Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

We consider N-multiple trigonometric series whose complex coefficients c _j1,...,j _N , (j ₁,...,j _N ) ∈ ? ^N , form an absolutely convergent series. Then the series $$ \sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N ) $$ converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $ \mathbb{T} $ ^N , $ \mathbb{T} $ := [?π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α₁,..., α _N ) and lip (α₁,..., α _N ) for some α₁,..., α _N > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.     

Subalgebras of free products of algebras of the variety\mathfrak{u}_{m,n}

The variety \(\mathfrak{u}_{m,n} \) is defined by the system of n-ary operations ω_i,..., ω_m, the system of m-ary operations ?_i,..., ?_n, 1≤ m ≤ n, and the system of identities $$\begin{gathered} x_1 ...x_n \omega _1 ...x_1 ...x_n \omega _m \varphi _i = x_i (i = 1,...,n), \hfill \\ x_1 ...x_m \varphi _1 ...x_1 ...x_m \varphi _n \omega _j = x_j (i = 1,...,m), \hfill \\ \end{gathered} $$ It is proved in this paper that the subalgebra U of the free product \(\Pi _{i \in I}^* A_i \) of the algebras A_i (i ε I) can be expanded as the free product of nonempty intersections U ∩ A_i (i ε I) and a free algebra.     

Dimension and superposition of continuous functions

We give a relatively short proof of the following theorem of Sternfeld: LetX be a compact metric space with dimX ≧ 2, and letX ?R ^m be an embedding such that everyf ∈C(X) can be represented as $$f(x_1 ,x_2 ,...,x_m ) = \sum\limits_{i = 1}^m {g_i (x_i ),} (x_1 ,x_2 ,...,x_m ) \in X,g_i \in $$ Thenm ≧ 2 dimX + 1.     

Transformations z(t) = L(t)y(ϕ(t)) of ordinary differential equations

The paper describes the general form of an ordinary differential equation of an order n + 1 (n ≥ 1) which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $f\left( {s,w_{00} \upsilon _0 ,...,\sum\limits_{j = 0}^n {w_{nj\upsilon _j } } } \right) = \sum\limits_{j = 0}^n {w_{n + 1j\upsilon j} + w_{n + 1n + 1} f\left( {x,\upsilon ,\upsilon _1 ,...,\upsilon _n } \right),}$ where $w_{n + 10} = h\left( {s,x,x_1 ,u,u_1 ,...,u_n } \right),w_{n + 11} = g\left( {s,x,x_1 ,...,x_n ,u,u_1 ,...,u_n } \right){\text{ and }}w_{ij} = a_{ij} \left( {x_i ,...,x_{i - j + 1} ,u,u_1 ,...,u_{i - j} } \right)$ for the given functions a _ij is solved on $\mathbb{R},u \ne {\text{0}}$ .     

Ultrapowers ofL 1(μ) and the subsequence splitting principle

We study the ultrapowers $L_1 (\mu )_\mathfrak{U} $ of aL ₁(μ) space, by describing the components of the well-known representation $L_1 (\mu )_\mathfrak{U} = L_1 (\mu _\mathfrak{U} ) \oplus _1 L_1 (\nu _\mathfrak{U} )$ , and we give a representation of the projection from $L_1 (\mu )_\mathfrak{U} $ onto $L_1 (\mu _\mathfrak{U} )$ . Moreover, the subsequence splitting principle forL ₁(μ) motivates the following question: if $\mathfrak{V}$ is an ultrafilter on ? and $[f_i ] \in L_1 (\mu )_\mathfrak{V} $ , is it possible to find a weakly convergent sequence (g _i ) ?L ₁(μ) following $\mathfrak{V}$ and a disjoint sequence (h _i ) ?L ₁(μ) such that [f _i ]=[g _i ]+[h _i ]? If $\mathfrak{V}$ is a selective ultrafilter, we find a positive answer by showing that $f = [f_i ] \in L_1 (\mu )_\mathfrak{V} $ belongs to $L_1 (\mu _{_\mathfrak{V} } )$ if and only if its representatives {f _i } are weakly convergent following $\mathfrak{V}$ and $f \in L_1 (\nu _\mathfrak{V} )$ if and only if it admits a representative consisting of pairwise disjoint functions. As a consequence, we obtain a new proof of the subsequence splitting principle. If $\mathfrak{V}$ is not a p-point then the above characterizations of $L_1 (\nu _{_\mathfrak{V} } )$ and $L_1 (\nu _{_\mathfrak{V} } )$ fail and the answer to the question is negative.     

