pts = {{0,0},{0,h},{0,-h},{h,0},{-h,0}};
c ={1/20,1/20};
$Assumptions={h>0,s>0};
scale=Norm[c-pts[[1]]];
sigma=1;
n = Length@pts;
values = {u1,u2,u3,u4,u5};
w[p_]:=Exp[-(p.p)/sigma/sigma]
basis={(1&),(#[[2]]&),(#[[1]]&)};
m = Length@basis;
uh[p_]:=Sum[basis[[i]][p] alpha[i],{i,1,m}];
W = DiagonalMatrix[Table[w[(pts[[i]]-c)/scale],{i,1,n}]];
(*W=IdentityMatrix[n];*)
eqs = Table[
uh[(pts[[i]]-c)/scale]-values[[i]],{i,1,n}
];

scale //N

W/.{h->0.1}

{rhs, M} = Normal@CoefficientArrays[eqs,Table[alpha[i],{i,1,m}]];
M //MatrixForm
W.M /.{h->0.1} //MatrixForm

coeff=LeastSquares[W.M,-W.rhs];
coeff//FullSimplify

coeff/.{h->0.1`18}//CForm

rl={u1->3.23432,u2->-2.3234,u3->0.12443498,u4->1.908432,u5->-0.98742532};
coeff /. rl/.{h->0.1`18}