > 0$$ .     

О некоторых свойства х частичных сумм рядо в Фурье—Уолша—Пэли

In this paper we consider the behaviour of partial sums of Fourier—Walsh—Paley series on the group62-01. We prove the following theorems: Theorem 1. Let {n _k } _k ^=1/∞ be some increasing convex sequence of natural numbers such that $$\mathop {\lim sup}\limits_m m^{ - 1/2} \log n_m< \infty $$ . Then for anyf∈L ^∞(G) $$\left( {\frac{1}{m}\sum\limits_{j = 1}^m {|Sn_j (f;0)|^2 } } \right)^{1/2} \leqq C \cdot \left\| f \right\|_\infty $$ . Theorem 2. Let {n _k } _k ^=1/∞ be a lacunary sequence of natural numbers,n _k+1/n _k≧q>1. Then for anyfεL ^∞(G) $$\sum\limits_{j = 1}^m {|Sn_j (f;0)| \leqq C_q \cdot m^{1/2} \cdot \log n_m \cdot \left\| f \right\|_\infty } $$ . Theorems. Let µ _k =2 ^k +2 ^k-2+2 ^k-4+...+2^α ₀,α ₀=0,1. Then $$\begin{gathered} \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in L^\infty (G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = 0(m)^2 \} .} \hfill \\ \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = o(m)^2 \} = } \hfill \\ = \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} \hfill \\ \end{gathered} $$ . Theorem 4. {{S ₂ k(f: 0)} _k ^=1/∞ ,f∈L ^∞(G)}=m. $$\{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = c. \{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} = c_0 $$ .     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

Asymptotic properties for the loglog laws under positive association

Let {X _n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X ₁|^2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x ₊ = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function.