(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 134042, 3145]*) (*NotebookOutlinePosition[ 135829, 3191]*) (* CellTagsIndexPosition[ 135491, 3181]*) (*WindowFrame->Normal*) Notebook[{ Cell["Basis and Orthogonal Functions", "Title"], Cell[CellGroupData[{ Cell[TextData[{ "This is a generalization of the Fourier series. The orthogonality of the \ trignometric functions\n(i.e., ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%1 Sin[2\ \[Pi]\ n\ \ x]\ \ Sin[ 2\ \[Pi]\ m\ x] \[DifferentialD]x\)]], " = ", Cell[BoxData[ FormBox[GridBox[{ {\(1\/2\), \(if\ \ \ \ n = m\)}, {"0", \(if\ \ \ n \[NotEqual] m\)} }], TraditionalForm]]], ")\nwas the key to the ``trick'' that permitted calculation of the Fourier \ coefficients.\nThis is a generalization of dot or scalar vector product, but \ for functions. The norm that is defined above for the trignometric functions \ is fairly simple, it is the l2-norm:\nf(x) \[Bullet] g(x) \[Congruent] ", StyleBox["\[Integral] ", FontSize->24], "f(x) g(x) dx (l2-norm)\nHowever, there can be many other types of \ norms, for instance a Gaussian weighted l2-norm\nf(x) \[Bullet] g(x) \ \[Congruent] ", StyleBox["\[Integral] ", FontSize->24], "f(x) g(x) ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(-x^2\)\)]], " dx (Gaussian weighted l2-norm)\nand many many others.\n\nBoth the \ norm and the domain of integration must be defined.\n\nTo represent a \ function in terms of a sum of basis functions with coefficients, a \ orthogonality relation must be obtained for the basis functions.\nWe will do \ a few examples of orthogonality relations, the coefficients can be found as a \ straightforward extension of the Fourier method." }], "Subsection"], Cell["\<\ Establish by example the orthogonality relations for the Lengendre \ Polynomials\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ Integrate[ LegendreP[i, x]\ LegendreP[5, x], {x, \(-1\), 1}], {i, \(-10\), 10}]\)], "Input", CellTags->"mmtag:27:LegendreP[]__demonstrating_orthogonality"], Cell[BoxData[ \({0, 0, 0, 0, 2\/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2\/11, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ Calculate the norm as a function of the order of the Legendre \ polynomial\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ Integrate[LegendreP[i, x]\ LegendreP[i, x], {x, \(-1\), 1}], {i, 0, 15}]\)], "Input"], Cell[BoxData[ \({2, 2\/3, 2\/5, 2\/7, 2\/9, 2\/11, 2\/13, 2\/15, 2\/17, 2\/19, 2\/21, 2\/23, 2\/25, 2\/27, 2\/29, 2\/31}\)], "Output"] }, Open ]], Cell["Make a guess at what the norm looks like", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[2\/\(2\ i\ + \ 1\), \ {i, 0, 15}]\)], "Input"], Cell[BoxData[ \({2, 2\/3, 2\/5, 2\/7, 2\/9, 2\/11, 2\/13, 2\/15, 2\/17, 2\/19, 2\/21, 2\/23, 2\/25, 2\/27, 2\/29, 2\/31}\)], "Output", CellTags->"mmtag:27:LegendreP[]__orthogonality_relations"] }, Open ]], Cell[TextData[{ "The orthogonality relation is plausibly: ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(-1\)\%1 LegendreP[i, x]\ \ LegendreP[ j, x] \[DifferentialD]x\)]], " = ", Cell[BoxData[ FormBox[GridBox[{ {\(2\/\(2 i\ + \ 1\)\), \(if\ | \ i | = \(\(|\)\(j\)\(|\)\)\)}, {"0", \(if\ | i\ | \(\(\[NotEqual]\)\(\(|\)\(j\)\(|\)\)\)\)} }], TraditionalForm]]], "\nIn fact, this is correct" }], "Text"], Cell["Representation of a function in terms of the Legendre basis", "Subtitle"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ This demonstrates how to take a specific set (basis) of orthogonal \ functions and represent an arbitrary function in terms of an infinite sum of \ terms involving the basis functions, each with its own coefficient. This is \ a direct analog to what we did with Fourier series.\ \>", "Section"], Cell["A function to calculate the coefficients", "Text"], Cell[BoxData[ \(LegendreCoefficient[f_, \ i_]\ \ := \ \(\(2\ i\ + \ 1\)\/2\) Integrate[f[x]\ LegendreP[i, x], {x, \(-1\), 1}]\)], "Input", CellTags->"mmtag:27:Legendre_series__determining_coefficients"], Cell[CellGroupData[{ Cell[TextData[{ "Use Legendre functions to represent the function ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(-x\^2\)\)]] }], "Subsubsection"], Cell["This is an example function:", "Text"], Cell[BoxData[ \(AFunc[x_]\ := Exp[\(-x^2\)]\)], "Input"], Cell["Calculate the first twenty coefficients", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Lcofs\ = \ Table[LegendreCoefficient[AFunc, i], {i, 0, 20}]\)], "Input"], Cell[BoxData[ \({1\/2\ \@\[Pi]\ Erf[1], 0, \(5\ \((\(-6\) + \[ExponentialE]\ \@\[Pi]\ Erf[1])\)\)\/\(8\ \ \[ExponentialE]\), 0, \(9\ \((\(-230\) + 57\ \[ExponentialE]\ \@\[Pi]\ Erf[1])\)\)\/\(64\ \ \[ExponentialE]\), 0, \(13\ \((\(-7938\) + 1955\ \[ExponentialE]\ \@\[Pi]\ \ Erf[1])\)\)\/\(256\ \[ExponentialE]\), 0, \(17\ \((\(-1579218\) + 388955\ \[ExponentialE]\ \@\[Pi]\ \ Erf[1])\)\)\/\(4096\ \[ExponentialE]\), 0, \(147\ \((\(-14740550\) + 3630537\ \[ExponentialE]\ \@\[Pi]\ Erf[1])\ \)\)\/\(16384\ \[ExponentialE]\), 0, \(25\ \((\(-16734510494\) + 4121641293\ \[ExponentialE]\ \@\[Pi]\ \ Erf[1])\)\)\/\(131072\ \[ExponentialE]\), 0, \(29\ \((\(-1621927519250\) + 399474095163\ \[ExponentialE]\ \@\[Pi]\ \ Erf[1])\)\)\/\(524288\ \[ExponentialE]\), 0, \(33\ \((\(-1463595204914978\) + 360477495590355\ \[ExponentialE]\ \ \@\[Pi]\ Erf[1])\)\)\/\(16777216\ \[ExponentialE]\), 0, \(37\ \((\(-188369607025634322\) + 46394661555265595\ \ \[ExponentialE]\ \@\[Pi]\ Erf[1])\)\)\/\(67108864\ \[ExponentialE]\), 0, \(41\ \((\(-54487481501236728090\) + 13420043196802227271\ \ \[ExponentialE]\ \@\[Pi]\ Erf[1])\)\)\/\(536870912\ \[ExponentialE]\)}\)], \ "Output"] }, Open ]], Cell["Construct a vector of the eigenfunctions", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Lfuncs\ = \ Table[LegendreP[i, x], {i, 0, 20}];\)\)], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(Lfuncs\\)\\\" is similar to existing symbol \ \\\"\\!\\(funcs\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"] }, Open ]], Cell["Visualize the approximation of the first twenty terms", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{Lcofs . 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