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|Random Linear Operators Arising from Piecewise Linear Interpolation on the Unit Interval
|Adell, José A.
|We introduce a sequence of random linear operators arising from piecewise linear interpolation at a set of random nodes on the unit interval. We show that such operators uniformly converge in probability to the target function, providing at the same time rates of convergence. Analogous results are shown for their deterministic counterparts, derived by taking expectations of the aforementioned random operators. Special attention is paid to the case in which the random nodes are the uniform order statistics, where an explicit form for their associated deterministic operators is provided. This allows us to compare the speed of convergence of the aforementioned operators with that of the random and deterministic Bernstein polynomials
|Random linear operator
Uniform convergence in probability
Ditzian–Totik modulus of smoothness
Uniform order statistics
Rate of convergence
|Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work is partially supported by the Spanish government, Research Project (PGC2018-097621-B-I00). The first author is also supported by Research Project DGA (E-64). The second author is also supported by Junta de Andalucía, Research Group, (FQM-0178).
|Birkhäuser - Springer
|Adell, J.A., Cárdenas-Morales, D. Random Linear Operators Arising from Piecewise Linear Interpolation on the Unit Interval. Mediterr. J. Math. 19, 223 (2022)
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