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Week
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Topics
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Study Metarials
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1
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The Hahn-Banach Theorem
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2
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Problems of approximations, existence and separation
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3
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Topologies on the dual
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4
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Lp spaces and examples
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5
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Radon measure, Generalized Functions
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6
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More Duals; Polynomials and Formal Power series
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7
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Transpose of continuous linear map, injections of duals, Differential Operators
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8
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Support and Structure of Generalized Functions
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9
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Fourier Transformation of Tempered Generalized Functions
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10
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Convolutions of Functions
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11
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Convolution of Generalized Functions
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12
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Approximations of Generalized Functions by regularizing
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13
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Fourier transforms of Generalized Functions with compact support, the Paley-Weiner Theorem
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14
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Fourier Transformations of Convolutions and Multiplications
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Prerequisites
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-
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Language of Instruction
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Turkish
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Responsible
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Assist. Prof. Dr. Gonca Durmaz
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Instructors
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-
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Assistants
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The related lecturers of the department
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Resources
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1) A. Wilansky, Modern Methods in Topological Vektör Spaces, ABD
2) R. Cristescu, Topological Vector Spaces,1977, Romanya
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Supplementary Book
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1) François Treves ; Topological Vector Spaces, Distributions and Kernels, Academic Press 1967
2) Juan Horvath ; Topological Vector Spaces and Distributions, Addison-Wesley, 1966
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Goals
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The aim of this course is to teach the dual of Topological Vector spaces with their structure and consequently the generalized functions spaces, and define convolution product and product of generalized and give their Fourier transformations.
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Content
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Hahn-Banach Theorem, problems of approximation, Existence and Separation
Topologies on the dual
Lp spaces and an examples
Generalized Functions, support and structure of generalized function
Transpose of continuous linear map, injections of duals, Differential Operators
Approximations of Generalized functions by regularizing
Fourier transformations of functions and of Generalized functions
Fourier transforms of generalized functions with compact support, the Paley-Weiener Theorem
Fourier transforms of Convolutions and Multiplications
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Program Learning Outcomes |
Level of Contribution |
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1
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Improve and deepen the gained knowledge in Mathematics in the speciality level
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5
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2
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Use gained speciality level theoretical and applied knowledge in mathematics
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5
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3
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Perform interdisciplinary studies by relating the gained knowledge in Mathematics with other fields.
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2
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4
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Analyze mathematical problems by using the gained research methods
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4
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5
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Conduct independently a study requiring speciliaty in Mathematics
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3
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6
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Develop different approaches and produce solutions by taking responsibility to problems encountered in applications
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5
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7
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Evaluate the gained speciality level knowledge and skills with a critical approach and guide the process of learning
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3
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8
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Transfer recent and own research related to mathematics to the expert and non-expert shareholders written, verbally and visually
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5
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9
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Uses computer software and information technologies related to the field of mathematics at an advanced level.
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-
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10
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Have the awareness of acting compatible with social, scientific, cultural and ethical values during the process of collecting, interpreting, applying and informing data related to Mathematics
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4
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