Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

(Bounded) continuous cohomology and Gromov��s proportionality principle

Let X be a topological space, and let C*(X) be the complex of singular cochains on X with coefficients in ${\mathbb{R}}$ . We denote by ${C^{\ast}_{c}(X) \subseteq C^{\ast}(X)}$ the subcomplex given by continuous cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous real function. We prove that at least for ??reasonable?? spaces the inclusion ${C^{\ast}_{c}(X) \hookrightarrow C^{\ast}(X)}$ induces an isomorphism in cohomology, thus answering a question posed by Mostow. We also prove that this isomorphism is isometric with respect to the L ^??-norm on cochains defined by Gromov. As an application, we clarify some details of Gromov??s original proof of the proportionality principle for the simplicial volume of Riemannian manifolds, also providing a self-contained exposition of Gromov??s argument.     

The morphism induced by Frobenius push-forward

Let X be a smooth projective curve of genus g 2 over an algebraically closed field k of characteristic p0,and F:X→X(1)the relative Frobenius morphism.Let M s X(r,d)(resp.M ss X(r,d))be the moduli space of(resp.semi-)stable vector bundles of rank r and degree d on X.We show that the set-theoretic map S ss Frob:M ss X(r,d)→M ss X(1)(rp,d+r(p-1)(g-1))induced by[E]→[F(E)]is a proper morphism.Moreover,the induced morphism S s Frob:M s X(r,d)→M s X(1)(rp,d+r(p-1)(g-1))is a closed immersion.As an application,we obtain that the locus of moduli space M s X(1)(p,d)consisting of stable vector bundles whose Frobenius pull backs have maximal Harder-Narasimhan polygons is isomorphic to the Jacobian variety Jac X of X.     

The Maximal Variation of Martingales of Probabilities and Repeated Games with Incomplete Information

The variation of a martingale $p_{0}^{k}=p_{0},\ldots,p_{k}$ of probabilities on a finite (or countable) set X is denoted $V(p_{0}^{k})$ and defined by $$ V\bigl(p_0^k\bigr)=E\Biggl(\sum_{t=1}^k\|p_t-p_{t-1}\|_1\Biggr). $$ It is shown that $V(p_{0}^{k})\leq\sqrt{2kH(p_{0})}$ , where H(p) is the entropy function H(p)=?∑ _x p(x)logp(x), and log stands for the natural logarithm. Therefore, if d is the number of elements of X, then $V(p_{0}^{k})\leq\sqrt{2k\log d}$ . It is shown that the order of magnitude of the bound $\sqrt{2k\log d}$ is tight for d≤2 ^k : there is C>0 such that for all k and d≤2 ^k , there is a martingale $p_{0}^{k}=p_{0},\ldots,p_{k}$ of probabilities on a set X with d elements, and with variation $V(p_{0}^{k})\geq C\sqrt{2k\log d}$ . An application of the first result to game theory is that the difference between v _k and lim _j v _j , where v _k is the value of the k-stage repeated game with incomplete information on one side with d states, is bounded by $\|G\|\sqrt{2k^{-1}\log d}$ (where ∥G∥ is the maximal absolute value of a stage payoff). Furthermore, it is shown that the order of magnitude of this game theory bound is tight.     

Supports of (\mathfrak{g},\mathfrak{k})-modules of finite type

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{k}$ be a reductive subalgebra in $\mathfrak{g}$ . We say that a $\mathfrak{g}$ -module M is a $(\mathfrak{g},\mathfrak{k})$ -module if M, considered as a $\mathfrak{k}$ -module, is a direct sum of finite-dimensional $\mathfrak{k}$ -modules. We say that a $(\mathfrak{g},\mathfrak{k})$ -module M is of finite type if all $\mathfrak{k}$ -isotopic components of M are finite-dimensional. In this paper we prove that any simple $(\mathfrak{g},\mathfrak{k})$ -module of finite type is holonomic. A simple $\mathfrak{g}$ -module M is associated with the invariants V(M), V(LocM), and L(M) reflecting the ??directions of growth of M.?? We also prove that for a given pair $(\mathfrak{g},\mathfrak{k})$ the set of possible invariants is finite.     

