Compute the Green's function for one dimensional Laplacian operator?

Questions
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1

I am trying to compute the Green’s function for one dimensional Laplacian operator.
The command I’m using is-

GreenFunction[Laplacian[u[x],x], u[x], x \[Element] FullRegion[1], m]

There is no result for this command,
However when I’m using this command for two dimensional case-

GreenFunction[  Laplacian[
  u[x, y], {x, y}], 
 u[x, y], {x , y} \[Element] FullRegion[2], {m, n}],

Mathematica is giving me the correct result.
Can someone please help?

2

This is not well defined on the whole axis

In[1]:= G = 
 GreenFunction[{Laplacian[u[x], {x}], u[a] == 0, u[b] == 0}, 
  u[x], {x, a, b}, m]

Out[1]= -(((b - m) (a - x) HeavisideTheta[m - x])/(
  a - b)) - ((a - m) (b - x) HeavisideTheta[-m + x])/(a - b)

In[2]:= G1 = Limit[G, {a -> -Infinity}]

Out[2]= -b HeavisideTheta[m - x] + m HeavisideTheta[m - x] - 
 b HeavisideTheta[-m + x] + x HeavisideTheta[-m + x]

In[3]:= G2 = Limit[G1, {b -> Infinity}]

Out[3]= \[Infinity] (-HeavisideTheta[m - x] - HeavisideTheta[-m + x])