Question

Let $\varphi(x)$ satisfy the Daubechies scaling equation (9.153). Prove that if $\varphi(i) \neq 0$ for any $i \leq 0$ or $i \geq p$, then supp $\varphi$ is unbounded.

    Let $\varphi(x)$ satisfy the Daubechies scaling equation (9.153). Prove that if $\varphi(i) \neq 0$ for any $i \leq 0$ or $i \geq p$, then supp $\varphi$ is unbounded.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 9, Problem 22

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The Daubechies scaling equation for a scaling function $\varphi(x)$ can be written as: \[ \varphi(x) = \sum_{k=0}^{N-1} h_k \varphi(2x - k) \] where $h_k$ are the coefficients of a low-pass filter, and $N$ is the number of coefficients. This equation is a Show more…

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Let $\varphi(x)$ satisfy the Daubechies scaling equation (9.153). Prove that if $\varphi(i) \neq 0$ for any $i \leq 0$ or $i \geq p$, then supp $\varphi$ is unbounded.

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