Commutators of Marcinkiewicz integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator [b, ??_??,?? ] from the homogeneous Sobolev space $\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)$ to the Lebesgue space L ^p (? ⁿ ) for 1 < p < ??, where ??_??,?? denotes the Marcinkiewicz integral with rough hypersingular kernel defined by $\mu _{\Omega ,\gamma } f\left( x \right) = \left( {\int_0^\infty {\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega \left( {x - y} \right)}} {{\left| {x - y} \right|^{n - 1} }}f\left( y \right)dy} } \right|^2 \frac{{dt}} {{t^{3 + 2\gamma } }}} } \right)^{\frac{1} {2}} ,$ , with ?? ?? L ¹(S ^n?1) for $0 < \gamma < min\left\{ {\frac{n} {2},\frac{n} {p}} \right\}$ or ?? ?? L(log⁺ L) ^?? (S ^n?1) for $\left| {1 - \frac{2} {p}} \right| < \beta < 1\left( {0 < \gamma < \frac{n} {2}} \right)$ , respectively.     

Inversion complexity of self-correcting circuits for a certain sequence of boolean functions

The asymptotics L _k ^? (f ₂ ⁿ ) ?? n min{k+1, p} is obtained for the sequence of Boolean functions $f_2^n \left( {x_1 , \ldots ,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n}$ for any fixed k, p ?? 1 and growing n, here L _k ^? (f ₂ ⁿ ) is the inversion complexity of realization of the function f ₂ ⁿ by k-self-correcting circuits of functional elements in the basis B = {&, ?}, p is the weight of a reliable invertor.     

Quasikonvexe Multiplikatoren starker Konvergenz

пУсть {f _k; f _k ^* ?X×X^* — пОлНАь БИОРтОгОНАльНАь сИс тЕМА В БАНАхОВОМ пРОстРАН стВЕ X (X^* — сОпРьжЕННОЕ пРОст РАНстВО). пУсть (?→+0) $$\begin{gathered} S_n f = \sum\limits_{k = 0}^n {f_k^* (f)f_k ,} K(f,t) = \mathop {\inf }\limits_{g \in Z} (\left\| {f - g} \right\|_x + t\left| g \right|_z ), \hfill \\ X_0 = \{ f \in X:\mathop {\lim }\limits_{n \to \infty } \left\| {S_n f - f} \right\|_x = 0\} ,X_\omega = \{ f \in X:K(f,t) = 0(\omega (t))\} , \hfill \\ \end{gathered} $$ гДЕZ?X — НЕкОтОРОЕ пОД пРОстРАНстВО с пОлУН ОРМОИ ¦·¦ И Ω — МОДУль НЕпРЕРыВНО стИ УДОВлЕтВОРьУЩИИ Усл ОВИУ sup Ω(t)/t=∞. пОслЕДОВАтЕ льНОстьΤ={Τ _k} кОМплЕксНых ЧИ сЕл НАжыВАЕтсь МНОжИтЕл ЕМ сИльНОИ схОДИМОст И ДльX _Τ, жАпИсьΤ?М[X _Τ,X _Τ], ЕслИ Д ль кАжДОгО ЁлЕМЕНтАf?X _Τ сУЩЕстВ УЕт тАкОИ ЁлЕМЕНтf ^τ?х₀, ЧтОf _k ^* (f ^τ)=Τ_kf _k ^* (f) Дль ВсЕхk. ДОкА жАНО сРЕДИ ДРУгИх слЕДУУЩ ЕЕ УтВЕРжДЕНИЕ. тЕОРЕМА. пУсmь {f_k; f _k ^* } —Н ЕкОтОРыИ (с, 1)-БАжИс тАк ОИ, ЧтО ВыпОлНьУтсь НЕРАВЕН стВА тИпА НЕРАВЕНстВА ДжЕ ксОНА с пОРьДкОМ O(?_n) u тИ пА НЕРАВЕНстВА БЕРНшmЕИ НА с пОРьДкОМ O(1/?_n). ЕслИ пОслЕДОВАтЕл ьНОсть Τ кВАжИВыпУкл А И ОгРАНИЧЕНА, тО $$\tau \in M[X_{\omega ,} X_0 ] \Leftrightarrow \omega (\varphi _n )\tau _n \left\| {S_n } \right\|_{[X,X]} = o(1).$$ ЁтОт ОБЩИИ пОДхОД НЕМ ЕДлЕННО ДАЕт клАссИЧ ЕскИЕ РЕжУльтАты, ОтНОсьЩИ Есь к ОДНОМЕРНыМ тРИгОНОМЕтРИЧЕскИМ РьДАМ. НО тЕпЕРь ВОжМО жНы ДАльНЕИшИЕ пРИлОжЕН Иь, НАпРИМЕР, к РАжлОжЕНИьМ пО пОлИ НОМАМ лЕжАНДРА, лАгЕР РА ИлИ ЁРМИтА.     

