Consider the dilation equation (9.138) with c0=0, c1=c2=1, so φ(x)= φ(2 x-1)+φ(2 x-2). Prove tha

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Consider the dilation equation (9.138) with c0=0, c1=c2=1,...

Consider the dilation equation (9.138) with $c_0=0, c_1=c_2=1$, so $\varphi(x)=$ $\varphi(2 x-1)+\varphi(2 x-2)$. Prove that $\psi(x)=\varphi(x+1)$ satisfies the Haar dilation equation (9.139). Generalize this result to prove that we can always, without loss of generality, assume that $c_0 \neq 0$ in the general dilation equation (9.138).

Key Concepts

Dilation Equation

A dilation equation, also known as a refinement equation, is a functional equation where a function is expressed as a linear combination of scaled and translated copies of itself. This is a cornerstone in the theory of multiresolution analysis and wavelets, and it defines how scaling functions can be constructed recursively via fixed coefficients.

Haar Dilation Equation

The Haar dilation equation is a special case of a dilation equation that characterizes the Haar scaling function. It typically takes on a form that simplifies into two pieces after a change of variables. This special case illustrates the fundamental properties of wavelets, such as orthogonality and compact support, in the simplest nontrivial example.

Shifting (Translation) of Functions

Shifting a function involves translating its argument by a fixed constant. This technique is used to transform the original dilation equation into a more convenient form, such as turning a case with a zero coefficient into an equivalent one with a nonzero coefficient. The translation preserves many of the essential properties of the function, allowing for the derivation of standard forms like the Haar dilation equation.

Without Loss of Generality Assumption

The 'without loss of generality' argument in the context of dilation equations allows one to assume certain conditions (like a nonzero first coefficient) by performing a translation of the function. Such a shift does not alter the underlying structure of the equation but simplifies the analysis and proofs by ensuring the non-degeneracy of the coefficients.

Normalization

Normalization in the context of dilation equations refers to the process of adjusting the coefficients or the function itself so that certain desirable properties are satisfied — for instance, making sure that one of the coefficients (typically the first one) is nonzero. This is particularly useful in ensuring the well-posedness of the refinement process and simplifying the study of the scaling function.

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