Approximation by q-Bernstein Polynomials in the Case q→1+
Let Bn,q(f;x), q∈(0,∞) be the q-Bernstein polynomials of a function f∈C[0,1]. It has been known that, in general, the sequence Bn,qn(f) with qn→1+ is not an approximating sequence for f∈C[0,1], in contrast to the standard case qn→1-. In this paper, we give the sufficient and necessary condition unde...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/259491 |
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| Summary: | Let Bn,q(f;x), q∈(0,∞) be the q-Bernstein polynomials of a function f∈C[0,1]. It has been known that, in general, the sequence Bn,qn(f) with qn→1+ is not an approximating sequence for f∈C[0,1], in contrast to the standard case qn→1-. In this paper, we give the sufficient and necessary condition under which the sequence Bn,qn(f) approximates f for any f∈C[0,1] in the case qn>1. Based on this condition, we get that if 1<qn<1+ln2/n for sufficiently large n, then Bn,qn(f) approximates f for any f∈C[0,1]. On the other hand, if Bn,qn(f) can approximate f for any f∈C[0,1] in the case qn>1, then the sequence (qn) satisfies lim¯n→∞n(qn-1)≤ln2. |
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| ISSN: | 1085-3375 1687-0409 |