The formal translation equation for iteration groups of type II

We investigate the translation equation $$F(s+t, x) = F(s, F(t, x)),\quad \quad s,t\in{\mathbb{C}},\qquad\qquad\qquad\qquad({\rm T})$$ in ${\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ , the ring of formal power series over ${\mathbb{C}}$ . Here we restrict ourselves to iteration groups of type II, i.e. to solutions of (T) of the form ${F(s, x) \equiv x + c_k(s)x^k {\rm mod} x^{k + 1}}$ , where k ≥ 2 and c _k ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions c _n (s) of $$F(s, x) = x + \sum_{n \ge q k}c_n(s)x^n$$ are polynomials in c _k (s). It is possible to replace this additive function c _k by an indeterminate. In this way we obtain a formal version of the translation equation in the ring ${(\mathbb{C}[y])\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ . We solve this equation in a completely algebraic way, by deriving formal differential equations or an Aczél–Jabotinsky type equation. This way it is possible to get the structure of the coefficients in great detail which are now polynomials. We prove the universal character (depending on certain parameters, the coefficients of the infinitesimal generator H of an iteration group of type II) of these polynomials. Rewriting the solutions G(y, x) of the formal translation equation in the form ${\sum_{n\geq 0}\phi_n(x)y^n}$ as elements of ${(\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right])\left[\kern-0.15em\left[{y}\right]\kern-0.15em\right]}$ , we obtain explicit formulas for ${\phi_n}$ in terms of the derivatives H ^(j)(x) of the generator ${H}$ and also a representation of ${G(y, x)}$ as a Lie–Gröbner series. Eventually, we deduce the canonical form (with respect to conjugation) of the infinitesimal generator ${H}$ as x ^k + hx ^2k-1 and find expansions of the solutions ${G(y, x) = \sum_{r\geq 0} G_r(y, x)h^r}$ of the above mentioned differential equations in powers of the parameter h.     

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. $$\{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = c. \{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} = c_0 $$ .     

The Quintic Grassmannian G_{1,4,2} in PG(9, 2)

The 155 points of the Grassmannian $G_{1,4,2}$ of lines of PG (4, 2) = $\mathbb{P}V\left( {5,2} \right)$ are those points $x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X $ \subset $ PG (9, 2) will be termed odd or even according as X intersects $G_{1,4,2}$ in an odd or even number of points. Let $Q^\ddag \left( {x_1 ,...,x_5 } \right)$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X ^# of a r-flat X $ \subset $ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$ . Because $Q^\ddag$ is quinquelinear, the associate X ^# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X $ \subseteq$ X ^# while if X is an even 4-flat then X ^# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X ^# = X. An example of an even 4-flat X such that $\left( {X^\# } \right)^\#$ = X is provided by any 4-flat X which is external to $G_{1,4,2}$ . However, it appears that the two possibilities just illustrated, namely X ^# = X for an odd 4-flat and $\left( {X^\# } \right)^\#$ = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X ^# = PG (9, 2) and of even 4-flats for which ${X^{\# \# \# } }$ = X.     

