Scaling function

In the theory of bursts, a scaling function is a function satisfying the equation

\varphi (x)={\sqrt {2}}\sum \limits _{k\in \mathbb {Z} }h_{k}\varphi (2x+k).

{\ displaystyle \ varphi (x) = {\ sqrt {2}} \ sum \ limits _ {k \ in \ mathbb {Z}} h_ {k} \ varphi (2x + k).}

{\ displaystyle \ varphi (x) = {\ sqrt {2}} \ sum \ limits _ {k \ in \ mathbb {Z}} h_ {k} \ varphi (2x + k).}

This equation is called a two-scale relation or a scaling equation in the time domain . Coefficient Set $\{h_{k}\}_{k\in \mathbb {Z} }$ ${\ displaystyle \ {h_ {k} \} _ {k \ in \ mathbb {Z}}}$ ${\ displaystyle \ {h_ {k} \} _ {k \ in \ mathbb {Z}}}$ called a mask or filter .

Marking $m_{0}(\xi )={\frac {1}{\sqrt {2}}}\sum \limits _{k\in \mathbb {Z} }h_{k}e^{2\pi in\xi }$ ${\ displaystyle m_ {0} (\ xi) = {\ frac {1} {\ sqrt {2}}} \ sum \ limits _ {k \ in \ mathbb {Z}} h_ {k} e ^ {2 \ pi in \ xi}}$ ${\ displaystyle m_ {0} (\ xi) = {\ frac {1} {\ sqrt {2}}} \ sum \ limits _ {k \ in \ mathbb {Z}} h_ {k} e ^ {2 \ pi in \ xi}}$ and applying the Fourier transform to both sides of the scaling equation we get

{\widehat {\varphi }}(\xi )=m_{0}(\xi /2){\widehat {\varphi }}(\xi /2).

{\ displaystyle {\ widehat {\ varphi}} (\ xi) = m_ {0} (\ xi / 2) {\ widehat {\ varphi}} (\ xi / 2).}

{\ displaystyle {\ widehat {\ varphi}} (\ xi) = m_ {0} (\ xi / 2) {\ widehat {\ varphi}} (\ xi / 2).}

This equation is called the scaling equation in the frequency domain .

Literature

Charles K. Chui, An Introduction to Wavelets , (1992), Academic Press, San Diego, ISBN 0585470901
Novikov I. Ya., Protasov V. Yu., Skopina M.A., Splash Theory , (2005), Fizmatlit, Moscow, ISBN 5922106422

Scaling function

Literature

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