(*Define data and Fourier matrix*)
pts1={1,2,-2,-3};
F4= Table[Exp[-2*Pi*I/4*i*j],{i,0,3},{j,0,3}]/4;

(*Multiply and get function*)
mult=F4.Transpose[pts1];
f4[x_]:=Sum[mult[[i]]*Exp[I*(i-1)*x],{i,1,4}];

(*Plot function with original points*)
Plot[Re[f4[x]],{x,0,2Pi},Epilog->{Red,
PointSize[Large],Table[Point[{(k-1)*Pi/2,pts1[[k]]}],{k,1,4}]},Ticks->{Table[k*2Pi/8,{k,1,8}],Table[y,{y,-4,3}]},GridLines->{Table[k*2Pi/8,{k,1,8}],Table[y,{y,-4,3}]}]

(*Define new data and Fourier matrix*)
pts2={1,.7,2,2.3,-2,-1.8,-3,-3.3};
F8= Table[Exp[-2*Pi*I/8*i*j],{i,0,7},{j,0,7}]/8;

(*Multiply and get function*)
mult=F8.Transpose[pts2];
f8[x_]:=Sum[mult[[i]]*Exp[I*(i-1)*x],{i,1,8}];

(*Plot function with original points*)
Plot[Re[f8[x]],{x,0,2Pi},Epilog->{Red,
PointSize[Large],Table[Point[{(k-1)*Pi/4,pts2[[k]]}],{k,1,8}]},Ticks->{Table[k*2Pi/8,{k,1,8}],Table[y,{y,-4,3}]},GridLines->{Table[k*2Pi/8,{k,1,8}],Table[y,{y,-4,3}]}]