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# HW8 - = δ β 3(1pt cos 2 d cx x f e x ax = = − 4(1pt =...

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Homework #8 Due: Tuesday, April 21, 2009 Note : Use of Matlab (or any other software) is not permitted. I. (1pt) Denote by D the dilation operator that maps a set of function into a set of functions via D:f g , where g(x)=af(ax) , for some a 0 , and by T the translation (shift) operator that maps a set of functions into a set of functions via T:h k , where k(x)=h(x-b) , for some fixed real parameter b . Compute g=DTf and h=TDf for some function f (that is, express g(x) in terms of a,b and the function f evaluated somewhere depending on a,b , and x ). II. (Exercises 7.14) Let a,b,c,d be real numbers with a>0 and c>0 and let m,n=0,1,2,… . Find a simple representation for the convolution product β∗ f when: 2. (1pt) ) cos( ) ( ), ( ) ( d cx x f b ax
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Unformatted text preview: + = δ β ; 3. (1pt) ) cos( ) ( , ) ( 2 d cx x f e x ax + = = − ; 4. (1pt) ) ( ), ( ) ( = Π = x f x x Ш (n) (x); 5. (1pt) ) ( ) ( ), ( ) ( ) ( ) ( x x f x x m n = = ; 6. (1pt) = + = ) ( ), ( ) ( x f b ax x Ш (x); 7. (1pt) ) sgn( ) ( ), ( ' ) ( x x f b ax x = + = . Recall the Ш (shah) generalized function : Ш (x)= ⎣ x ⎦ ’ (i.e. the derivative of the floor function ⎣ x ⎦ ). III. (Exercise 7.26) Find all generalized functions f that satisfy each of the following homogeneous equations. 8. (1pt) ) ( ) 1 ( 2 = ⋅ − x f x ; 9. (1pt) ) ( ) 1 ( 4 = ⋅ − x f x ; 10. (1pt) ) ( ) 1 ( 2 4 = ⋅ − x f x Total : 10 pts (1 point each)...
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