h7color - ECE 580 Math 587 SPRING 2011 Correspondence 18...

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ECE 580 / Math 587 SPRING 2011 Correspondence # 18 April 23, 2011 ASSIGNMENT 7 Reading Assignment: Text: Chapters 7 (7.1 - 7.8), 8 , and 9 (9.1-9.5). Recommended Reading: Curtain & Pritchard: Chapter 12; Balakrishnan: Chapter 2. Problems (to be handed in): Due Date: Tuesday, May 3 . 48. Let f be a functional on C [0 , 1], defined by f ( x ) = x := max 0 t 1 | x ( t ) | Determine a class of functions, D C [0 , 1], so that if x D the Gateaux differential δf ( x ; h ) exists for all h , and is linear in h . Hint : First show that if x C [0 , 1] has a unique maximum at a point t o (0 , 1), which is also the unique maximum of | x ( t ) | , and h is an arbitrary element of C [0 , 1], then lim α 0 1 α {∥ x + αh ∥ − | x ( t o ) + αh ( t o ) |} = 0 . 49. i) Repeat Problem 48 above for the functional f ( x ) = max 0 t 1 x ( t ) ii) Do the same for the functional f ( x ) = 1 0 | x ( t ) | d t 50. Obtain the Gateaux differentials and Gateaux derivatives of the following transformations, and indicate in each case whether the Gateaux derivative is a linear operator or not.

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