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Every algebra character of the Wiener algebra is evaluation at a unique point of the unit circle. Writing $u(w)=w$, both $u$ and $u^{-1}$ have norm one. Thus $z=\varphi (u)$ satisfies $|z|\le 1$ and $|z^{-1}|\le 1$, hence $|z|=1$. Continuity then gives $\varphi ({\sum }_{n\in \mathbb{Z}}{a}_n{u}^n)=\sum a_n{z}^n$. Conversely, evaluation defines a continuous algebra character. The resulting bijection is a homeomorphism for the Gelfand topology.

LetThe Fourier coefficients of $\theta$ are absolutely summable, so $a$ belongs to the Wiener algebra, as does $a^{-1}$. The identity $\theta (2t)=\theta (t)+\theta (t+\pi )$ implies $a(2t)=a(t)a(t+\pi )$. Starting with a compactly supported Daubechies wavelet having $p\ge 3$ vanishing moments, the periodic phase change of a scaling function gives $m(t)=a(t+\pi )m_0(t)$, while the canonical high-pass construction leaves $\hat{\psi }(2t)=e^{-it}\overline{m_0(t+\pi )}{\hat{\phi }}_0(t)$ unchanged.

At a dyadic point $t_0=2\pi k/2^j$, the terms of the difference quotient with $n\ge j$ and $2^n|h|\le 1$ each contribute $1+O((2^{n}h{)}^2)$. Their number tends to infinity, their total error is bounded, and the remaining tail contributes a bounded amount. Hence $(\theta (t_0+h)-\theta (t_0))/h={\log }_2(1/|h|)+O(1)$, so neither $\theta$ nor $a$ has a finite derivative there. These points are dense. Away from the isolated zero of $m_0$ near $\pi$, multiplication by its nonzero smooth value cannot remove this nondifferentiability. Thus the new MRA low-pass filter is not differentiable throughout any neighborhood of $\pi$, despite integrability of the scaling function and unchanged vanishing moments. Ordinary higher derivatives at $\pi$ cannot be inferred, although the corresponding Peano zero survives.

Use the following general Banach algebra facts: in a complex commutative unital Banach algebra, maximal ideals are exactly the kernels of algebra characters, meaning nonzero complex multiplicative linear functionals. Such a character satisfies $\varphi (1)=1$ and $|\varphi (f)|\le \Vert f\Vert$, hence is continuous. The maximal ideal space is given the Gelfand topology, that is, pointwise convergence of characters on algebra elements. These facts follow from closedness of maximal ideals, the Gelfand-Mazur theorem for their quotients, and the spectral bound for character values. They do not identify any of the three spaces by themselves.

For the Wiener algebra, write $f={\sum }_{n\in \mathbb{Z}}{a}_n{u}^n$, where $u(e^{it})=e^{it}$ and $\Vert f{\Vert }_A=\sum |a_n|$. The Fourier series converges absolutely in the algebra norm. If $\varphi$ is a character, set $z=\varphi (u)$. Since $u$ is invertible and $\Vert u{\Vert }_A=\Vert u^{-1}{\Vert }_A=1$, multiplicativity and the norm bound give $|z|\le 1$, $|z^{-1}|\le 1$. Thus $|z|=1$. Continuity now givesConversely, for every $z\in \mathbb{T}$, evaluation is nonzero, linear and multiplicative, and $|f(z)|\le \sum |a_n|$ makes it continuous. Distinct $z$ give distinct characters because they have different values on $u$. Hence the character space of the Wiener algebra is the circle.

For the analytic Wiener algebra, the same expression uses only $n\ge 0$. A character is again determined by $z=\varphi (u)$, but $u^{-1}$ is no longer in this algebra, so only $|z|\le 1$ is forced. Continuity gives $\varphi (f)={\sum }_{n\ge 0}{a}_n{z}^n$. Conversely, every $z$ in the closed unit disk defines evaluation on this absolutely convergent power series. It is bounded and multiplicative, since multiplication of the series is absolutely convergent discrete convolution. Distinct $z$ again differ on $u$. Therefore the character space of the analytic Wiener algebra is the closed disk, including its interior and boundary, not just the original boundary circle.

For $C(\mathbb{T})$ with the supremum norm, let $M=\ker \varphi$ for a character. Suppose the functions in $M$ had no common zero. At each point choose some $f\in M$ nonzero there. Continuity supplies a neighborhood where it remains nonzero; compactness gives a finite subcover associated to $f_1,\dots ,f_k\in M$. Thenis continuous and strictly positive everywhere. It belongs to $M$ because $M$ is an ideal, but it is invertible in $C(\mathbb{T})$ with continuous inverse $1/g$. This would imply $1\in M$, a contradiction. There is therefore a common zero $z$ of $M$. The point-evaluation kernel $I_z=\{f:f(z)=0\}$ is maximal, and $M\subseteq I_z$ forces equality. Since $\varphi (1)=1$, writing $f=f(z)1+(f-f(z)1)$ then proves $\varphi (f)=f(z)$. Conversely every evaluation is a character, and $u$ distinguishes the points. Thus the character space of continuous functions on the circle is the circle.

