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Integral Kernel Operators in the Cochran -Kuo - Sengupta Space

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Integral Kernel Operators in the Cochran -Kuo - Sengupta Space Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1999 Integral Kernel Operators in the Cochran -Kuo - Sengupta Space. John Joseph Whitaker Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Whitaker, John Joseph, "Integral Kernel Operators in the Cochran -Kuo -Sengupta Space." (1999). LSU Historical Dissertations and Theses. 7061. https://digitalcommons.lsu.edu/gradschool_disstheses/7061 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6“ x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. Bell & Howell Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA UMI”800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INTEGRAL KERNEL OPERATORS IN THE COCHRAN-KUO-SENGUPTA SPACE A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by John J. Whitaker B.S., Birmingham-Southern College, 1991 December, 1999 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number 9951621 ___ ® UMI UMI Microform9951621 Copyright 2000 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments I would like to thank my wife, erica j. Whitaker, for her constant support and innovative time saving techniques. My parents, Mr. and Mrs. Joseph R. Whitaker, have been a continuous source of help and encouragement. My advisor, Dr. Hui Kuo, has provided me with inspiration and great suggestions, while remaining patient with my progress. I would also like to give special thanks to Dr. George Cochran, who was always willing to listen and give comments on my work. Finally, I would like to acknowledge the support I recieved from both graduate students and faculty members at Louisiana State University. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Acknowledgments ................................................................................................... ii A b s tra c t ................................................ iv Introduction .............. 1 C h a p te r 1 Test Functions and Generalized Functions ........................................... 3 1.1 Background ................................................................................................. 3 1.2 Standard Construction .............................................................................. 5 1.3 Kubo and Takenaka’s Construction ...................................................... 7 1.4 Kondratiev and Streit’s Construction ................................................... 8 1.5 Cochran, Kuo and Sengupta’s Construction . ................................ 9 2 Integral Kernel Operators ............................................................................. 11 2.1 Integral Kernel Operator .......................................................................... 12 2.2 Restriction and Extension of Integral Kernel Operators I ................. 15 2.3 Restriction and Extension of Integral Kernel Operators I I ..................... 21 2.4 Examples of Integral Kernel O perators .................................................. 23 2.5 The Symbol of a Continuous O perator ......................................................27 References .................................................................................................................. 33 V ita ............................................................................................................................... 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract This dissertation, contains several results about integral kernel operators in white noise analysis. The results found here apply to the space of test functions and generalized functions that were constructed in the paper of Cochran, Kuo, and Sengupta, based on a sequence of numbers (ar(n) }^o- We shall prove results about existence, restrictions, and extensions of integral kernel operators based on the conditions on {a(n)}£L 0 contained in the paper of Kubo, Kuo, and Sengupta. Also, we shall prove an analytic property and growth condition of the symbol of a continuous operator in CKS space. Our results are similiar to results Chung, Ji, and Obata have found for the CKS space. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Introduction White noise analysis is a branch of probability that originated from. Hida’s work [3] in 1975. The general focus of white noise analysis is to research properties of test functions and generalized functions. Since the creation of white noise analysis, several types of test function spaces and corresponding generalized functions have been introduced and analyzed. Kubo and Takenaka [ 8 ] [9] [10] [11] introduced and studied the space of test functions and generalized functions in the following Gel’fand triple (£) C (L2) C (£ )'. Kondratiev and Streit [5] [ 6] introduced a family of Gel’fand triples (£), C (£2) C (f)J corresponding to a parameter 0 < /? < 1. If /? = 0, then the Gel’fand triple resulting from the Kondratiev and Streit construction is the Kubo-Takenaka Gel’­ fand triple. More recently, Cochran, Kuo, and Sengupta [ 2] have introduced the Gel’fand triple [£]„ C (£2) c [£]; corresponding to a sequence {< 2(71) }£10 of positive numbers satisfying certain con­ ditions. If a(n) = (n!)^, then the CKS space is the same as the Kondratiev-Streit space. Obata [13] introduced the concept of integral kernel operator for the Kubo- Takenaka space. For the Kondratiev-Streit space, Kuo [12] proved results including (1) the existence of an integral kernel operator from the space of test functions to 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the space of generalized functions, ( 2) extensions of the integral kernel operator to the space of generalized functions, (3) restrictions of integral kernel operators to the space of test functions, and (4) representations of continuous linear operators from the space of test functions to the space of generalized functions through integral kernel operators. Naturally, we wondered if similiar results could be achieved in the more general CKS space. Using the CKS space with additional conditions introduced by Kubo, Kuo, and Sengupta [7], we shall present results about the existence of integral kernel oper­ ators, extensions and restrictions of integral kernel operators, and two properties of the symbol of a continuous operator. Chung, Ji, and Obata [ 1] proved results about norm estimates for integral kemal operators and the symbol in the CKS space based on the conditions on {a:(n)}£L 0 given by Kubo, Kuo and Sengupta [7]. Chung, Ji, and Obata’s [ 1] results about integral kernel operators are sometimes more general than the similiar results listed here, but they often use different conditions on {or(n)}^L0. Their work on the symbol is more extensive than what we independently found. Further information on white noise analysis can be found in [4] and [14]. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Test Functions and Generalized Functions In this introductory chapter, we shall present a brief overview of constructions
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