Characterization of Besov-Triebel-Lizorkin spaces via approximation by Whittaker's cardinal series and related unconditional Schauder bases

Let

Global blow-up for a heat system with localized sources and absorptions

In this paper there are established the global existence and finite time blow-up results of nonnegative solution for the following parabolic systemut=Δu vp(x0, t)-aur, x∈Ω, t＞0,vt=Δv uq(x0,t)-bvs, x∈Ω, t＞0subject to homogeneous Dirichlet conditions and nonnegative initial data, where x0 ∈Ω is a fixed point, p, q, r, s ≥ 1 and a, b ＞ 0 are constants. In the situation when nonnegative solution (u, v) of the above problem blows up in finite time, it is showed that the blow-up is global and this differs from the local sources case. Moreover, for the special case r = s = 1,are obtained uniformly on compact subsets of Ω, where T* is the blow-up time.     

A combinatorial proof of an identity of Ramanujan using tilings

Recently, Little and Sellers proved combinatorially a variety of Rogers-Ramanujan type of identities. They interpreted both sides of the equalities as enumerating the same collection of “weighted tilings”. They tiled an infinite 1 × ∞ board, with squares and dominos. In their articles, the concept of “projection” of tiles was defined. In this article, we use and extend some of their ideas and give a combinatorial proof to the following identity proposed in a survey of Pak

Une Formule Approximative de Dimension pour Certains Produits de Riesz

Consider the Riesz product $\mu _a = \mathop \prod \limits_{n = 1}^\infty (1 + r\cos (q^n t + \varphi _n ))$ . We prove the following approximative formula for the dimension ofμ _a. $$\dim \mu _a = 1 - \frac{1}{{\log q}}\int_0^{2\pi } {(1 + r\cos x)\log (1 + r\cos x)\frac{{dx}}{{2\pi }} + 0\left( {\frac{r}{{q^2 \log q}}} \right).}$$      

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.     

The ergodicity of service systems with an infinite number of servomechanisms

Existence, uniqueness, and ergodicity are proved for a stationary distribution for a service system having countably many servomechanisms with input flow rate μ_k depending on the number k of servomechanisms occupied, and with arbitrary (identical) distribution of the service time with finite mean μ, under the condition \(\mu \mathop {\overline {\lim } }\limits_{k \to \infty } \frac{{\lambda _k }}{{k + 1}}< 1\) . For this system we have, in particular, Erlang's formula $$p_k (t)\mathop \to \limits_{k + \infty } p_k = \frac{{\lambda _0 ...\lambda _{k - 1} \mu ^k }}{{k!}}p_0 ,k = 0,1,...,p_0^{ - 1} = \sum\nolimits_{k = 0}^\infty {\frac{{\lambda _0 ...\lambda _{k - 1} \mu ^k }}{{k!}}} ,\lambda _{ - 1} = 1.$$      

Precise rate in the law of iterated logarithm for <Emphasis Type="Italic">ρ</Emphasis>-mixing sequence

Let {X, X _n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set . Suppose lim _n→∞ and , where d=2, if −1<b<0 and d>2(b+1), if b≥0. It is proved that, for any b>−1,

Some Lambert Series Expansions of Products of Theta Functions

Let q be a complex number satisfying |q| < 1. The theta function (q) is defined by (q) = . Ramanujan has given a number of Lambert series expansions such as

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.     

Oscillation theorems for second-order nonlinear delay difference equations

By means of Riccati transformation technique, we establish some new oscillation criteria for second-order nonlinear delay difference equation $$\Delta (p_n (\Delta x_n )^\gamma ) + q_n f(x_{n - \sigma } ) = 0,\;\;\;\;n = 0,1,2,...,$$ when $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} = \infty }$ . When $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} < \infty }$ we present some sufficient conditions which guarantee that, every solution oscillates or converges to zero. When $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} = \infty }$ holds, our results do not require the nonlinearity to be nondecreasing and are thus applicable to new classes of equations to which most previously known results are not.     