Convergence of simultaneous Hermite-Padé approximants to then-tuple ofq-hypergeometric series\{ {}_1\Phi _1 (_{(c,\gamma _j )}^{(1,1)} ;z)\} _{j = 1}^n

We investigate the convergence of simultaneous Hermite-Padé approximants to then-tuple of power series $$f_i (z) = \sum\limits_{k = 0}^\infty {C_k^{(i)} z^k ,} i = 1,2,...,n,$$ where $$C_0^{(i)} = 1;C_k^{(i)} = \prod\limits_{p = 0}^{k - 1} {\frac{1}{{(C - q^{\gamma i + p} )}},} k \ge 1.$$ HereC, q∈?, γ _i ∈?,i=1, 2,...,n. For |C|≠1, ifq=e^iθ, θ∈(0, 2π) and θ/2π is irrational, eachf _i (z),i=1,...,n, has a natural boundary on its circle of convergence. We show that “close-to-diagonal” and other sequences of Hermite-Padé approximants converge in capacity to (f ₁(z),..., f_n (z)) inside the common circle of convergence of eachf _i (z),i=1,...,n.     

Weighted restriction theorems for Bessel potentials

Пустьk-мерное евклид ово пространствоR ^k рассматривается как подмножествоR ⁿ . Зафиксируемр, 1<р<∞ иα >(n?k)/p, α≠п. Как обычно, бесселев потенциалJαf обобщенной функции Шварцаf наR ⁿ определяется с помощ ью ее преобразования Фурь е \((\widehat{G_\alpha f})(\xi ) = (2\pi )^{ - n/2} [1 + |\xi |^2 ]^{\alpha /2} f(\xi ), \xi \in R^n .B\) , ξ∈R ⁿ . В работе характ еризуются положител ьные весовые функцииw(x ₁,...,x _k ), которые при продолжении наR ⁿ с помощью равенстваw(x ₁,...,x _k ,...,x _n )=w(x ₁, ...,x _k ) обладают с ледующим свойством: существует числос>0, не зависящее отf, такое, что $$\begin{gathered} \int\limits_{R^k } {|(G_\alpha f)(x_1 ,...,x_k ,0,...,0)w(x_1 ,...,x_k )|^p dx_1 ...dx_k \leqq } \hfill \\ \leqq C\int\limits_{R^n } {|f(x_1 ,...,x_n )w(x_1 ,...,x_n )|^p dx_1 ...dx_n } \hfill \\ \end{gathered} $$      

Finite groups ofOD-conjugates

Given any setx, define the set of conjugates ofx to be Given any subgroupG of the group of permutations of {1, ?,n}, it is consistent with ZFC that there exists an orderedn-tuple 〈x ₁, ?, x_n〉 such that $$c(\langle x_1 ,...,x_n \rangle ) = \{ \langle x_{\pi 1} ,...,x_{\pi n} \rangle |\pi \in G\} .$$      

Global solutions of the heat equation with a nonlinear boundary condition

We consider the heat equation with a nonlinear boundary condition, $$(P) \left\{\begin{array}{ll} \partial_t u = \Delta u, & x \in \Omega, \quad t > 0, \\ \partial_\nu u=u^p, & x \in \partial \Omega,\quad t > 0,\\ u (x,0) = \phi (x),& x\in\Omega, \end{array}\right.$$ where ${\Omega = \{x = (x^{\prime},x_N) \in {\bf R}^{N} : x_N > 0\}, N \ge 2, \partial_t = \partial{/}\partial t , \partial_\nu = -\partial{/}\partial x_{N}}$ , p > 1 + 1/N, and (N ? 2)p < N. In this paper we give a complete classification of the large time behaviors of the nonnegative global solutions of (P).     

Stability of a mixed additive and cubic functional equation in several variables in non-Archimedean spaces

Consider the functional equation ${\Im_1(f ) = \Im_2(f )\,\,(\Im)}$ in a certain general setting. A function g is an approximate solution of ${(\Im)}$ if ${\Im_1(g)}$ and ${\Im_2(g)}$ are close in some sense. The Ulam stability problem asks whether or not there is a true solution of ${(\Im)}$ near g. In this paper, we achieve the general solution and the stability of the following functional equation $$\begin{array}{ll}f\left(\sum\limits^{n}_{i=1}x_{i} \right)+f\left(\sum\limits^{n-1}_{i=1} x_{i}-x_{n} \right)\\\quad=2f\left(\sum\limits^{n-1}_{i=1}x_{i} \right)+\sum\limits^{n-1}_{i=1}(f(x_{i}+x_{n}) +f(x_{i}-x_{n})-2f(x_{i}))\end{array}$$ for all x _i (i =? 1,2, . . . , n), in non-Archimedean spaces.     