These are topological identifications as well as bijections. The evaluation map $z\mapsto {\varphi }_z$ is continuous for the Gelfand topology, because each fixed algebra element is a continuous function of $z$: absolute convergence gives uniform convergence on the circle or closed disk in the first two cases. The parameter spaces are compact, while the character spaces are Hausdorff as subspaces of the pointwise topology. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Consequently the three maximal ideal spaces are, respectively,In each case the maximal ideal associated to a point is the kernel of the evaluation just described. The spelling “Weiner” in the source refers to the Wiener algebra.

A wavelet with $p$ vanishing moments satisfies $x^k\psi \in L^1$ and $\int x^k\psi (x)dx=0$ for $0\le k<p$. The moment differentiation of the Fourier transform theorem states that these weighted integrability hypotheses make $\hat{\psi }\in C^{p-1}$, with ${\hat{\psi }}^{(k)}(\xi )=\int (-ix{)}^k\psi (x)e^{-ix\xi }dx$. Hence ${\hat{\psi }}^{(k)}(0)=0$. The MRA projection Fourier identity, together with density of the approximation spaces and continuity of $\hat{\phi }$ at zero, gives $|\hat{\phi }(0)|=1$. These are the two analytic theorems needed in addition to the quadrature mirror filter construction.

For the ordinary derivative conclusion, assume also that the MRA low-pass filter is $C^{p-1}$ near $\pi$ and $\hat{\phi }$ is $C^{p-1}$ near zero. These hypotheses hold, for example, for a compactly supported scaling function with a finite refinement filter. The wavelet identity givesThe denominator is nonzero near zero. Its reciprocal and the exponential are $C^{p-1}$, so the product rule and the zero Taylor polynomial of $\hat{\psi }$ give the smooth-mask vanishing-moment criterion

The printed integrability hypothesis alone does not guarantee ordinary higher derivatives. It gives only a continuous $\hat{\phi }$. Without added smoothness, Taylor theorem for $\hat{\psi }$ still gives $m(\pi +t)=o(|t{|}^{p-1})$ for the representative defined by the quotient: a Peano zero. This establishes the value at $\pi$, and for $p\ge 2$ the first derivative there, but it does not automatically establish iterated derivatives.

Here is a lacunary scaling-phase regularity counterexample for $p\ge 3$. Start with a compactly supported Daubechies wavelet having $p$ vanishing moments, with scaling function ${\phi }_0$ and finite MRA low-pass filter $m_0$. Define the periodic phaseThe Fourier coefficients of $\theta$ are absolutely summable. The Wiener algebra is closed under multiplication and the exponential series, so $a$ and $a^{-1}$ have absolutely summable Fourier coefficients. Thus $\phi$ is an absolutely summable combination of integer function translations of ${\phi }_0$, belongs to $L^1\cap L^2$, and generates the same $V_0$. Unimodularity preserves orthogonality and unit norm of its integer function translations; the inverse phase preserves their complete span. The same approximation spaces therefore give a multiresolution analysis.

The identity $\theta (2t)=\theta (t)+\theta (t+\pi )$ gives $a(2t)=a(t)a(t+\pi )$, so the periodic phase change of a scaling function produces $m(t)=a(t+\pi )m_0(t)$. In the high-pass formula the phases cancel:The associated wavelet is therefore exactly the original one, with unchanged vanishing moments.

At any dyadic point $t_0=2\pi k/2^j$, the difference quotient of the terms with $j\le n\le {\log }_2(1/|h|)$ is $1+O((2^{n}h{)}^2)$ per term. The sum of these errors and the remaining tail divided by $h$ are bounded; the finitely many earlier terms have finite limits. Hence $(\theta (t_0+h)-\theta (t_0))/h={\log }_2(1/|h|)+O(1)$, which has no finite limit. The exponential has the same failure of differentiability. Dyadic points are dense, and $m_0(\pi +t)$ is nonzero for sufficiently small $t\ne 0$. Thus $m$ is not differentiable on any neighborhood of $\pi$, so an ordinary second derivative at $\pi$, understood as the derivative of a locally defined first derivative, need not exist. The Peano zero remains valid. This separates the intended smooth-mask theorem from what the literal assumptions establish.

A finite collection of Euclidean open balls has a disjoint subcollection whose concentric triples cover the original union. Greedily retain a largest remaining ball and discard all intersecting ones. Each discarded ball has radius no greater than the retained one and lies in its triple. Lebesgue measure consequently bounds the original union by $3^d$ times the sum of the retained volumes. This covering lemma is distinct from inverse-closedness results for the Wiener algebra.