On Some Degenerate Differential Operators on Weighted Function Spaces

Altomare, Francesco; Carbone, Ingrid

doi:10.1006/jmaa.1997.5540

Cited by 18 publications

(14 citation statements)

References 12 publications

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“…then the semigroup (T * (t)) t 0 can be represented in terms of the above powers in the following way (see [4,Theorem 4.3]):…”

Section: Notation and Main Definitionsmentioning

confidence: 99%

“…In two recent papers [3,4] the case of unbounded intervals has been also considered. It was introduced a sequence of positive linear operators acting on weighted function spaces on the interval [0, + [, denoted by (M n, * ) n 1 , where * is a continuous function on [0, + [ such that 0 * 1.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, generalizing some results obtained by Becker and Nessel [5], it was shown that the differential operator A * u(x) :=(x*(x)Â2) u"(x) (x>0) defined on a suitable domain of W 0 2 , generates a C 0 -semigroup (T * (t)) t 0 which can be expressed in terms of the sequence (M n, * ) n 1 , via formula (III) [4]. Here W 0 2 denotes the space of all continuous functions f on [0, + [ satisfying lim x Ä + ( f (x)Â(1+x 2 ))=0.…”

Section: Introductionmentioning

confidence: 99%

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Shape Preserving Properties of Some Positive Linear Operators on Unbounded Intervals

Carbone¹

1998

Journal of Approximation Theory

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In this paper we study some shape preserving properties of particular positive linear operators acting on spaces of continuous functions defined on the interval [0, + [, which are strongly related to the semigroups generated by a large class of degenerate elliptic second order differential operators. We study the conditions under which these operators leave invariant the class of increasing functions, as well as the class of convex functions and Ho lder continuous functions. As a consequence, we derive some regularity results concerning the related semigroups.

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“…then the semigroup (T * (t)) t 0 can be represented in terms of the above powers in the following way (see [4,Theorem 4.3]):…”

Section: Notation and Main Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Shape Preserving Properties of Some Positive Linear Operators on Unbounded Intervals

Carbone¹

1998

Journal of Approximation Theory

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“…In the papers [4,6,10] the authors showed, among other things, that the iterates of Szász-Mirakjan operators, Baskakov operators and Post-Widder operators converge to C 0 -semigroups of positive operators acting on suitable weighted spaces of continuous functions on [0, +∞[. The generators of these semigroups are showed to be the differential operators A i (u) = 1 2 p i u defined on suitable domains, where p 1 (x) = x, p 2 (x) = x(1 + x) and p 3 (x) = x 2 (x 0).…”

Section: Introductionmentioning

confidence: 98%

On a class of exponential-type operators and their limit semigroups

Altomare

Raşa

2005

Journal of Approximation Theory

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The paper is mainly focused upon the study of a class of second order degenerate elliptic operators on unbounded intervals.We show that these operators generate strongly continuous semigroups in suitable weighted spaces of continuous functions.Furthermore, we represent the semigroups as limits of iterates of the so-called exponential-type operators.In a particular case, starting from the stochastic differential equations associated with these operators, we also find an integral representation of the semigroup and determine its asymptotic behaviour.

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“…In the setting of unbounded intervals, we also refer to [6][7][8], and [14] for additional results about the representation of semigroups by positive linear operators.…”

Section: Introductionmentioning

confidence: 99%

APPROXIMATION BY POSITIVE OPERATORS OF THE C₀–SEMIGROUPS ASSOCIATED WITH ONE-DIMENSIONAL DIFFUSION EQUATIONS: PART II

Altomare

Amiar

2005

Numerical Functional Analysis and Optimization

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In this second part of the paper, we state some general conditions that guarantee that the C 0 -semigroups generated by special classes of one-dimensional second-order differential operators acting on weighted spaces of continuous functions on an arbitrary real interval can be represented as limits of iterates of the positive linear operators constructively defined according to the method developed in the first part. The particular case where the interval is compact is treated in full detail. Finally, an application to the Black-Scholes equation, which appears in some models arising from mathematical finance, is discussed as well.

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On Some Degenerate Differential Operators on Weighted Function Spaces

Cited by 18 publications

References 12 publications

Shape Preserving Properties of Some Positive Linear Operators on Unbounded Intervals

Shape Preserving Properties of Some Positive Linear Operators on Unbounded Intervals

On a class of exponential-type operators and their limit semigroups

APPROXIMATION BY POSITIVE OPERATORS OF THE C₀–SEMIGROUPS ASSOCIATED WITH ONE-DIMENSIONAL DIFFUSION EQUATIONS: PART II

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On Some Degenerate Differential Operators on Weighted Function Spaces

Cited by 18 publications

References 12 publications

Shape Preserving Properties of Some Positive Linear Operators on Unbounded Intervals

Shape Preserving Properties of Some Positive Linear Operators on Unbounded Intervals

On a class of exponential-type operators and their limit semigroups

APPROXIMATION BY POSITIVE OPERATORS OF THE C0–SEMIGROUPS ASSOCIATED WITH ONE-DIMENSIONAL DIFFUSION EQUATIONS: PART II

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APPROXIMATION BY POSITIVE OPERATORS OF THE C₀–SEMIGROUPS ASSOCIATED WITH ONE-DIMENSIONAL DIFFUSION EQUATIONS: PART II