Tauberian Remainder Theory
This chapter deals with real Tauberian remainder theory, for both series and integrals, in the case where one has information on their transforms only in the real domain. There are three main topics.
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1 Introduction This chapter deals with real Tauberian remainder theory, for both series and integrals, in the case where one has information on their transforms only in the real domain. There are three main topics. The first part of the chapter, comprising Sections 1-12, deals with remainder theory for the case of Abel summability, so that the relevant transforms are power series or Laplace transforms. The remainders are obtained with the aid of polynomial approximation and, in some cases, complex analysis. The principal contributors to this theory were Freud, the author and Ganelius; see Sections 2-4 for the method and a detailed account of its development. At first sight, the remainders provided by the real theory would seem rather weak. For example, in the case (Xl
A
f(x) = "anx n = - ~ I-x
+ 0(1)
as x ./ 1,
an:::: -C,
n=O
Freud obtained the estimate
From the point of view of possible applications (cf. Section 22), the result was disappointing, and Freud's first article was held up for several years. When it became clear from Erdos-inspired examples of the author that the estimate was optimal, Turan quickly gave the green light for publication! Freud obtained his results by an important improvement of Karamata' s method for Hardy-Littlewood theorems (Section 1.11): he introduced the technique of optimal one-sided L 1 approximation by polynomials. Independently using L 1 approximation and helped by Freud's work, the author obtained a variety of more general remainder theorems; cf. Section 2 and Section 3 on Laplace transforms. Shortly thereafter, J. Korevaar, Tauberian Theory © Springer-Verlag Berlin Heidelberg 2004
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VII Tauberian Remainder Theory
Ganelius formulated the very general Theorem 2.5 below for Laplace-Stie1tjes transforms. The theory of one-sided L 1 approximation by polynomials takes up Sections 5-8. Examples to show the optimality of various remainder estimates are given in Section 10. Some interesting results of real remainder theory are derived by a complex method, which involves harmonic-measure arguments. We mention the proof of Theorem 3.3 on vanishing remainders in Section 9 and the remainder estimates for general Dirichlet series in Sections 11, 12. The latter extend work of Freud and Halasz. Our second topic, taking up Sections 13-20, is the powerful Fourier integral method developed mostly by Ganelius, which applies to large classes of Wiener kernels K. Ganelius took his cue from Beurling, who proposed to use complex-analytic properties of the reciprocal of the Fourier transform, 1/ K, for remainder estimates. Beurling treated a special case, which was later worked out by Lyttkens. Starting at about the same time (1954), Ganelius greatly enlarged the scope of the analytic theory; cf. Section 13. Sections 14-18 contain a thorough discussion of selected remainder theorems of Ganelius; additional results can be found in his Springer Lecture Notes [1971]. Section 19 is devoted to Frennemo's application of Ganelius's method to Laplace transforms. Later, Lyttkens replaced Ganelius'
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