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See the ", StyleBox["Mathematica ", FontSlant->"Italic"], "Help browser or wolfram.com for more information on using ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell["\<\ This notebook and those for the other seven chapters use a common stylesheet \ that has been imported into the notebook. This style sheet supports \ chapters, section, subsections, numbered equations, numbered figures and \ citations. A separate copy of the stylesheet can be download from the same \ page where this notebook was downloaded. \ \>", "Text"], Cell[TextData[{ StyleBox["COPYRIGHT ISSUES:", FontWeight->"Bold"], " The authors reserve all copyrights associated with this work. Any of \ the material appearing in these notebooks (such as the polyhedral meshes in \ chapter 7) can be used and modified without restriction as long as the use is \ non\[Hyphen]commercial. We simply ask that you acknowledge the authors when \ using material from these notebooks. For those readers interested in \ commercial use of the material in these notebooks, please contact \ jwarren@cs.rice.edu. " }], "Text"], Cell[CellGroupData[{ Cell["Helper functions from previous chapters", "Subsubsection"], Cell[BoxData[ \(plotCoeff[p_, opts___] := Show[Graphics[{Line[p], PointSize[0.02], RGBColor[1, 0, 0], Map[Point, p]}], Join[{opts}, {Axes \[Rule] True, Ticks \[Rule] {Automatic, Automatic}}]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(makeGenFun[coeff_, a_, x_] := Sum[coeff\[LeftDoubleBracket]i\[RightDoubleBracket] x\^\(i - 1 + a\), {i, 1, Length[coeff]}]\)], "Input", InitializationCell->True], Cell[BoxData[ \(subdivide[s_, p_, k_] := If[k \[Equal] 0, p, subdivide[s, s*\((p /. {x \[Rule] x\^2})\), k - 1]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(makeCoeff[genFun_, {min_, max_}, k_, x_] := With[{expFun = Together[genFun]}, \n\t Table[Coefficient[expFun, x, i], {i, \(2\^k\) min, \(2\^k\) max}]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(plotGen[genFun_, {min_, max_}, k_, x_, opts___] := With[{expFun = Together[genFun]}, \n\t plotCoeff[ Table[{i\/2\^k, Coefficient[expFun, x, i]}, {i, \(2\^k\) min, \(2\^k\) max}], opts]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(makeGenFun[coeff_, {a_, b_}, x_, y_] := Sum[coeff\[LeftDoubleBracket]i, j\[RightDoubleBracket] \(x\^\(i - 1 + a\)\) y\^\(j - 1 + b\), {i, \(Dimensions[ coeff]\)\[LeftDoubleBracket]1\[RightDoubleBracket]}, {j, \ \(Dimensions[coeff]\)\[LeftDoubleBracket]2\[RightDoubleBracket]}]\)], "Input",\ InitializationCell->True], Cell[BoxData[ \(subdivide2D[s_, p_, k_] := If[k \[Equal] 0, p, subdivide2D[s, s*\((p /. {x \[Rule] x\^2, y \[Rule] y\^2})\), k - 1]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(makeCoeff[genFun_, {{minX_, maxX_}, {minY_, maxY_}}, k_, x_, y_] := With[{expFun = Expand[genFun]}, \n\t Table[With[{rowY = Table[Coefficient[expFun, y, j], {j, \(2\^k\) minY, \(2\^k\) maxY}]}, \n\t\t\t\tTable[ Coefficient[rowY, x, i], {i, \(2\^k\) minX, \(2\^k\) maxX}]]]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(plotCoeff3D[p_, plotArgs___] := Show[Graphics3D[\[IndentingNewLine]Table[ Polygon[{p\[LeftDoubleBracket]i, j\[RightDoubleBracket], p\[LeftDoubleBracket]i + 1, j\[RightDoubleBracket], p\[LeftDoubleBracket]i + 1, j + 1\[RightDoubleBracket], p\[LeftDoubleBracket]i, j + 1\[RightDoubleBracket]}], {i, \(Dimensions[ p]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - 1}, {j, \(Dimensions[ p]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - 1}]], Join[{plotArgs}, {Axes \[Rule] True}]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(plotGen2D[genFun_, {{minX_, maxX_}, {minY_, maxY_}}, k_, x_, y_, opts___] := Module[{expFun = Expand[genFun], rowY}, \[IndentingNewLine]plotCoeff3D[ Table[rowY = Coefficient[expFun, y, j]; \[IndentingNewLine]Table[{i\/2\^k, j\/2\^k, Coefficient[rowY, x, i]}, {i, \(2\^k\) minX, \(2\^k\) maxX}], {j, \(2\^k\) minY, \(2\^k\) maxY}], opts]]\)], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell["Convergence of a sequence of functions", "Section", CellTags->"SEC convergence sequence of functions"], Cell[CellGroupData[{ Cell["Pointwise convergence", "Subsection"], Cell[CellGroupData[{ Cell["Convergence figures", "Subsubsection"], Cell[TextData[{ "Sawtooth text function ", Cell[BoxData[ \(\[ScriptP]\_k[\[ScriptX]]\)]] }], "Text"], Cell[BoxData[ \(\[ScriptP][\[ScriptX]_, k_] := \(2\^\(-k\)\) With[{i = Floor[\(2\^k\) \[ScriptX]]}, If[EvenQ[i], \(2\^k\) \[ScriptX] - i, 1 + i - \(2\^k\) \[ScriptX]]]\)], "Input", InitializationCell->True], Cell[TextData[{ "Derivative of sawtooth function ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[ScriptP]", "k", "\[Prime]", MultilineFunction->None], "[", "\[ScriptX]", "]"}]]], " " }], "Text"], Cell[BoxData[ \(\[ScriptD]\[ScriptP][\[ScriptX]_, k_] := With[{i = Floor[\(2\^k\) \[ScriptX]]}, If[EvenQ[i], 1, \(-1\)]]\)], "Input", InitializationCell->True], Cell[TextData[{ "Plots of ", Cell[BoxData[ \(\[ScriptP]\_k[\[ScriptX]]\)]], " for increasing ", Cell[BoxData[ \(k\)]], ". 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