On the continuity of functions in connection with a problem of V. G. Krotov

В РАБОтЕ ДАЕтсь ОтВЕт НА ОДИН ВОпРОс, пОстАВ лЕННыИ В. г. кРОтОВыМ. УстАНОВлЕН О, ЧтО ЕслИ Ф(х) — МОНОтОННО ВО жРАстАУЩАь ФУНкцИь,Ф (0)=0, Ф(2х)≦кФ(х), х[0, ∞), тО $$\left\{ {f:\left\| {\sum\limits_{k = 1}^\infty {\mu _k \Phi (\lambda _k \left| {S_k - f} \right|)} } \right\|_c< \infty } \right\} \subseteqq C \Leftrightarrow \sum\limits_{k = 1}^\infty {\mu _k } \Phi (\lambda _k ) = \infty $$ Дль пРОИжВОльНых НЕО тРИцАтЕльНых ЧИслОВ ых пОслЕДОВАтЕльНОстЕ И {Μ_k} И {λ_k}. (жДЕсьS _k ОБОжНАЧАЕт ЧАстНУУ с УММУ пОРьДкАk РьДА ФУ РьЕ ФУНкцИИf). УстАНОВлЕН О тАкжЕ, ЧтО ВО МНОгИх слУЧАьх $$\left\{ {f:\left\| {\sum\limits_{k = 1}^\infty {\mu _k \Phi (\lambda _k \left| {\tilde S_k - \tilde f} \right|)} } \right\|_c< \infty } \right\} \subseteqq C \Leftrightarrow \sum\limits_{k = 1}^\infty {\frac{1}{{k\lambda _k }}} \Phi ^{ - 1} \left( {\frac{1}{{k\mu _k }}} \right)< \infty .$$      

Strong approximation by rectangular partial sums of double orthogonal series

Пусть {? _ik(x):i, k=1, 2,...} — орто нормированная систе ма в пространстве с полож ительной мерой и {a _ik} — последов ательность действит ельных чисел, для которой $$\mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik}^2 \kappa ^2 (i,k)< \infty ,$$ где {x(i, K)} — определенна я неубывающая последовательность положительных чисел. Тогда суммаf(x) двойног о ортогонального ряд а \(\mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik} \varphi _{ik} (x)\) существует в смысле с ходимости в метрикеL ² и сходимос ти почти всюду. Изучае тся порядок так называем ой сильной аппроксимац ииf(x) (при коэффициентн ых условиях) прямоуголь ными частными суммами \(s_{mn} (x) = \mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik} \varphi _{ik} (x)\) . Основной ре зультат состоит в сле дующем. Если {λ_j(m):m=1, 2,...} — неубывающи е последовательност и положительньк чисел, стремящиеся к ∞ и такие, что \(\mathop {\lim \sup }\limits_{m \to \infty } \lambda _j (2m)/\lambda _j (m)< \sqrt 2 \) дляj=1,2, и если $$\mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik}^2 \left[ {\log log (i + 3)} \right]^2 \left[ {\log log (k + 3)} \right]^2 (\lambda _1^2 (i) + \lambda _2^2 (k))< \infty ,$$ TO ПОЧТИ ВСЮДУ $$\left\{ {\frac{1}{{mn}}\mathop \sum \limits_{i = 1}^m \mathop \sum \limits_{\kappa = 1}^m \left[ {s_{ik} (x) - f(x)} \right]^2 } \right\}^{1/2} = o_x (\lambda _1^{ - 1} (m) + \lambda _2^{ - 1} (n))$$ при min (m, n) → ∞.     

Approximating periodic functions in Hölder type metrics by singular integrals with positive Kernels

Let M be either the space of 2π-periodic functions L_p, where 1 ≤ p < ∞, or C; let ω_r(f, h) be the continuity modulus of order r of the function f, and let

Geometric interpretation of the Euler-Knopp summation method in the problem of inverting the Laplace transform

In this paper, we consider a method for inverting the Laplace transform F(s) = \(\int\limits_0^\infty {e^{ - st} f(t)dt} \), which consists in representing the original function by the Laguerre series

$f(t) = \sum\limits_{k = 0}^\infty {a_k L_k (bt).} $

(1)

