Optimal methods for the approximate calculation of functional on classesW r L ∞

Пусть T_n(f)={L₁(f), ..., L_n(f)} — набор линейных функционал ов, заданных на простран стве \(C_{(r - 1)} (\parallel f\parallel _{C_{(r - 1)} } = \mathop {\max }\limits_{0 \leqq i \leqq r - 1} \parallel f^{(i)} \parallel _C );A_{n,r}\) — множество всех так их наборов функцио налов; С_{2n, 2} — множество всех н аборов из 2n функциона лов вида $$T_{2n} (f) = \{ f(x_1 ), \ldots ,f(x_n ),f'(x_1 ), \ldots ,f'(x_n )\}$$ и s: Е_n→Е₁. Доказано, что е слиW _∞ ^r множество всех 2π-периодических функ цийfεW∞0, 2π_r, то приr=1,2,3,... ирε(1, ∞) и $$\begin{gathered} \mathop {\inf }\limits_{T_{2n} \in A_{2n,r} } \parallel \mathop {\inf }\limits_s \mathop {\sup }\limits_{f \in W_\infty ^r } |f( \cdot ) - s(T_{2n} ,f, \cdot )|\parallel _p = \parallel \varphi _{n,r} \parallel _p \hfill \\ \mathop {\inf }\limits_{T_{2n} \in C_{2n,2} } \parallel \mathop {\inf }\limits_s \mathop {\sup }\limits_{f \in W_\infty ^r } |f( \cdot ) - s(T_{2n} ,f, \cdot )|\parallel _p = \parallel \parallel \varphi _{n,r} \parallel _\infty - \varphi _{n,r} \parallel _p , \hfill \\ \end{gathered}$$ где ?_n,r —r-й периодичес кий интеграл, в средне м равный нулю на периоде, от фун кции ?_n, 0t=sign sinnt. При этом указан ы оптимальные методы приближенного вычис ления.     

H ∞-calculus for the Stokes operator on unbounded domains

We consider the Stokes operator A on unbounded domains of uniform C ^1,1-type. Recently, it has been shown by Farwig, Kozono and Sohr that – A generates an analytic semigroup in the spaces , 1 < q < ∞, where for q ≥ 2 and for q ∈ (1, 2). Moreover, it was shown that A has maximal L ^p -regularity in these spaces for p ∈ (1,∞). In this paper we show that ɛ + A has a bounded H ^∞-calculus in for all q ∈ (1, ∞) and ɛ > 0. This allows to identify domains of fractional powers of the Stokes operator. Received: 12 October 2007     

Bergman kernel function on the third Hua Construction

The Bergman kernel function for Hua Construction of the third type is given in an explicit formula.     

Banach-valued holomorphic functions on the maximal ideal space of H ∞

We study Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of the Banach algebra H ^∞ of bounded holomorphic functions on the unit disk $\mathbb{D}\subset \mathbb{C}$ with pointwise multiplication and supremum norm. In particular, we establish vanishing cohomology for sheaves of germs of such functions and, solving a Banach-valued corona problem for H ^∞, prove that the maximal ideal space of the algebra $H_{\mathrm{comp}}^{\infty}(A)$ of holomorphic functions on $\mathbb{D}$ with relatively compact images in a commutative unital complex Banach algebra A is homeomorphic to the direct product of maximal ideal spaces of H ^∞ and A.     

Some integral operators acting on H ∞

Let f and g be analytic on the unit disk \({\mathbb{D}}\) . The integral operator T _g is defined by \({ T_g f(z) = \int_0^z f(t)g'(t) \,dt, z \in \mathbb{D}}\) . The problem considered is characterizing those symbols g for which T _g acting on H ^∞, the space of bounded analytic functions on \({\mathbb{D}}\) , is bounded or compact. When the symbol is univalent, these become questions in univalent function theory. The corresponding problems for the companion operator, \({ S_g f(z)= \int_0^z f'(t)g(t) \,dt}\) , acting on H ^∞ are also studied.     

Maximizing the L∞ norm of the gradient of solutions to the Poisson equation

We find the maximum of ¦Du _f ¦ _L∞ when u_f is the solution, which vanishes at infinity, of the Poisson equation Δu =f on ? ⁿ in terms of the decreasing rearrangement off. Hence, we derive sharp estimates for ¦Du _f ¦ _L∞ in terms of suitable Lorentz orL ^p norms off. We also solve the problem of maximizing ¦Du _f ^B (0)¦ whenu _f ^B is the solution, vanishing on?B, to the Poisson equation in a ballB centered at 0 and the decreasing rearrangement off is assigned.     

A note on the equal division kernel and the α‐power model

The equal division kernel and the α‐power solution are two recently proposed theories of coalition formation in n‐person characteristic function games with sidepayments. A closed form solution for payoff disbursement is derived from the α‐power model for n‐person games with only 1, n‐1, and n‐person permissible coalitions and no weak players. It is shown that for this class of games, which are frequently employed to experimentally test theories of coalition formation, the equal division kernel is a special case of the α‐power model with α = 1/2.     

A correction theorem and the dyadic space H1, ∞

It is shown that for every l-function f and for every , >0, there exists a function g such that mes {t=g} <, while the partial sums of the Fourier and Fourier-Walsh series of the function g are uniformly bounded by the number C log (^–1)f. In the proof we make use of the characterization of the dyadic space H^1, in terms of atomic decompositions (it is, apparently, new).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 67–75, 1986.     

Bergman kernel function on Hua Construction of the second type

In this paper, we give an explicit formula of the Bergman kernel function on Hua Construction of the second type when the parameters 1/p1,…,1/(pr-1) are positive integers and 1/pr is an arbitrary positive real number.     

A note on the approximate amenability of semigroup algebras

It is known that the bicyclic semigroup S ₁ is an amenable inverse semigroup. In this note we show that the convolution semigroup algebra ℓ ¹(S ₁) is not approximately amenable.     

Weighted Composition Operators of H∞ into α-Bloch Spaces on the Unit Ball

In this paper we characterize the boundedness and compactness of weighted composition operators of H∞ into α-Bloch spaces on the unit ball in C^n.     

Interpolating sequences for H∞ in the ball

We give a sufficient condition for a sequence of points in the unit ball of ℂⁿ to be an interpolating sequence for H^∞. This result generalizes Carleson's interpolation theorem in one complex variable, but an example shows that the condition is not necessary when n≥1.     

Thin sequences in the corona of H ∞

In this paper we consider several conditions for sequences of points in M(H ^∞) and establish relations between them. We show that every interpolating sequence for QA of nontrivial points in the corona $M(H^\infty )\backslash \mathbb{D}$ of H ^∞ is a thin sequence for H ^∞, which satisfies an additional topological condition. The discrete sequences in the Shilov boundary of H ^∞ necessarily satisfy the same condition.     

Well-conditioned computation for H ∞ controller near the optimum

The present paper gives a procedure for determining a H optimal controller in the assumption that the game Riccati equations have stabilizing positive definite solutions at the optimum value. A specific feature of the construction is its first step consisting in balancing with respect to the positive definite stabilizing solutions of the Riccati equations. The justification is based on singular perturbations reduction.