Asymptotic properties of powers of Kantorovič operators

Nagel, Josef

doi:10.1016/0021-9045(82)90046-6

Cited by 6 publications

(6 citation statements)

References 7 publications

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“…Indeed, this result has first been shown by Nagel [16] but only for f ∈ L 2 ([0, 1]). In contrast, our result not only extends the uniform convergence to the space L 1 ([0, 1]) we also derive the convergence in the uniform operator norm.…”

Section: Examples In C([0 1]) and L 1 ([0 1])mentioning

confidence: 74%

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On the spectrum of positive linear operators with a partition of unity property

Nagler

2015

Journal of Mathematical Analysis and Applications

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We characterize the spectrum of positive linear operators between Banach function spaces having finite rank and a partition of unity property. Our main result states that all the points in the spectrum are eigenvalues and 1 is the only eigenvalue on the unit circle. Finally, we show that the iterates converge in the uniform operator topology to a projection operator that reproduces constant functions and we provide a simple criterion to obtain the limiting projection operator.We study positive linear operators that have finite rank on a general infinite-dimensional complex Banach function space X that contains the constant function 1 with norm equal to one. In addition, we assume that the associated basis functions of the positive finite-rank operator form a partition of unity. Operators of this kind are used in many applications to approximate functions where only a finite number of samples are available. The partition of unity property guarantees the exact reconstruction of constant functions. Of our interest here is the asymptotic behaviour of iterative applications of the operator and the question whether the limit exists.The asymptotic behaviour of the iterates of positive linear operators has extensively been discussed by many authors. Kelisky and Rivlin [12] have first been considering the limit of iterates of the classical Bernstein operator on the space C ([0, 1]). This result has been extended by Karlin and Ziegler [10] to a more general setting. In [15,16], J. Nagel has examined the asymptotic behaviour of the Bernstein and the Kantorovič operators. Using a contraction principle, Rus [19] has shown an alternative way to prove the convergence of the iterates of the Bernstein operator. The iterates of the Bernstein operators have been also revisited by Badea [2] using spectral properties. Recently, contributions have been made by and by Altomare [1] using methods based on Korovkin-type approximation theory. However, all these results are restricted to the space of continuous functions, i.e., are not applicable for the L p spaces, and there is no general theory that guarantees the existence of the limit of the iterates.

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Section: Examples In C([0 1]) and L 1 ([0 1])mentioning

confidence: 74%

“…This result has been extended by Karlin and Ziegler [10] to a more general setting. In [15,16], J. Nagel has examined the asymptotic behaviour of the Bernstein and the Kantorovič operators. Using a contraction principle, Rus [19] has shown an alternative way to prove the convergence of the iterates of the Bernstein operator.…”

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confidence: 99%

On the spectrum of positive linear operators with a partition of unity property

Nagler

2015

Journal of Mathematical Analysis and Applications

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“…(ii) follows from (17) and Proposition 4. The proof of (iii) will take a turn which is completely parallel with the proof of Theorem 2 in [8]. Again without loss of generality we may restrict ourselves to the subspace (~g2) [ ' (x) (d,,.…”

Section: --+ O0mentioning

confidence: 99%

“…In a preceding publication [8] we have studied the limit behaviour of pk,,f The results from [8] now can be used to establish parallel theorems for the Kantorovie operators of second order. First we deal only with the limit behaviour of Q~"f for polynomials f We apply Proposition 1 from [8] to the integral function F, which is also a polynomial. Then passing to the derivative we obtain the following proposition.…”

Section: P Oc; X) = (D/dx) B+ 1 (F; X)mentioning

confidence: 99%