First, we perform a conformal mapping of the plane (s), which depends on parameter ξ. The value of the parameter is determined by the location of the singular points of the given representation. Under this mapping, series (1) takes the form

$f(t) = \frac{{b - \xi }}{b}\exp (\xi t)\sum\limits_{k = 0}^\infty {c_k L_k ((b - \xi )t).} $

It is demonstrated that such inverting scheme is equivalent to applying the Picone-Tricomi method with further acceleration of the rate of convergence of series (1) using the Euler-Knopp nonlinear procedure

$\sum\limits_{k = 0}^\infty {a_k z^k } = \sum\limits_{k = 0}^\infty {A_k (p)\frac{{z^k }}{{(1 - pz)^{k + 1} }},} A_k (p) = \sum\limits_{j = 0}^k {\left( \begin{gathered} k \hfill \\ j \hfill \\ \end{gathered} \right)( - p)^{k - j} a_j } .$

Under this approach, the original function is represented by the series

$f(t) = \exp \left( {\frac{{bpt}}{{p - 1}}} \right)\sum\limits_{k = 0}^\infty {\frac{{A_k (p)}}{{(1 - p)^{k + 1} }}L_k } \left( {\frac{{bpt}}{{1 - p}}} \right),$

where parameters ξ and p are related by the formula p = x/(ξ ? b). Unlike many other methods for summation of series, in the scheme suggested, there is no need to investigate the regularity conditions.

     

Compactification of supports of energy weak solutions to quasilinear parabolic equations of nonstationary filtration type with strong convection

The paper deals with localization properties of solutions to the Cauchy problem with the initial data u₀(x) ∈ L₂(ℝ_n) for a wide class of equations in the divergence form. This class contains, e.g., the following equation:

Inequalities for derivatives of functions in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

  We obtain a new sharp inequality for the local norms of functions x ∈ L _{∞, ∞} ^r (R), namely,

An accurate approximation of zeta-generalized-Euler-constant functions

Zeta-generalized-Euler-constant functions,

О раэложении мероморфных функций специального вида на простейщие дроби

The paper is devoted to study the entire functions L(λ) with simple real zeros λ_k, k = 1, 2, ..., that admit an expansion of Krein’s type: $$\frac{1}{{\mathcal{L}(\lambda )}} = \sum\limits_{k = 1}^\infty {\frac{{c_k }}{{\lambda - \lambda _k }}} ,\sum\limits_{k = 1}^\infty {\left| {c_k } \right| < \infty } .$$ We present a criterion for these expansions in terms of the sequence {L′ (λ _k )} _k=1 ^∞ . We show that this criterion is applicable to certain classes of meromorphic functions and make more precise a theorem of Sedletski? on the annihilating property in L ² systems of exponents.     

A note on parameterized Marcinkiewicz integrals with variable kernels

In this paper,the parameterized Marcinkiewicz integrals with variable kernels defined by μΩ^ρ（f）（x）=（∫0^∞│∫│1-y│≤t Ω（x,x-y）/│x-y│^n-p f（y）dy│^2dt/t1＋2p）^1/2 are investigated.It is proved that if Ω∈ L∞（R^n） × L^r（S^n-1）（r〉（n-n1p＇/n） is an odd function in the second variable y,then the operator μΩ^ρ is bounded from L^p（R^n） to L^p（R^n） for 1 〈 p ≤ max{（n＋1）/2,2}.It is also proved that,if Ω satisfies the L^1-Dini condition,then μΩ^ρ is of type（p,p） for 1 〈 p ≤ 2,of the weak type（1,1） and bounded from H1 to L1.     

A note on certain bilinear and bilateral generating relations for the laguerre polynomial

The author presents here a systematic analysis of certain bilinear and bilateral generating relations associated with the generalized Laguerre polynomials {L_n ^(a) (x)} defined by $$(1 - t)^{ - 1 - \alpha } \exp \left( { - \frac{{xt}}{{1 - t}}} \right) = \sum\limits_{n = 0}^\infty {L_n ^{(\alpha )} (x) t^n .} $$ It is observed that the four formulas proved recently by Jain [7] are only specialized or limiting forms of the results of Erdélyi,[4] Meixner [8] and the author [12–14].