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## Homework Statement

I need to prove this

<< [itex]\Phi[/itex] (x,y) >> = [itex]\frac{4}{5}[/itex] [itex]< \Phi (x,y)>_{c}[/itex] + [itex]\frac{1}{5}[/itex] [itex]< \Phi (x,y)>_{d}[/itex]

## Homework Equations

[itex]< \Phi (x,y)>_{c}[/itex] = [itex]\frac{1}{4}[/itex] [itex]\left[< \Phi (x,y+h)> + < \Phi (x-h,y)> + < \Phi (x,y-h)> \right][/itex]

[itex]< \Phi (x,y)>_{d}[/itex] = [itex]\frac{1}{4}[/itex] [itex]\left[< \Phi (x+h,y+h)> + < \Phi (x-h,y+h)> + < \Phi (x-h,y-h)> + < \Phi (x+h,y-h)> \right][/itex]

## The Attempt at a Solution

I tried to substitute the formula for [itex]< \Phi (x,y)>_{c}[/itex] and [itex]< \Phi (x,y)>_{d}[/itex] to no avail. I don't know how to work on this kind of problem. Can somebody please tell me how should I do this? I'm willing to learn. Help much appreciated. Thanks.