scholarly journals Discrete quadratic estimates and holomorphic functional calculi in Banach spaces

1998 ◽  
Vol 58 (2) ◽  
pp. 271-290 ◽  
Author(s):  
Edwin Franks ◽  
Alan McIntosh
Keyword(s):  
Functional Calculus ◽  
Sufficient Conditions ◽  
Uniform Norm ◽  
Commuting Operators ◽  
Orthonormal Sequence ◽  
Quadratic Estimates ◽  
Holomorphic Functional Calculi ◽  
Necessary And Sufficient ◽  
Dyadic Decomposition ◽  
Uniformly Bounded

We develop a discrete version of the weak quadratic estimates for operators of type w explained by Cowling, Doust, McIntosh and Yagi, and show that analogous theorems hold. The method is direct and can be generalised to the case of finding necessary and sufficient conditions for an operator T to have a bounded functional calculus on a domain which touches σ(T) nontangentially at several points. For operators on Lp, 1 < p < ∞, it follows that T has a bounded functional calculus if and only if T satisfies discrete quadratic estimates. Using this, one easily obtains Albrecht's extension to a joint functional calculus for several commuting operators. In Hilbert space the methods show that an operator with a bounded functional calculus has a uniformly bounded matricial functional calculus.The basic idea is to take a dyadic decomposition of the boundary of a sector Sv. Then on the kth ingerval consider an orthonormal sequence of polynomials . For h ∈ H∞(Sν), estimates for the uniform norm of h on a smaller sector Sμ are obtained from the coefficients akj = (h, ek, j). These estimates are then used to prove the theorems.

scholarly journals Modified cauchy kernels and functional calculus for operators on Banach space

1997 ◽  
Vol 63 (1) ◽  
pp. 91-99 ◽  
Author(s):  
Edwin Franks
Keyword(s):  
Banach Space ◽  
Functional Calculus ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Cauchy Kernel ◽  
Necessary And Sufficient ◽  
Banach Space Operators

AbstractIn Banach space operators with a bounded H∞ functional calculus, Cowling et al. provide some necessary and sufficient conditions for a type-ω operator to have a bounded H∞ functional calculus. We provide an alternate development of some of their ideas using a modified Cauchy kernel which is L1 with respect to the measure ]dz]/]z]. The method is direct and has the advantage that no transforms of the functions are necessary.

scholarly journals Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

2018 ◽  
Vol 50 (01) ◽  
pp. 178-203 ◽  
Author(s):  
Nicolas Champagnat ◽  
Denis Villemonais
Keyword(s):  
Sufficient Conditions ◽  
Initial Distribution ◽  
Initial Position ◽  
Local Martingale ◽  
One Dimensional ◽  
Positive Time ◽  
Strict Local Martingale ◽  
Necessary And Sufficient ◽  
Stationary Behavior ◽  
Uniformly Bounded

Abstract In this paper we study the quasi-stationary behavior of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one-dimensional strict local martingale diffusions coming down from infinity. We prove, under mild assumptions, that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps.

Orthonormal expansions and the principle of uniform boundedness

1991 ◽  
Vol 43 (2) ◽  
pp. 341-347
Author(s):  
S.A. Husain ◽  
V.M. Sehgal
Keyword(s):  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Orthonormal Sequence ◽  
Necessary And Sufficient ◽  
Complex Valued ◽  
Orthonormal Expansions

Let {φν: ν ∈ N (non-negative integers)} ⊆ C[0, 1] be a complete orthonormal sequence of complex-valued functions in L2[0, 1], {λν: ν ∈ N} and {λνμ: ν, μ ∈ N} be sequences of complex numbers. In this paper, the necessary and sufficient conditions are developed for the series to converge and also to exist, in C[0, 1] for each f ∈ L1[0, 1] where .