On the Non-Commutative Neutrix Product of the Distributions x＋^λ and x^＋μ

Let f and g be distributions and let gn = （g ＊ δn）（x）, where δn （x） is a certain converging to the Dirac delta function. The non-commutative neutrix product fog of f and g to be the limit of the sequence {fgn }, provided its limit h exists in the sense that sequence is defined N-lim n-∞（f（x）g,, （x）, φ（x）〉 = （h（x）, φ（x）},for all functions p in 2. It is proved that （x^λ＋1n^px＋）0（x^μ＋1n^qx＋）=x＋^λμ1n^p＋qx＋,（x^λ-1n^qx-）=x-^λ＋μ1n^p＋qx-,for λ＋μ〈-1; λ,μ, λ＋μ≠-1,-2…and p,q=0,1,2……     

On the regular summability of multiple function series and Lebesgue functions

Основной целью работ ы является обобщение одного результата Кратца и Т раутнера [4], известного для одном ерных функциональны х рядов, на кратные ряды. Этот рез ультат касается суммируемо сти функционального ряда почти всюду при слабых пред положениях. В частности, он примен им к суммируемости по Чезаро и по Риссу. Мы рассматриваемd-кр атный ряд $$\mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x), \mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty c_{k_1 ,...,k_d }^2< \infty $$ и предполагается, что функции \(f_{k_1 ,...,k_d } (x)\) интегрируе мы по пространству с полож ительной мерой и имеют почти вс юду ограниченные фун кции Лебега для метода суммирова ния Т. Метод Т определяетсяd-мерной матрицей \(T = \{ a_{m_1 ,...,m_d ;k_1 ,...,k_d } \} \) сл едующим образом: $$t_{m_1 ,...,m_d } (x) = \mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty a_{m_1 ,...,m_d ;k_1 ,...,k_d } c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x).$$ Эти средние существу ют, поскольку мы предп олагаем, что \(a_{m_1 ,...,m_d ;k_1 ,...,k_d } = 0\) ,если max(k ₁,...,k _d) достаточно вели к (в зависимости, конеч но, отm ₁,...,m _d). При некоторых дополнительных усло виях на матрицуТ (см. (7)– (9) в разделе 3) устанавлива ется почти всюду регулярная схо димость средних \(t_{m_1 ,...,m_d } (x) \user2{} \user2{(}m_1 \user2{,}...\user2{,}m_d \user2{)} \to \infty \) . Как вспомогательный результат, в работе об общается теорема Алексича [1] о сх одимости почти всюду некоторы х подпоследовательн остей частных сумм функцио нального ряда.     

Criterion of polynomial increase of a function and its derivatives

ДОкАжАНО, ЧтО Дль тОгО, ЧтОБы Дльr РАж ДИФФЕРЕНцИРУЕМОИ НА пРОМЕжУткЕ [А, + ∞) ФУНкцИИf сУЩЕстВОВА л тАкОИ МНОгОЧлЕН (1) $$P(x) = \mathop \Sigma \limits_{\kappa = 0}^{r - 1} a_k x^k ,$$ , ЧтО (2) $$\mathop {\lim }\limits_{x \to + \infty } (f(x) - P(x))^{(k)} = 0,k = 0,1,...,r - 1,$$ , НЕОБхОДИМО И ДОстАтО ЧНО, ЧтОБы схОДИлсь ИН тЕгРАл (3) $$\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{r - 1} }^{ + \infty } {f^{(r)} (t)dt.}$$ ЕслИ ЁтОт ИНтЕгРАл сх ОДИтсь, тО Дль кОЁФФИц ИЕНтОВ МНОгОЧлЕНА (1) ИМЕУт МЕс тО ФОРМУлы $$\begin{gathered} a_{r - m} = \frac{1}{{(r - m)!}}\left( {\mathop \Sigma \limits_{j = 1}^m \frac{{( - 1)^{m - j} f^{(r - j)} (x_0 )}}{{(m - j)!}}} \right.x_0^{m - j} + \hfill \\ + ( - 1)^{m - 1} \left. {\mathop \Sigma \limits_{l = 0}^{m - 1} \frac{{x_0^l }}{{l!}}\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{m - l - 1} }^{ + \infty } {f^{(r)} (t_{m - 1} )dt_{m - 1} } } \right),m = 1,2,...,r. \hfill \\ \end{gathered}$$ ДОстАтОЧНыМ, НО НЕ НЕОБхОДИМыМ Усл ОВИЕМ схОДИМОстИ кРА тНОгО ИНтЕгРАлА (3) ьВльЕтсь схОДИМОсть ИНтЕгРАл А \(\int\limits_a^{ + \infty } {x^{r - 1} f^{(r)} (x)dx}\)      

On some boundary value problems with integral conditions for functional differential equations

For the functional differential equationu ⁽ⁿ⁾ (t)=f(u)(t) we have established the sufficient conditions for solvability and unique solvability of the boundary value problems $$u^{(i)} (0) = c_i (i = 0,...,m - 1), \smallint _0^{ + \infty } |u^{(m)} (t)|^2 dt< + \infty $$ and $$\begin{gathered} u^{(i)} (0) = c_i (i = 0),...,m - 1, \hfill \\ \smallint _0^{ + \infty } t^{2j} |u^{(j)} (t)|^2 dt< + \infty (j = 0,...,m), \hfill \\ \end{gathered} $$ wheren≥2,m is the integer part of $\tfrac{n}{2}$ ,c _i ∈R, andf is the continuous operator acting from the space of (n?1)-times continuously differentiable functions given on an interval [0,+∞] into the space of locally Lebesgue integrable functions.     

