Minimal Cubature Formulae for Spherically Symmetric Integrals and Tight Euclidean Designs

Acknowledgements I would like to express my deepest appreciation to my adviser Professor Hiroyuki Mat-sumoto. He has supervised my study for six and a half years and his continuous encouragement , patience, and excellent guidance had led me to successfully complete this thesis. I am especially indebted to Professor Eiichi Bannai, Professor Etsuko Bannai for fruitful discussion and support of my understanding of the theories on Euclidean designs in algebraic combinatorics. Without their kind support, I could not have completed this thesis. I owe a very important debt to Professor Masanori Sawa. I am grateful for his advices not only on our study but also on my private life. I have learned a lot through joint works with him. I would like to thank Professor Masakazu Jimbo and Professor Hidehiro Kaise for their knowledge and valuable advice. I am very thankful for everything to Dr. Koji Momihara, Ms. Yuanyuan Zhou and all our laboratory members. Finally, I wish to express my gratitude to my parents for their warm supports. I could not have completed this work without their hearty encouragement.