scholarly journals Holomorphic functional calculi and sums of commuting opertors

1998 ◽  
Vol 58 (2) ◽  
pp. 291-305 ◽  
Author(s):  
David Albrecht ◽  
Edwin Franks ◽  
Alan McIntosh
Keyword(s):  
Banach Space ◽  
Holomorphic Function ◽  
Functional Calculus ◽  
Holomorphic Functional Calculus ◽  
Regularity Problem ◽  
Commuting Operators ◽  
Consistent Manner ◽  
Holomorphic Functional Calculi ◽  
The Individual

Let S and T be commuting operators of type ω and type ϖ in a Banach space X. Then the pair has a joint holomorphic functional calculus in the sense that it is possible to define operators f(S, T) in a consistent manner, when f is a suitable holomorphic function defined on a product of sectors. In particular, this gives a way to define the sum S + T when ω + ϖ < π. We show that this operator is always of type μ where μ = max{ω, ϖ}. We explore when bounds on the individual functional calculi of S and T imply bounds on the functional calculus of the pair (S, T), and some implications for the regularity problem of when ∥(S + T)u∥ is equivalent to ∥Su∥ + ∥Tu∥.

Quasi-Monte Carlo tractability of integration problem in function spaces defined over products of balls

2019 ◽  
Vol 17 (06) ◽  
pp. 1950043
Author(s):  
Jie Zhang ◽  
Yongping Liu
Keyword(s):  
Monte Carlo ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Weighted Sobolev Spaces ◽  
Sufficient Conditions ◽  
Quasi Monte Carlo ◽  
Integration Problem ◽  
Kernel Hilbert Spaces ◽  
Necessary And Sufficient ◽  
Uniformly Bounded

Recently, quasi-Monte Carlo (QMC) rules for multivariate integration in some weighted Sobolev spaces of functions defined over unit cubes [Formula: see text], products of [Formula: see text] copies of the simplex [Formula: see text] and products of [Formula: see text] copies of the unit sphere [Formula: see text] have been well-studied in the literature. In this paper, we consider QMC tractability of integrals of functions defined over product of [Formula: see text] copies of the ball [Formula: see text]. The space is a tensor product of [Formula: see text] reproducing kernel Hilbert spaces defined by positive and uniformly bounded weights [Formula: see text] for [Formula: see text]. We obtain matching necessary and sufficient conditions in terms of this weights for various notions of QMC tractability, including strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniformly weak tractability and [Formula: see text]-weak tractability.

scholarly journals EXPONENTIAL POLYNOMIAL APPROXIMATION OF WEIGHTED BANACH SPACE ON ℝn

2012 ◽  
Vol 55 (1) ◽  
pp. 115-121 ◽  
Author(s):  
XIANGDONG YANG
Keyword(s):  
Banach Space ◽  
Polynomial Approximation ◽  
Sufficient Conditions ◽  
Uniform Norm ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Exponential Polynomial ◽  
Exponential System ◽  
Weighted Banach Space ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions for the incompleteness of exponential system in Cα are characterised, where Cα is the weighted Banach space of complex continuous functions f defined on ℝn with f(t)exp(−α(t)) vanishing at infinity in the uniform norm.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Budgeting in Socio-economic Development Strategies (Raising a Problem)

2008 ◽  
pp. 134-151
Author(s):  
A. Shastitko ◽  
M. Ovchinnikov
Keyword(s):  
Economic Policy ◽  
Economic Development ◽  
Social Change ◽  
Eastern Europe ◽  
Statistical Data ◽  
Sufficient Conditions ◽  
Transitional Period ◽  
Socio Economic Development ◽  
Necessary And Sufficient ◽  
Economic Development Strategies

The article proposes an approach to the analysis of social change and contributes to the clarification of concepts of economic policy. It deals in particular with the notion of "change of system". The author considers positive and normative aspects of the analysis of capitalist and socialist systems. The necessary and sufficient conditions for the system to be changed are introduced, their fulfillment is discussed drawing upon the historical and statistical data. The article describes both economic and political peculiarities of the transitional period in different countries, especially in Eastern Europe.

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