Scaling Vector

Matrices

G[-2]={{0,(1/80) (Sqrt[(7/3) (23-6 (Sqrt[14]))]),-(1/80) (Sqrt[3]) (-1+(Sqrt[14])),-(1/80) (Sqrt[(151/3)-10 (Sqrt[14])])},{0,0,0,0},{0,0,0,0},{0,0,0,0}};

G[-1]={{(1/40) (Sqrt[(151/2)-15 (Sqrt[14])]),-(1/80) (Sqrt[(71/3)-4 (Sqrt[14])]),(1/80) (-(Sqrt[3])-4 (Sqrt[42])),-(1/80) (Sqrt[(1351/3)+70 (Sqrt[14])])},{0,0,0,0},{0,0,0,0},{0,0,0,0}};

G[0]={{(1/(Sqrt[2])),-(1/80) (Sqrt[(71/3)-4 (Sqrt[14])]),(1/80) ((Sqrt[3])+4 (Sqrt[42])),-(1/80) (Sqrt[(1351/3)+70 (Sqrt[14])])},{0,(7/(8 (Sqrt[2]))),-((Sqrt[(7/2)])/8),-(1/8)},{0,((3 (Sqrt[(7/2)]))/8),(1/(8 (Sqrt[2]))),0},{0,(1/80) (-16-3 (Sqrt[14])),-(1/40) (Sqrt[(905/2)-24 (Sqrt[14])]),-((3 (Sqrt[7]))/40)}};

G[1]={{(1/40) (Sqrt[(151/2)-15 (Sqrt[14])]),(1/80) (Sqrt[(7/3) (23-6 (Sqrt[14]))]),(1/80) (Sqrt[3]) (-1+(Sqrt[14])),-(1/80) (Sqrt[(151/3)-10 (Sqrt[14])])},{((Sqrt[(3/2)])/4),(7/(8 (Sqrt[2]))),((Sqrt[(7/2)])/8),-(1/8)},{0,-((3 (Sqrt[(7/2)]))/8),(1/(8 (Sqrt[2]))),0},{((3 (Sqrt[(21/2)]))/20),(1/80) (-16-3 (Sqrt[14])),(1/40) (Sqrt[(905/2)-24 (Sqrt[14])]),-((3 (Sqrt[7]))/40)}};

Function Values

Clear[phi];

phi[0]={(Sqrt[10]),0,0,0}

phi[x_]:=If[x≤-1||x≥1,{0,0,0,0},phi[x]];

phi[3]

fill[j_]:=Do[phi[x]=(Sqrt[2])Sum[G[i].phi[2x-i],{i,-2,1}],{x,-1,1,(2^(-j))}];

fill[1]

fill[2]

fill[3]

fill[4]

fill[5]

fill[6]

fill[7]

pts[i_]:=Table[{x,phi[x][[i]]},{x,-1,1,(2^(-7))}];

phi[-(3/4)]//FullSimplify

phi[-(1/2)]//FullSimplify

phi[-(1/4)]//FullSimplify

phi[0]

phi[(1/4)]//FullSimplify

phi[(1/2)]//FullSimplify

phi[(3/4)]//FullSimplify

Graphs

Show[Graphics[{Black,Thickness[.008],Line[pts[1]//N]}],Axes->True,PlotRange->{{-1.1,1.1},{-0.7,(Sqrt[10])+.1}},Ticks->{{-1,-.5,.5,1},{-1.5,-1,-.5,.5,1,1.5,2,2.5,3}}]

Show[Graphics[{Black,Thickness[.017],Line[pts[2]//N]}],Axes->True,PlotRange->{{0,1.1},{-0.7,(Sqrt[10])+.1}},Ticks->{{-1,-.5,.5,1},{-.5,.5,1,1.5,2,2.5,3}}]

Show[Graphics[{Black,Thickness[.017],Line[pts[3]//N]}],Axes->True,PlotRange->{{0,1.1},{-1.5,2.3}},Ticks->{{-1,-.5,.5,1},{-1.5,-1,-.5,.5,1,1.5,2,2.5,3}}]

Show[Graphics[{Black,Thickness[.017],Line[pts[4]//N]}],Axes->True,PlotRange->{{0,1.1},{-1.5,2.3}},Ticks->{{-1,-.5,.5,1},{-1.5,-1,-.5,.5,1,1.5,2,2.5,3}}]

