Abstract
For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form \(L_n (\psi ) = \sum\limits_{k = 0}^n {a_k \psi ^{(k)} } \), where the constant coefficients a k ∈ R may be adapted to f.
We prove that for each f ∈ C (n)(I), there is a selection of coefficients {a 1, …, a n } and a corresponding linear combination
of functions ψ k (t) = e λ k t in the nullity of L which satisfies the following Jackson’s type inequality:
where \(\left| {\lambda _n } \right| = \mathop {\max }\limits_k \left| {\lambda _k } \right|\), 0 ≤ m ≤ n −1, p, q ≥ 1, and \(\frac{1} {p} + \frac{1} {q} = 1 \).
For the particular operator M n (f) = f +1/(2n)! f (2n) the rate of approximation by the eigenvalues of M n for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.
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Vatchev, V. On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients. Anal. Theory Appl. 27, 187–200 (2011). https://doi.org/10.1007/s10496-011-0187-3
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DOI: https://doi.org/10.1007/s10496-011-0187-3