h25sol1 - Math 2025, Homework, Due Tuesday, Sept. 17 Name:...

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Math 2025, Homework, Due Tuesday, Sept. 17 Name: We denote by ϕ [ a,b ) ( t ) the indicator function of the interval [ a,b ). Thus ϕ [ a,b ) ( t ) = ( 1 t [ a,b ) 0 otherwise . We let ϕ ( t ) = ϕ [0 , 1) ( t ) be the Haar scaling function and ψ ( t ) = ϕ (2 t ) - ϕ (2 t - 1). For the solution of (1) and (2) you use the following facts: ϕ ( λt - r ) = ϕ [ r/λ, ( r +1) ( t ) ϕ [ a,b ) ( t ) = ϕ ± 1 b - a ( t - a ) ) 1) Write the following translate and dilate ϕ ( λt - r ) of the Haar scaling function as an indicator function ϕ [ a,b ) ( t ) of an interval. (1) ϕ (8 t - 5) = ϕ [ 5 8 , 3 4 ) ( t ) (2) ϕ ( t 5 - 1) = ϕ [5 , 10) 2) Write the following indicator functions as translate and dilate of the Haar scaling function: (1) ϕ [1 , 3) ( t ) = ϕ ± 1 2 ( t - 1) ) = ϕ ± 1 2 t - 1 2 ) (2) ϕ [0 . 5 , 2) ( t ) = ϕ ± 2 3 ± t - 1 2 )) = ϕ ± 2 3 t - 1 3 ) 3) Write the step function f ( t ) = 3 ϕ (4 t ) + ϕ (4 t - 1) + 2 ϕ (4 t - 2) + 4 ϕ (4 t - 3) as a linear combination of the function
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h25sol1 - Math 2025, Homework, Due Tuesday, Sept. 17 Name:...

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