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Register a new userIntersection Bodies and Generalized Cosine Transforms
مجالس التقاطعات والتحويلات الجيبية العامة
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Boris Rubin
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جسمات التقاطع تمثل فئة ملهمة من الأجسام الهندسية المرتبطة بأجزاء من الجسمات النجمية والتي تشمل التحويلات الرادون والتحويلات الكوسية المحلولة وتحليل الفوريير المتعلق. التركيز الرئيسي لهذه المقالة هو العلاقة بين التحويلات الكوسية المحلولة من أنواع مختلفة في سياق تطبيقها في التحقيق في عائلة معينة من أجسام التقاطع، التي نسميها أجسام التقاطع $lam$. وتشمل هذه الأخيرة أجسام التقاطع $k$ (في سياق أ. كولدوبسكي) وكرات الوحدات للأبعاد المحدودة من مساحات $L_p$. وبالخصوص، نظرنا في أن القيود على الأبعاد الأقل في التحويلات الرادون الكروية والتحويلات الكوسية المحلولة تحافظ على بنيتها الهندسية الإجمالية. ونطبق هذا النتيجة في دراسة الأجزاء من أجسام التقاطع $lam$. وتم الحصول على تشريحات جديدة لهذه الفئة من الأجسام وتم إعطاء أمثلة. كما أعدنا مراجعة بعض الحقائق المعروفة وأعطيناها دلائل جديدة.
Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. The main focus of this article is interrelation between generalized cosine transforms of different kinds in the context of their application to investigation of a certain family of intersection bodies, which we call $lam$-intersection bodies. The latter include $k$-intersection bodies (in the sense of A. Koldobsky) and unit balls of finite-dimensional subspaces of $L_p$-spaces. In particular, we show that restrictions onto lower dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms preserve their integral-geometric structure. We apply this result to the study of sections of $lam$-intersection bodies. New characterizations of this class of bodies are obtained and examples are given. We also review some known facts and give them new proofs.
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