Estimation of the remainder of a cubature formula on a Chebyshev grid

Let C(Q) denote the space of continuous functions f(x, y) in the square Q = [?1, 1] × [?1, 1] with the norm $\begin{gathered} \left\| f \right\| = \max \left| {f(x,y)} \right|, \hfill \\ (x,y) \in Q. \hfill \\ \end{gathered} $ On a Chebyshev grid, a cubature formula of the form $\int\limits_{ - 1}^1 {\int\limits_{ - 1}^1 {\frac{1} {{\sqrt {(1 - x^2 )(1 - y^2 )} }}f(x,y)dxdy = \frac{{\pi ^2 }} {{mn}}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {f\left( {\cos \frac{{2i - 1}} {{2n}}\pi ,\cos \frac{{2j - 1}} {{2m}}\pi } \right)} + R_{m,n} (f)} } } $ is considered in some class H(r ₁, r ₂) of functions f ?? C(Q) defined by a generalized shift operator. The remainder R _{m, n} (f) is proved to satisfy the estimate $\mathop {\sup }\limits_{f \in H(r_1 ,r_2 )} \left| {R_{m,n} (f)} \right| = O(n^{ - r_1 + 1} + m^{ - r_2 + 1} ), $ where r ₁, r ₂ > 1; ??^?1 ?? n/m ?? ?? with ?? > 0; and the constant in O(1) depends on ??.     

On Isomorphisms of Hardy Spaces Associated with Schrödinger Operators

Let L=?Δ+V is a Schrödinger operator on $\mathbb{R}^{d}$ , d≥3, V≥0. Let $H^{1}_{L}$ denote the Hardy space associated with L. We shall prove that there is an L-harmonic function w, 0<δ≤w(x)≤C, such that the mapping $$H_L^1 \ni f\mapsto wf\in H^1\bigl(\mathbb{R}^d\bigr) $$ is an isomorphism from the Hardy space $H_{L}^{1}$ onto the classical Hardy space $H^{1}(\mathbb{R}^{d})$ if and only if $\Delta^{-1}V(x)=-c_{d}\int_{\mathbb{R}^{d}} |x-y|^{2-d} V(y) dy$ belongs to $L^{\infty}(\mathbb{R}^{d})$ .     

Isothermic Submanifolds

We propose an answer to a question raised by F. Burstall: Is there any interesting theory of isothermic submanifolds of ? ⁿ of dimension greater than two? We call an n-immersion f(x) in ? ^m isothermic _k if the normal bundle of f is flat and x is a line of curvature coordinate system such that its induced metric is of the form $\sum_{i=1}^{n} g_{ii}\,\mathrm{d} x_{i}^{2}$ with $\sum_{i=1}^{n} \epsilon_{i} g_{ii}=0$ , where ?? _i =1 for 1??i??n?k and ?? _i =?1 for n?k<i??n. A smooth map (f ₁,??,f _n ) from an open subset ${\mathcal{O}}$ of ? ⁿ to the space of m×n matrices is called an n-tuple of isothermic _k n-submanifolds in ? ^m if each f _i is an isothermic _k immersion, $(f_{i})_{x_{j}}$ is parallel to $(f_{1})_{x_{j}}$ for all 1??i,j??n, and there exists an orthonormal frame (e ₁,??,e _n ) and a GL(n)-valued map (a _ij ) such that $\mathrm{d}f_{i}= \sum_{j=1}^{n} a_{ij} e_{j}\,\mathrm {d} x_{j}$ for 1??i??n. Isothermic₁ surfaces in ?³ are the classical isothermic surfaces in ?³. Isothermic _k submanifolds in ? ^m are invariant under conformal transformations. We show that the equation for n-tuples of isothermic _k n-submanifolds in ? ^m is the $\frac{O(m+n-k,k)}{O(m)\times O(n-k,k)}$ -system, which is an integrable system. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries.     

Summation of weakly dependent sequences by Cesàro methods

Пусть (X, A, u) — пространст во с конечной мерой, (ξ_k) ₁ ^∞ — последовательност ь функций, \(\xi _k \varepsilon L_{2r} (X), r > 1, \int\limits_X {\xi _k d\mu = 0} \) . Изучаются условия, п ри которых справедли вgа - у. з. б.ч., т. e. (ξ _k) суммируется к ну лю почти всюду методо м (С, а),а > 0. Приведем два резу льтата. 1) Если (ξ _k) — слабо мульт ипликативная систем а (в частности, мартингал-разности или независимая сист ема), то условие $$\mathop \sum \limits_1^\infty \mathop {\smallint }\limits_X \left| {\xi _k } \right|^{2r} d\mu \cdot c_r (k,\alpha )< \infty $$ влечетβ - у.з.б.ч. Здесьc _r(k,α)=k ^-2rα при 0<α<(r+1)/2r, c_r=k^?(r+1) In3r-1 k приа=(r+1)/2r, с_r=k^?(r+1) при а >(r+1)/2r. 2) Если (ξ _k) независимы,Mξ _k=0, (r+1)/2r<α=1, то условия $$\mathop \sum \limits_{k = 1}^\infty \frac{{(M\xi _k^2 )^r }}{{k^{r + 1} }}< \infty ,\mathop \sum \limits_{k = 1}^\infty \frac{{M|\xi _k |^{2r} }}{{k^{2r\alpha } }}< \infty $$ влекут за собой а - у. з. б. ч.     

