eBook ISBN:  9781470400644 
Product Code:  MEMO/102/487.E 
List Price:  $36.00 
MAA Member Price:  $32.40 
AMS Member Price:  $21.60 
eBook ISBN:  9781470400644 
Product Code:  MEMO/102/487.E 
List Price:  $36.00 
MAA Member Price:  $32.40 
AMS Member Price:  $21.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 102; 1993; 90 ppMSC: Primary 33; 40; 41; 42
This work completely characterizes the behavior of Cesaro means of any order of the Jacobi polynomials. In particular, pointwise estimates are derived for the Cesaro mean kernel. Complete answers are given for the convergence almost everywhere of partial sums of Cesaro means of functions belonging to the critical \(L^p\) spaces. This characterization is deduced from weak type estimates for the maximal partial sum operator. The methods used are fairly general and should apply to other series of special functions.
ReadershipResearch mathematicians.

Table of Contents

Chapters

1. Introduction

2. Facts and definitions

3. An absolute value estimate for $3(1y) \leq 2(1x)$

4. A basic estimate for $3(1y) \leq 2(1x)$

5. A kernel estimate for $3(1y) \leq 2(1x)$ and $1 \leq \theta \leq 0$

6. A reduction lemma

7. A kernel estimate for $3(1y) \leq 2(1x)$ and $\theta \geq $1

8. A Cesaro kernel estimate for $t \leq s/2$

9. A basic estimate for separated arguments

10. A reduction lemma for separated arguments

11. A kernel estimate for separated arguments

12. Cesaro kernel estimate for $t \leq s  b$

13. Cesaro kernel estimate for $s$ near $t$

14. Kernel estimates

15. A weak type lemma

16. Lemmas for the upper critical value

17. Proofs of theorems (1.1)–(1.3)

18. Norm estimates for $p$ not between the critical values

19. A polynomial norm inequality

20. A lower bound for a norm of the kernel

21. Some limitations of the basic results

22. Growth of Cesaro means


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This work completely characterizes the behavior of Cesaro means of any order of the Jacobi polynomials. In particular, pointwise estimates are derived for the Cesaro mean kernel. Complete answers are given for the convergence almost everywhere of partial sums of Cesaro means of functions belonging to the critical \(L^p\) spaces. This characterization is deduced from weak type estimates for the maximal partial sum operator. The methods used are fairly general and should apply to other series of special functions.
Research mathematicians.

Chapters

1. Introduction

2. Facts and definitions

3. An absolute value estimate for $3(1y) \leq 2(1x)$

4. A basic estimate for $3(1y) \leq 2(1x)$

5. A kernel estimate for $3(1y) \leq 2(1x)$ and $1 \leq \theta \leq 0$

6. A reduction lemma

7. A kernel estimate for $3(1y) \leq 2(1x)$ and $\theta \geq $1

8. A Cesaro kernel estimate for $t \leq s/2$

9. A basic estimate for separated arguments

10. A reduction lemma for separated arguments

11. A kernel estimate for separated arguments

12. Cesaro kernel estimate for $t \leq s  b$

13. Cesaro kernel estimate for $s$ near $t$

14. Kernel estimates

15. A weak type lemma

16. Lemmas for the upper critical value

17. Proofs of theorems (1.1)–(1.3)

18. Norm estimates for $p$ not between the critical values

19. A polynomial norm inequality

20. A lower bound for a norm of the kernel

21. Some limitations of the basic results

22. Growth of Cesaro means