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On Hahn polynomial expansion of a continuous function of bounded variation | | René Goertz
; Philipp Öffner
; | | Date: |
21 Oct 2016 | | Abstract: | We consider the well-known method of least squares on an equidistant grid
with $N+1$ nodes on the interval $[-1,1]$. We investigate the following
problem: For which ratio $N/n$ and which functions, do we have pointwise
convergence of the least square operator
${LS}_n^N:mathcal{C}left[-1,1
ight]
ightarrowmathcal{P}_n$? To solve this
problem we investigate the relation between the Jacobi polynomials
$P_k^{alpha,eta}$ and the Hahn polynomials
$Q_kleft(cdot;alpha,eta,N
ight)$. Thereby we describe the least square
operator ${LS}_n^N$ by the expansion of a function by Hahn polynomials. In
particular we present the following result: The series expansion
$sum_{k=0}^n{hat{f} Q_k}$ of a function $f$ by Hahn polynomials $Q_k$
converges pointwise, if the series expansion $sum_{k=0}^n{hat{f} P_k}$ of the
function $f$ by Jacobi polynomials $P_k$ converges pointwise and if
${n^4}/N
ightarrow 0$ for $n,N
ightarrowinfty$. Furthermore we obtain the
following result: Let
$finleft{ginmathcal{C}^1left[-1,1
ight]:g^primeinmathcal{BV}left[-1,1
ight]
ight}$
and let $(N_n)_{n}$ be a sequence of natural numbers with
${n^4}/{N_n}
ightarrow 0$. Then the least square method ${LS}_n^{N_n}[f]$
converges for each $xin[-1,1]$. | | Source: | arXiv, 1610.6748 | | Services: | Forum | Review | PDF | Favorites | |
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