HARDY : Divergent Series, 1963

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Godfrey Harold HARDY

DIVERGENT

SERIES

3rd corrected printing

Oxford University Press
1963

Auteur :
Godfrey Harold HARDY

Thèmes :
MATHÉMATIQUES
Analyse

Reprint 1992
16 x 24 cm
416 p.
Broché
ISBN : 978-2-87647-131-3



S O M M A I R E

1 - Introduction.
- The sum of a series.
- Some calculations with divergent series.
- First definitions.
- Regularity of a method.
- Divergent integrals and generalized limits of functions of a continuous variable.
- Some historical remarks.
- A note on the British analysts of the early nineteenth century.

2- Some historical examples.
- Introduction.
- A. Euler and the functional equation of Riemann's zeta-function.
- B. Euler and the series 1 - 1!x + 2!x2 - ...
- C. Fourier and Fourier's theorem.
- D. Heaviside's exponential series.

3 - General theorems.
- Generalities concerning linear transformations.
- Regular transformations.
- Proof of Theorems.
- Variants and analogues.
- Positive transformations.
- Knopp's kernel theorem.
- Dilution of series.

4 - Special methods of summation.
- Nörlund means.
- Regularity and consistency of Nörlund means.
- Inclusion.
- Equivalence.
- Another theorem concerning inclusion.
- Euler means.
- Abelian means.
- A theorem of inclusion for Abelian means.
- Complex methods.
- Summability of 1 - 1 + 1 - ... by special Abelian methods.
- Lindelöf's and Mittag-Leffler's methods.
- Means defined by integral functions.
- Moment constant methods.
- A theorem of consistency.
- Methods ineffective for the series 1 - 1 + 1 - ...
- Riesz's typical means.
- Methods suggested by the theory of Fourier series.
- A general principle.

5 - Arithmetic means (1).
- Introduction.
- Hölder's means.
- Simple theorems concerning Hölder summability.
- Cesàro means.
- Means of non-integral order.
- A theorem concerning integral resultants.
- Simple theorems concerning Cesàro summability.
- The equivalence theorem.
- Mercer's theorem and Schur's proof of the equivalence theorem.
- Other proofs of Mercer's theorem.
- Infinite limits.
- Cesàro and Abel summability.
- Cesàro means as Nörlund means.
- Integrals.
- Theorems concerning summable integrals.
- Riesz's arithmetic means.
- Uniformly distributed sequences.

6 - Arithmetic means (2).
- Tauberian theorems for Cesàro summability.
- Slowly oscillating and slowly decreasing functions.
- Another Tauberian condition.
- Convexity theorems.
- Convergence factors.
- The factor (n + 1)-s.
- Another condition for summability.
- Integrals.
- The binomial series.

7 - Tauberian theorems for power series.
- Abelian and Tauberian theorems.
- Tauber's first theorem.
- Applications to general Dirichlet's series.
- The deeper Tauberian theorems.
- Further remarks.
- Slowly oscillating and slowly decreasing functions.
- The method of Hardy and Littlewood.
- The 'high indices' theorem.

8 - The methods of Euler and Borel (1).
- Introduction.
- The (E, q) method.
- Simple properties of the (E, q) method. 
- The formal relations between Euler's and Borel's methods.
- Borel's methods.
- Normal, absolute, and regular summability.
- Abelian theorems for Borel summability.
- Analytic continuation of a function regular at the origin ; the polygon of summability.
- Series representing functions with a singular point at the origin.
- Analytic continuation by other methods.
- The summability of certain asymptotic series.

9 - The methods of Euler and Borel (2).
- Some elementary lemmas.
- Proof of theorems.
- Another elementary lemma.
- Ostrowski's theorem on over-convergence.
- Tauberian theorems for Borel summability.
- Examples of series not summable (B).
- A theorem in the opposite direction.
- The (e, c) method of summation.
- The circle methods of summation.
- Further remarks.
- The principal Tauberian theorem.
- Generalizations.
- The geometric series.
- Valiron's methods.

10 - Multiplication of series.
- Formal rules for multiplication.
- The classical theorems for multiplication by Cauchy's rule.
- Multiplication of summable series.
- Another theorem concerning convergence.
- Alternating series.
- Formal multiplication.
- Multiplication of integrals.
- Euler summability.
- Borel summability.
- Dirichlet multiplication.
- Series infinite in both directions.
- The analogues of Cauchy's and Mertens's theorems.
- Further theorems.
- The analogue of Abel's theorem.

11 - Hausdorff means.
- Expression of the (E, q) and (C, 1) transformations.
- Hausdorff's general transformation.
- The general Hölder and Cesàro transformations.
- Conditions for the regularity of a real Hausdorff transformation.
- Totally monotone sequences.
- Final form of the conditions for regularity.
- Moment constants.
- Hausdorff's theorem.
- Inclusion and equivalence.
- Mercer's theorem and the equivalence theorem for Hölder and Cesàro means.
- Some special cases.
- Logarithmic cases.
- Exponential cases.
- The Legendre series.
- The moment constants of functions of particular classes.
- An inequality for Hausdorff means.
- Continuous transformations.
- Quasi-Hausdorff transformations.
- Regularity of a quasi-Hausdorff transformation.
- Examples.

12 - Wiener's Tauberian theorems.
- Introduction.
- Wiener's condition.
- Lemmas concerning Fourier Transforms.
- Lemmas concerning the class U.
- Final lemmas.
- Wiener's second theorem.
- Some special kernels.
- Application of the general theorems to some special kernels.
- Applications to the theory of primes.
- One-sided conditions.
- Vijayaraghavan's theorem.
- Borel summability.
- Summability (R, 2).

13 - The Euler-Maclaurin sum formula.
- Introduction.
- The Bernoullian numbers and functions.
- The associated periodic functions.
- The Euler-Maclaurin sum formula.
- The sign and magnitude of the remainder term.
- Poisson's proof of the Euler-Maclaurin formula.
- A formula of Fourier.
- The case f(x) = x-s and the Riemann zeta-function.
- The case f(x) = log(x + c) and Stirling's theorem.
- Generalization of the formulae.
- Other formulae for C.
- Investigation of the Euler-Maclaurin formula by complex integration.
- Summability of the Euler-Maclaurin series.
- Additional remarks.

Appendixes.
1 - On the evaluation of certain definite integrals by means of divergent series.
2 - The Fourier kernels of certain methods of summation.
3 - On Riemann and Abel summability.
4 - On Lambert and Ingham summability.
5 - Two theorems of M. L. Cartwright.

 

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