The convolution transform 卷积变换

In studies of general operators of the same nature, general convolution transforms are immediately encountered as the objects of inversion. The relation between differential operators and integral transforms is the basic theme of this work. Discusses finite and non-finite kernels, variation diminishing transforms, asymptotic behavior of kernels, real inversion theory, representation theory, the Weierstrass transform, and complex inversion theory.

CHAPTER Ⅰ INTRODUCTION
SECTION
　 1. IntrodUction
　 2. Convolutions
　 3. Operational calculus
　 4. Green's functions
　 5. Operational calculus continued
　 6. The generation of kernels
　 7. Variation diminishing convolutions
　 8. Outline of program
　 9. Summary
CHAPTER Ⅱ THE FINITE KERNELS
　1. Introduction
　2. Distribution functions
　3. Frequency functions
4. Characteristic functions
　5. Convolutions
　6. The finite kernels
　7. Inversion
　8. Exponential polynomials
　9. Green's functions
　10. Examples
　11. Summary
CHAPTER Ⅲ THE NON-FINITE KERNELS
　1. Introduction
　2. Limits of distribution functions
　3. PSlya's class of entire functions
4. The closure of a class of distribution functions
5. The non-finite kernels
　SECTION
　　6. Properties of the non-finite kernels
　　7. Inversion
　　8. Green's functions
　　9. Examples
　　10. Associated kernels
　　11. Summary
CHARTER Ⅳ VARIATION DIMINISHING TRANSFORMS
1. Introduction
2. Generation of variation diminishing frequency functions
3. Logarithmic convexity
4. Characterization of variation diminishing functions
5. The changes of sign of G(n)(t)
6. Intersection properties
7. Generation of totally positive functions
8. Matrix transformations
9. Totally positive frequency functions
　10. Summary
CHAPTER Ⅴ ASYMPTOTIC BEHAVIOUR OF KERNELS
1. Introduction
2. Asymptotic estimates
3. Asymptotic estimates continued
4. Summary
CHAPTER Ⅵ REAL INVERSION THEORY
1. Introduction
2. Some preliminary results
3. Convergence
4. The sequence of kernels
5. The inversion theorem
6. Stieltjes integrals
7. Relaxation of continuity conditions
8. Factorization
　9. Summary
CHAPTER Ⅶ REPRESENTATION THEORY
CHAPTER Ⅷ THE WEIERSTRASS TRANSFORM
CHAPTER Ⅸ COMPLEX INVERSION THEORY
CHAPTER Ⅹ MISCELLANEOUS TOPICS
BIBLIOGRAPHY
SYMBOLS AND NOTATIONS
INDEX