Eine Bemerkung zu Verallgemeinerungen der Ungleichung vom arithmetischen und geometrischen Mittel

In the literature there are known homogeneous polynomialsP(x ₁,...,x _n) with real coefficients, for which \(P(x_1 ,...,x_n ) \leqslant P(\bar x,...,\bar x)\) for allx _i≥0, and \(\bar x = (x_1 + ... + x_n )/n\) . This paper gives two theorems, which lead to new polynomials of this kind.     

On the uniqueness of symmetric bases in finite dimensional Banach spaces

Suppose{e _i} _i=1 ⁿ and{f _i} _i=1 ⁿ are symmetric bases of the Banach spacesE andF. Letd(E,F)≦C andd(E,l _n ² )≧n' for somer>0. Then there is a constantC _r=C_r(C)>0 such that for alla _i∈Ri=1,...,n $$C_r^{ - 1} \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\| \leqq \left\| {\sum\limits_{i = 1}^n {a_i f_i } } \right\| \leqq C_r \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\|$$ We also give a partial uniqueness of unconditional bases under more restrictive conditions.     

A Weighted Isoperimetric Inequality in an Orthant

Let c, k ₁ , ..., k _N be non-negative numbers, and define a measure μ in the orthant W :?=?{x?∈??by \(d\mu = e^{c|x|^2 } x_1 ^{k_1 }\cdots x_N ^{k_N } \, dx \) . It is shown that among all measurable subsets of W with fixed μ-measure, the intersection of W with a ball centered at the origin renders the weighted perimeter relative to W a minimum. Norm inequalities in weighted Sobolev spaces and a comparison result for solutions to boundary value problems for degenerate elliptic equations are also given.     

Complete unitary invariant for some subnormal operator

IfS is a subnormal operator with minimal normal extensionN satisfying the conditions that (i) \(\left[ {S^* ,S} \right]^{\frac{1}{2}} \in \mathcal{L}^1\) , (ii) sp (S) is the unit disk and (iii) sp (N)={N: |z|=1 orz=a ₁,...,a _k then $$tr\left( {\left[ {S^* ,(\lambda I - S)^{ - 1} } \right]\left[ {S^* ,(\mu I - S)^{ - 1} } \right]} \right) = \frac{n}{{\lambda ^2 \mu ^2 }} + \sum\limits_{i,j = 1}^k {\frac{{\gamma ij}}{{\lambda \mu (\lambda - a_i )(\mu - a_j )}}} $$ . wheren=index ( \(S^* - \bar zI\) ) forz∈sp (S)/sp (N) and (γij) is a real symmetric matrix. The set {n, γij,i,j = 1,...,k} is a complete unitary invariant for an operator in the class of all irreducible subnormal operators satisfying conditions (i), (ii), (iii) and that there is at least one positive simple eigenvalue of [S ^*,S].     

Chebyshev's inequalities for functions monotone in the mean

LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ?,n} →R. Choosew ∈ F withw _i = w(i) > 0. PutW _i = w₁ + ? + w_i. Givenf ∈ F, define \(\bar f\) ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if \(\bar f\) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$      

On oscillation of solutions of second-order systems of deviated differentaial equations

Sufficient conditions are found for the oscillation of proper solutions of the system of differential equations $$\begin{array}{*{20}c} {u'_1 (t) = f_1 (t,u_1 (\tau _1 (t)),...,u_1 (\tau _m (t)),u_2 (\sigma _1 (t)),...,u_2 (\sigma _m (t))),} \\ {u'_2 (t) = f_2 (t,u_1 (\tau _1 (t)),...,u_1 (\tau _m (t)),u_2 (\sigma _1 (t)),...,u_2 (\sigma _m (t))),} \\ \end{array}$$ wheref _i: R₊×R^2m→R (i=1,2) satisfy the local Carathéodory conditions andσ _i , τ _i :R ₊ →R (i=1,...,m) are continuous functions such that $\sigma _i (t) \leqslant t for t \in R_ + ,\mathop {\lim }\limits_{t \to + \infty } \sigma _i (t) = + \infty ,\mathop {\lim }\limits_{t \to + \infty } \tau _i (t) = + \infty (i = 1,...,m)$