scholarly journals On an interpolation process of Lagrange-Hermite type

2012 ◽  
Vol 91 (105) ◽  
pp. 163-175
Author(s):  
Giuseppe Mastroianni ◽  
Gradimir Milovanovic ◽  
Incoronata Notarangelo
Keyword(s):  
Hermite Polynomial ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Necessary And Sufficient Conditions ◽  
Interpolation Process ◽  
Lp Spaces ◽  
Related Operator ◽  
Optimal Estimates ◽  
Necessary And Sufficient

We consider a Lagrange-Hermite polynomial, interpolating a function at the Jacobi zeros and, with its first (r?1) derivatives, at the points ?1. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator in certain suitable weighted Lp-spaces, 1 < p < 1, proving a Marcinkiewicz inequality involving the derivative of the polynomial at ?1. Moreover, we give optimal estimates for the error of this process also in the weighted uniform metric.

Hermite-Fejér interpolation with Jacobi weights

2011 ◽  
Vol 48 (3) ◽  
pp. 408-420
Author(s):  
G. Mastroianni ◽  
J. Szabados
Keyword(s):  
Error Estimates ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Interpolation Process ◽  
Weighted Norm ◽  
Jacobi Weights ◽  
Jacobi Weight ◽  
Necessary And Sufficient

We consider the weighted Hermite-Fejér interpolation process based on Jacobi nodes for classes of locally continuous functions defined by another Jacobi weight. Necessary and sufficient conditions for the weighted norm boundedness and for the convergence, as well as error estimates of the approximation, are given.

Entropy Numbers of Certain Summation Operators

2001 ◽  
Vol 8 (2) ◽  
pp. 245-274
Author(s):  
J. Creutzig ◽  
W. Linde
Keyword(s):  
Wiener Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nondecreasing Sequence ◽  
Entropy Numbers ◽  
Optimal Estimates ◽  
Summation Operator ◽  
Necessary And Sufficient

Abstract Given nonnegative real sequences and we study the generated summation operator regarded as a mapping from ℓ p (ℤ) to ℓ q (ℤ). We give necessary and sufficient conditions for the boundedness of S 𝑎, 𝑏 and prove optimal estimates for its entropy numbers relative to the summation properties of 𝑎 and 𝑏. Our results are applied to the investigation of the behaviour of as ε → 0, where is some nondecreasing sequence in [0, ∞) and (W(t)) t ≥ 0 denotes the Wiener process.

scholarly journals Basic properties of finite sum of weighted composition operators

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 4005-4019
Author(s):  
Abolghasem Alishahi ◽  
Saeedeh Shamsigamchi ◽  
Ali Ebadian
Keyword(s):  
Composition Operators ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weighted Composition Operators ◽  
Lp Spaces ◽  
Basic Properties ◽  
Weighted Composition ◽  
Finite Sums ◽  
Finite Sum ◽  
Necessary And Sufficient

In this paper,we continue the study of finite sum of weighted composition operators between different Lp-spaces that was investigated by Jabbarzadeh and Estaremi in 2012. Indeed, we first obtain some necessary and sufficient conditions for boundedness of the finite sums of weighted composition operators between distinct Lp-spaces. In the sequel, we investigate the compactness of finite sum of weighted composition operators. By using theorems of boundedness and compactness, we estimate the essential norms of these operators. Finally, some examples to illustrate the main results are given.

scholarly journals On properties of multiplication conditional type operators between Lp-space

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1933-1940
Author(s):  
Yousef Estaremi
Keyword(s):  
Measure Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closed Range ◽  
Lp Space ◽  
Lp Spaces ◽  
Necessary And Sufficient

In this paper, firstwe give some necessary and sufficient conditions for multiplication conditional type operators between two Lp-spaces to have closed range. Then we investigate Fredholm ones when the underlying measure space is non-atomic. Finally we give some examples.

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 .

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.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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