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WaveletPhi
✖
WaveletPhi

✖
WaveletPhi[wave,x]

gives the scaling function for the symbolic wavelet wave evaluated at x.

✖
WaveletPhi[wave]

gives the scaling function as a pure function.

Details and Options

Examples

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Basic Examples  (2)Summary of the most common use cases

Haar scaling function:

In[1]:=1
✖
https://wolfram.com/xid/0mlijs723hoz-4lu6z4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mlijs723hoz-kcilqx
Out[2]=2

Symlet scaling function:

In[1]:=1
✖
https://wolfram.com/xid/0mlijs723hoz-ottmes
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0mlijs723hoz-x9588r
Out[2]=2

Scope  (4)Survey of the scope of standard use cases

Compute primal scaling function:

In[1]:=1
✖
https://wolfram.com/xid/0mlijs723hoz-34r0y7
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0mlijs723hoz-jrnvtl
Out[2]=2

Dual scaling function:

In[1]:=1
✖
https://wolfram.com/xid/0mlijs723hoz-rqnscj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mlijs723hoz-wgpsnb
Out[2]=2

Scaling function for HaarWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0mlijs723hoz-5qil82
Out[1]=1

DaubechiesWavelet:

In[2]:=2
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https://wolfram.com/xid/0mlijs723hoz-8a3e1y
Out[2]=2

SymletWavelet:

In[3]:=3
✖
https://wolfram.com/xid/0mlijs723hoz-6m8se7
Out[3]=3

CoifletWavelet:

In[4]:=4
✖
https://wolfram.com/xid/0mlijs723hoz-uikmj
Out[4]=4

BiorthogonalSplineWavelet:

In[5]:=5
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https://wolfram.com/xid/0mlijs723hoz-slohlo
Out[5]=5

ReverseBiorthogonalSplineWavelet:

In[6]:=6
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https://wolfram.com/xid/0mlijs723hoz-isy48o
Out[6]=6

CDFWavelet:

In[7]:=7
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https://wolfram.com/xid/0mlijs723hoz-m6y6cq
Out[7]=7

ShannonWavelet:

In[8]:=8
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https://wolfram.com/xid/0mlijs723hoz-cylqwd
Out[8]=8

BattleLemarieWavelet:

In[9]:=9
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https://wolfram.com/xid/0mlijs723hoz-2pig47
Out[9]=9

MeyerWavelet:

In[10]:=10
✖
https://wolfram.com/xid/0mlijs723hoz-tg8sc8
Out[10]=10

Multivariate scaling and wavelet functions are products of univariate ones:

In[1]:=1
✖
https://wolfram.com/xid/0mlijs723hoz-rk8e1w
In[2]:=2
✖
https://wolfram.com/xid/0mlijs723hoz-49wf0
Out[2]=2

Options  (3)Common values & functionality for each option

MaxRecursion  (1)

Plot scaling function using different levels of recursion:

In[1]:=1
✖
https://wolfram.com/xid/0mlijs723hoz-si6qgc
Out[1]=1

WorkingPrecision  (2)

By default WorkingPrecision->MachinePrecision is used:

In[1]:=1
✖
https://wolfram.com/xid/0mlijs723hoz-jj9fjg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mlijs723hoz-6vi03j
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0mlijs723hoz-qtqwdb
Out[3]=3

Use higher-precision filter computation:

In[1]:=1
✖
https://wolfram.com/xid/0mlijs723hoz-oxpk71
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mlijs723hoz-f6wruc
Out[2]=2

Properties & Relations  (4)Properties of the function, and connections to other functions

Scaling function integrates to unity :

In[1]:=1
✖
https://wolfram.com/xid/0mlijs723hoz-f8fecv
Out[1]=1

In particular, :

In[2]:=2
✖
https://wolfram.com/xid/0mlijs723hoz-cckg8j
Out[2]=2

satisfies the recursion equation :

In[1]:=1
✖
https://wolfram.com/xid/0mlijs723hoz-yjbxzh
In[2]:=2
✖
https://wolfram.com/xid/0mlijs723hoz-h6wmys

Plot the components and the sum of the recursion:

In[3]:=3
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https://wolfram.com/xid/0mlijs723hoz-c986c4
In[4]:=4
✖
https://wolfram.com/xid/0mlijs723hoz-jgy4v3
Out[4]=4

Frequency response for is given by :

In[1]:=1
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https://wolfram.com/xid/0mlijs723hoz-y5x0mm

The filter is a lowpass filter:

In[2]:=2
✖
https://wolfram.com/xid/0mlijs723hoz-za6vn3
Out[2]=2

Fourier transform of is given by :

In[1]:=1
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https://wolfram.com/xid/0mlijs723hoz-idptzz
In[2]:=2
✖
https://wolfram.com/xid/0mlijs723hoz-u0b6k1
In[3]:=3
✖
https://wolfram.com/xid/0mlijs723hoz-z65kgr
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

Plot translates and dilations of scaling function:

In[1]:=1
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https://wolfram.com/xid/0mlijs723hoz-yz9dxl
In[2]:=2
✖
https://wolfram.com/xid/0mlijs723hoz-evsjio
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0mlijs723hoz-p5xgws
Out[3]=3
Wolfram Research (2010), WaveletPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/WaveletPhi.html.
✖
Wolfram Research (2010), WaveletPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/WaveletPhi.html.

Text

Wolfram Research (2010), WaveletPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/WaveletPhi.html.

✖
Wolfram Research (2010), WaveletPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/WaveletPhi.html.

CMS

Wolfram Language. 2010. "WaveletPhi." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WaveletPhi.html.

✖
Wolfram Language. 2010. "WaveletPhi." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WaveletPhi.html.

APA

Wolfram Language. (2010). WaveletPhi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WaveletPhi.html

✖
Wolfram Language. (2010). WaveletPhi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WaveletPhi.html

BibTeX

@misc{reference.wolfram_2025_waveletphi, author="Wolfram Research", title="{WaveletPhi}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/WaveletPhi.html}", note=[Accessed: 01-April-2025 ]}

✖
@misc{reference.wolfram_2025_waveletphi, author="Wolfram Research", title="{WaveletPhi}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/WaveletPhi.html}", note=[Accessed: 01-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_waveletphi, organization={Wolfram Research}, title={WaveletPhi}, year={2010}, url={https://reference.wolfram.com/language/ref/WaveletPhi.html}, note=[Accessed: 01-April-2025 ]}

✖
@online{reference.wolfram_2025_waveletphi, organization={Wolfram Research}, title={WaveletPhi}, year={2010}, url={https://reference.wolfram.com/language/ref/WaveletPhi.html}, note=[Accessed: 01-April-2025 ]}