Multiwavelet

Matrices

H[-2]={{0,-(1/80) (Sqrt[(7/3) (23-6 (Sqrt[14]))]),(1/80) (Sqrt[3]) (-1+(Sqrt[14])),(1/80) (Sqrt[(151/3)-10 (Sqrt[14])])},{0,-(1/40) (Sqrt[(161/6)-7 (Sqrt[14])]),(1/40) (Sqrt[(3/2)]) (-1+(Sqrt[14])),(1/40) (Sqrt[(151/6)-5 (Sqrt[14])])},{0,0,0,0},{0,0,0,0}};

H[-1]={{-(1/40) (Sqrt[(151/2)-15 (Sqrt[14])]),(1/80) (Sqrt[(71/3)-4 (Sqrt[14])]),(1/80) ((Sqrt[3])+4 (Sqrt[42])),(1/80) (Sqrt[(1351/3)+70 (Sqrt[14])])},{(1/40) (5-3 (Sqrt[14])),(1/40) (Sqrt[(71/6)-2 (Sqrt[14])]),(1/80) ((Sqrt[6])+8 (Sqrt[21])),(1/40) (Sqrt[(1351/6)+35 (Sqrt[14])])},{0,0,0,0},{0,0,0,0}};

H[0]={{(1/(Sqrt[2])),(1/80) (Sqrt[(71/3)-4 (Sqrt[14])]),(1/80) (-(Sqrt[3])-4 (Sqrt[42])),(1/80) (Sqrt[(1351/3)+70 (Sqrt[14])])},{0,-(1/40) (Sqrt[(71/6)-2 (Sqrt[14])]),(1/80) ((Sqrt[6])+8 (Sqrt[21])),-(1/40) (Sqrt[(1351/6)+35 (Sqrt[14])])},{0,(1/40) (3-(Sqrt[14])),(1/40) (Sqrt[65+6 (Sqrt[14])]),(1/20) (Sqrt[(239/2)-15 (Sqrt[14])])},{0,-(1/(8 (Sqrt[6]))),((Sqrt[(21/2)])/8),(1/(Sqrt[3]))}};

H[1]={{-(1/40) (Sqrt[(151/2)-15 (Sqrt[14])]),-(1/80) (Sqrt[(7/3) (23-6 (Sqrt[14]))]),-(1/80) (Sqrt[3]) (-1+(Sqrt[14])),(1/80) (Sqrt[(151/3)-10 (Sqrt[14])])},{(1/40) (-5+3 (Sqrt[14])),(1/40) (Sqrt[(161/6)-7 (Sqrt[14])]),(1/40) (Sqrt[(3/2)]) (-1+(Sqrt[14])),-(1/40) (Sqrt[(151/6)-5 (Sqrt[14])])},{(1/20) (Sqrt[3]) (5+(Sqrt[14])),(1/40) (3-(Sqrt[14])),(1/40) (-(Sqrt[2])-3 (Sqrt[7])),(1/20) (Sqrt[(239/2)-15 (Sqrt[14])])},{0,(1/(8 (Sqrt[6]))),((Sqrt[(21/2)])/8),-(1/(Sqrt[3]))}};

Graphs

Do[psi[x]=(Sqrt[2])Sum[H[i].phi[2x-i],{i,-2,1}],{x,-1,1,(2^(-7))}];

pts[i_]:=Table[{x,psi[x][[i]]}//N,{x,-1,1,(2^(-7))}];

Show[Graphics[{Black,Thickness[.01],Line[pts[1]]}],Axes->True,PlotRange->{{-1.1,1.1},{-1.75,(Sqrt[10])+.1}},Ticks->{{-1,-.5,.5,1},{-1.5,-1,-.5,.5,1,1.5,2,2.5,3}}]

Show[Graphics[{Black,Thickness[.01],Line[pts[2]]}],Axes->True,PlotRange->{{-1.1,1.1},{-1.75,(Sqrt[10])+.1}},Ticks->{{-1,-.5,.5,1},{-1.5,-1,-.5,.5,1,1.5,2,2.5,3}}]

Show[Graphics[{Black,Thickness[.02],Line[pts[3]]}],Axes->True,PlotRange->{{-.1,1.1},{-2,(Sqrt[10])+.3}},Ticks->{{-1,-.5,.5,1},{-2,-1.5,-1,-.5,.5,1,1.5,2,2.5,3}}]

Show[Graphics[{Black,Thickness[.02],Line[pts[4]]}],Axes->True,PlotRange->{{-.1,1.1},{-2,(Sqrt[10])+.3}},Ticks->{{-1,-.5,.5,1},{-2,-1.5,-1,-.5,.5,1,1.5,2,2.5,3}}]