Algebraic equations on the adèlic closure of a Drinfeld module

Let k be a field of positive characteristic and K = k(V) a function field of a variety V over k and let A _K be the ring of adèles of K with respect to the places on K corresponding to the divisors on V. Given a Drinfeld module $\Phi :\mathbb{F}[t] \to End_K (\mathbb{G}_a )$ over K and a positive integer g we regard both K ^g and A _K ^g as $\Phi \left( {\mathbb{F}_p [t]} \right)$ -modules under the diagonal action induced by Φ. For Γ ? K ^g a finitely generated $\Phi \left( {\mathbb{F}_p [t]} \right)$ -submodule and an affine subvariety $X \subseteq \mathbb{G}_a^g$ defined over K, we study the intersection of X(A _K ), the adèlic points of X, with $\bar \Gamma$ , the closure of Γ with respect to the adèlic topology, showing under various hypotheses that this intersection is no more than X(K) ∩ Γ.     

Convolutions of random measures on compact groups

LetG be a compact group andM ₁(G) be the convolution semigroup of all Borel probability measures onG with the weak topology. We consider a stationary sequence {μ _n } _n=?∞ ^+∞ of random measures μ _n =μ _n (ω) inM ₁(G) and the convolutions $$v_{m,n} (\omega ) = \mu _m (\omega )* \cdots *\mu _{n - 1} (\omega ), m< n$$ and $$\alpha _n^{( + k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n,n + i} (\omega ),} \alpha _n^{( - k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n - i,n} (\omega )} $$ We describe the setsA _m ⁺ (ω) andA _n ⁺ (ω) of all limit points ofv _m,n(ω) asm→?∞ orn→+∞ and the setA _∞(ω) of its two-sided limit points for typical realizations of {μ _n (ω)} _n=?∞ ^+∞ . Using an appropriate random ergodic theorem we study the limit random measures ρ _n ^(±) (ω)=lim _k→∞ α _n ^(±k) (ω).     

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 $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

Module amenability of the second dual and module topological center of semigroup algebras

In this paper we define the module topological center of the second dual $\mathcal{A}^{**}$ of a Banach algebra $\mathcal{A}$ which is a Banach $\mathfrak{A}$ -module with compatible actions on another Banach algebra $\mathfrak{A}$ . We calculate the module topological center of ? ¹(S)^**, as an ? ¹(E)-module, for an inverse semigroup S with an upward directed set of idempotents E. We also prove that ? ¹(S)^** is ? ¹(E)-module amenable if and only if an appropriate group homomorphic image of S is finite.     

On the Equivariant Cohomology of Subvarieties of a \mathfrak{B}-Regular Variety

By a $\mathfrak{B}$ -regular variety, we mean a smooth projective variety over $\mathbb{C}$ admitting an algebraic action of the upper triangular Borel subgroup $\mathfrak{B} \subset {\text{SL}}_{2} {\left( \mathbb{C} \right)}$ such that the unipotent radical in $\mathfrak{B}$ has a unique fixed point. A result of Brion and the first author [4] describes the equivariant cohomology algebra (over $\mathbb{C}$ ) of a $\mathfrak{B}$ -regular variety X as the coordinate ring of a remarkable affine curve in $X \times \mathbb{P}^{1}$ . The main result of this paper uses this fact to classify the $\mathfrak{B}$ -invariant subvarieties Y of a $\mathfrak{B}$ -regular variety X for which the restriction map i _Y : H ^*(X) → H ^*(Y) is surjective.