ON THE CONVERGENCE OF SEQUENCES OF KANTOROVICH OPERATORS IN LEBESGUE SPACES
According to the Weierstrass Approximation Theorem, any continuous function on the closed and bounded interval can be approximated by polynomials. A constructive proof of this theorem uses the so-called Bernstein polynomials. For the approximation of integrable functions, we may consider Kantorov...
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| Format: | Final Project |
| Language: | Indonesia |
| Online Access: | https://digilib.itb.ac.id/gdl/view/71845 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | According to the Weierstrass Approximation Theorem, any continuous function
on the closed and bounded interval can be approximated by polynomials. A
constructive proof of this theorem uses the so-called Bernstein polynomials. For
the approximation of integrable functions, we may consider Kantorovich operators
as certain modifications for Bernstein polynomials. In this final project, we investigate
the behaviour of Kantorovich operators in Lebesgue spaces. We first give a
rather ad hoc proof of the uniform boundedness of Kantorovich operators. We then
provide two alternative proofs by using the Riesz-Thorin Interpolation Theorem and
the boundedness of Hardy-Littlewood maximal operator respectively. In addition,
we discuss the convergence of the sequence of Kantorovich operators in Lebesgue
spaces of finite exponents. We provide a counterexample to the convergence of
Kantorovich operators in the space of essentially bounded functions. We also
consider the rate of convergence of Kantorovich operators in a certain subspace
of Lebesgue spaces. |
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