Direct Approximation Theorems for Dirichlet Series in the Norm of Uniform Convergence

B. Forster

Journal of Approximation Theory, vol. 132, no. 1, pp. 1–14, January 2005.

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We consider functions ƒ ∈ AC(D‾) on a convex polygon D ⊂ C and their regularity in terms of P.M. Tamarazov's moduli of smoothness. Using the relation between Fourier and Leont′ev coefficients given in [1] we prove direct approximation theorems of Jackson type for the Dirichlet expansion

ƒ(z) ∼ ∑_λ∈Λ κ_ƒ(Λ) e^{Λ z} ⁄ L′(Λ),

where L(z) = ∑_k=1^N d_k e^{a_k z} is a quasipolynomial with respect to the vertices a₁,…,a_N of D and Λ its set of zeros. We show by an example that our results Improve Mel′nik's estimates in [2] on the rate of convergence.

References

1. B. Forster, "On the Relation Between Fourier and Leont′ev Coefficients with Respect to the Space AC(D‾)," Computational Methods and Function Theory, vol. 1, no. 1, pp. 193-204, 2001.

2. Y.I. Mel′nik, "Approximation of Functions Regular in Convex Polygons by Exponential Polynomials," Ukrainian Mathematical Journal, vol. 40, no. 4, pp. 382-387, April 1988.

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AUTHOR="Forster, B.",
TITLE="Direct Approximation Theorems for {D}irichlet Series in the Norm
	of Uniform Convergence",
JOURNAL="Journal of Approximation Theory",
YEAR="2005",
volume="132",
number="1",
pages="1--14",
month="January",
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DOI: 10.1016/j.jat.2004.10.008