Uniform linear splines is my equation correct?

[SMA_01]

The question as stated: Define the uniform linear splines H_k:=H₀(x−k), k=−1,0,1,2,3, where

H₀={x, 0≤x<1,

2−x, 1≤x<2,

0, otherwise.

For k=−1, I would get:

H₋₁={x+1, 1≤x<2,

1−x, 2≤x<3,

0, otherwise.

Is that correct?

Thank you.

 

[LCKurtz]

No. $H_{-1}$ would be $H_0$ translated one unit to the left. You moved it to the right.

 

[haruspex]

H₋₁(x)=H₀(x+1), right?
H₀(x)={x+1, 0≤x<1,

2−x, 1≤x<2,
0, otherwise}​

To get H₀(x+1), wouldn't you write x+1 for x everywhere in the definition of H₀(x)?

 

[SMA_01]

LCKurtz-Wouldn't I just plug in k=-1, into H_{_0}(x-(-1))=H_{_0}(x+1)?

 

[SMA_01]

haruspex-

I accidently put x+1 for the first function in H₀, it's supposed to be x.

Yes, I see what you mean. I thought I did that though:
H₋₁={x+1, 1≤x<2,

1−x, 2≤x<3,

0, otherwise.

I don't see what I'm doing wrong...

 

[haruspex]

No, you should have got:
H₀(x+1)={x+1, 0≤x+1<1,

1−x, 1≤x+1<2,​

 

[SMA_01]

Oh okay, I see what you mean. I though I could evaluate it at the inequalities. Thanks!