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Coefficients calculation of the best linear method for recovery of bounded analytic functions in a circle
This paper considers the problem of optimal recovery of bounded analytic functions. Namely, the values of these functions are determined at the point from their values at n given points lying in the unit circle. At first, we recall the necessary basic concepts: error of approximation by some method (which is a complex function of n complex variables), the best approximation method. Some theorems from the works of K.U. Osipenko are discussed: on the existence of a best linear approximation method and on calculating the error of best recovery method. After that we write out the formula for finding the error of best approximation method of bounded analytic functions in a unit circle. The lemma of conformal invariance of optimal recovery problem of these functions follows. We prove that under conformal mapping of the unit circle onto itself the error of the best approximation method before mapping coincides with the error of the best approximation method after mapping. It is also proved that a linear best method after conformal mapping coincides with the linear best restore method before this mapping (wherein the problem of optimal recovery after mapping is considered on the images of n given points lying in the original unit circle). Finally, we consider the problem of optimal recovery of bounded analytic functions in a circle in special case when the given points coincide with the vertices of a regular n-gon, and the point itself coincides with its center (which coincides with the origin). We prove that all the coefficients of the best linear method in this case are identical (wherein we apply the lemma of conformal invariance of optimal recovery problem of bounded analytic functions). The formulas for calculating these coefficients are given (for this purpose we write out an integral). The result is the smart, simple formulas for calculating the coefficients of the best linear approximation method for this particular case.
Coefficients calculation of the best linear method for recovery of bounded analytic functions in a circle
This paper considers the problem of optimal recovery of bounded analytic functions. Namely, the values of these functions are determined at the point from their values at n given points lying in the unit circle. At first, we recall the necessary basic concepts: error of approximation by some method (which is a complex function of n complex variables), the best approximation method. Some theorems from the works of K.U. Osipenko are discussed: on the existence of a best linear approximation method and on calculating the error of best recovery method. After that we write out the formula for finding the error of best approximation method of bounded analytic functions in a unit circle. The lemma of conformal invariance of optimal recovery problem of these functions follows. We prove that under conformal mapping of the unit circle onto itself the error of the best approximation method before mapping coincides with the error of the best approximation method after mapping. It is also proved that a linear best method after conformal mapping coincides with the linear best restore method before this mapping (wherein the problem of optimal recovery after mapping is considered on the images of n given points lying in the original unit circle). Finally, we consider the problem of optimal recovery of bounded analytic functions in a circle in special case when the given points coincide with the vertices of a regular n-gon, and the point itself coincides with its center (which coincides with the origin). We prove that all the coefficients of the best linear method in this case are identical (wherein we apply the lemma of conformal invariance of optimal recovery problem of bounded analytic functions). The formulas for calculating these coefficients are given (for this purpose we write out an integral). The result is the smart, simple formulas for calculating the coefficients of the best linear approximation method for this particular case.
Coefficients calculation of the best linear method for recovery of bounded analytic functions in a circle
Ovchintsev Mikhail Petrovich (author) / Gusakova Ekaterina Mikhaylovna (author)
2014
Article (Journal)
Electronic Resource
Unknown
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