# 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

• The scaling function satisfies the recursion equation , where are the lowpass filter coefficients.
• WaveletPhi[wave,x,"Dual"] gives the dual scaling function for biorthogonal wavelets such as BiorthogonalSplineWavelet and ReverseBiorthogonalSplineWavelet.
• The dual scaling function satisfies the recursion equation , where are the dual lowpass filter coefficients.
• The following options can be used:
•  MaxRecursion 8 number of recursive iterations to use WorkingPrecision MachinePrecision precision to use in internal computations

# Examples

open allclose all

## Basic Examples(2)

Haar scaling function:

Symlet scaling function:

## Scope(4)

Compute primal scaling function:

Dual scaling function:

Scaling function for HaarWavelet:

Multivariate scaling and wavelet functions are products of univariate ones:

## Options(3)

### MaxRecursion(1)

Plot scaling function using different levels of recursion:

### WorkingPrecision(2)

By default is used:

Use higher-precision filter computation:

## Properties & Relations(4)

Scaling function integrates to unity :

In particular, :

satisfies the recursion equation :

Plot the components and the sum of the recursion:

Frequency response for is given by :

The filter is a lowpass filter:

Fourier transform of is given by :

## Neat Examples(1)

Plot translates and dilations of scaling function:

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.

#### CMS

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

#### BibTeX

@misc{reference.wolfram_2022_waveletphi, author="Wolfram Research", title="{WaveletPhi}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/WaveletPhi.html}", note=[Accessed: 01-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_waveletphi, organization={Wolfram Research}, title={WaveletPhi}, year={2010}, url={https://reference.wolfram.com/language/ref/WaveletPhi.html}, note=[Accessed: 01-June-2023 ]}