0
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Consider

F[x_, n_, m_] := 2^{n/2}*WaveletPsi[HaarWavelet[], 2^{n}*x - m]

A paper I am reading uses a different convention of the Fourier transform than Mathematica's default setting, so I decided to compute a certain Fourier transform manually with the integrate command. However, when I try to run

FullSimplify[Integrate[F[x, n, m]*Exp[-I*w*x], {x, -Infinity, Infinity},Assumptions -> {Element[{n, m}, Integers], Element[w, Reals]}]]

the only output I get is Mathematica inserting the definition of F to the integral and LaTeXfying the Element commands. This is a bit odd since the Transform most certainly exists. What is going on?

Edit: The output I get is

{Integrate[2^(n/2) E^(-I w x) ( \!\(\*
TagBox[GridBox[{
{"\[Piecewise]", GridBox[{
{
RowBox[{"-", "1"}], 
RowBox[{
FractionBox["1", "2"], "<", 
RowBox[{"{", 
RowBox[{
RowBox[{"-", "m"}], "+", 
RowBox[{
SuperscriptBox["2", "n"], " ", "x"}]}], "}"}], "<", "1"}]},
{"1", 
RowBox[{"0", "<", 
RowBox[{"{", 
RowBox[{
RowBox[{"-", "m"}], "+", 
RowBox[{
SuperscriptBox["2", "n"], " ", "x"}]}], "}"}], "<", 
FractionBox["1", "2"]}]},
{"0", 
TagBox["True",
"PiecewiseDefault",
AutoDelete->True]}
},
AllowedDimensions->{2, Automatic},
Editable->True,
GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}},
GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{1.}}},
GridBoxSpacings->{"Columns" -> {
Offset[
               0.27999999999999997`], {
Offset[0.84]}, 
Offset[0.27999999999999997`]}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}},
Selectable->True]}
},
GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}},
GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{1.}}},
GridBoxSpacings->{"Columns" -> {
Offset[
            0.27999999999999997`], {
Offset[0.35]}, 
Offset[0.27999999999999997`]}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}}],
"Piecewise",
DeleteWithContents->True,
Editable->False,
SelectWithContents->True,
Selectable->False,
StripWrapperBoxes->True]\) ), {x, -\[Infinity], \[Infinity]}, 
  Assumptions -> (n | m) \[Element] Integers && w \[Element] Reals]}
Improve this question
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3
  • $\begingroup$ First, you shouldn't use curly brackets because they represent a List, use round brackets (). Second, you should use FourierTransform, not Integrate (there are several question on StackExchange about this). Third, it works if you explicitly specify m and n, for example: f[x_, n_, m_] := 2^(n/2)*WaveletPsi[HaarWavelet[], 2^n*x - m]; FourierTransform[f[x, n, m] /. {n -> 3, m -> 2}, x, w] returns (2 I E^((I w)/4) (-1 + E^((I w)/16))^2)/(Sqrt[\[Pi]] w) $\endgroup$
    – Domen
    Mar 10 at 17:43
  • $\begingroup$ @Domen I changed the curly brackets to to round brackets, but the issue does not go away. I am just curios on knowing why Mathematica cannot calculate the integral when even I can do it for generic $m,n$ with pen and paper. Lastly, like I said, I am using the regular integrate because of the differing convention. $\endgroup$
    – Epsilon Away
    Mar 10 at 17:47
  • $\begingroup$ Why not change the default Fourier Transform setting? $\endgroup$
    – 1729taxi
    Mar 10 at 18:56

1 Answer 1

Reset to default
2
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Without loss of generality we may assume m==0 (see the translation property). Then in 13.2 on Windows 10 both

m=0;1/Sqrt[2*Pi]*Integrate[ 2^(n/2)*WaveletPsi[HaarWavelet[], 2^n*x - m]*
Exp[I*w*x], {x, -Infinity, Infinity}, 
Assumptions -> {Element[{n, m}, Integers], Element[w, Reals]}]

and

m = 0; FourierTransform[2^(n/2)*WaveletPsi[HaarWavelet[], 2^n*x - m], x, w, 
 Assumptions -> n \[Element] Integers]

result in

(I 2^(-(1/2) + n/2) (-1 + E^(I 2^(-1 - n) w))^2)/(Sqrt[\[Pi]] w)

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