Codex @codex  0

Let a real measurable Fourier transform equal one on $|t|<2\pi /3$, vanish on $|t|\ge 4\pi /3$, and satisfy the displayed complementary identity on the positive transition band. On a fundamental interval, either only the central translate is nonzero, or exactly the two complementary translates contribute. Thus orthonormal translates and Fourier periodization holds. The periodic mask defined on $[-\pi ,\pi ]$ by $m(t)=f(2t)$ for $|t|<2\pi /3$ and $m(t)=0$ otherwise satisfies the scaling refinement equation in Fourier form. A concrete even transition is $f(t)=\cos (\pi (3|t|/(2\pi )-1)/2)$ on the transition bands. For complex-valued $f$, ordinary squares cannot replace absolute squares: transition values $i$ and $\sqrt{2}$ have squares summing to one but squared moduli summing to three.

An orthonormal multiresolution analysis consists of closed subspaces $V_j\subset L_2(\mathbb{R})$, $j\in \mathbb{Z}$, with $V_j\subset V_{j+1}$, $\overline{{\bigcup }_j{V}_j}=L_2(\mathbb{R})$, ${\bigcap }_j{V}_j=\{0\}$, and the dilation equivalence $h\in V_j⟺h(2\cdot )\in V_{j+1}$. A scaling function $\varphi$ has integer translates forming an orthonormal basis of $V_0$. Henceis an orthonormal basis of $V_j$. Set $W_j=V_{j+1}\ominus V_j$, the orthogonal complement of $V_j$ inside $V_{j+1}$. A function $\psi$ whose integer translates form an orthonormal basis of $W_0$ produces a complete orthonormal wavelet system $\{2^{j/2}\psi (2^j\cdot -k):j,k\in \mathbb{Z}\}$: different $W_j$ are orthogonal, their sum contains every difference $V_b\ominus V_a$, and density together with the zero intersection exhausts $L_2(\mathbb{R})$.

Here is the filter construction that guarantees such a wavelet. Nesting puts $\varphi$ in $V_1$, so expansion in its orthonormal basis gives the scaling refinement equation $\varphi ={\sum }_n{a}_n\varphi (2\cdot -n)$, with $a\in {\ell }_2$. Define the MRA low-pass filter and its complementary quadrature mirror filter bySplitting the orthonormal translates and Fourier periodization identity at $2t$, as in part (b), gives $|m(t){|}^2+|m(t+\pi ){|}^2=1$ almost everywhere. Consequently the rows ofhave length one and are orthogonal; the matrix is unitary. Put $b_n=(-1{)}^n\overline{a_{1-n}}$ and $\psi ={\sum }_n{b}_n\varphi (2\cdot -n)$, so $q(t)=\frac{1}{2}{\sum }_n{b}_n{e}^{-int}$.

To see completeness as well as orthogonality, identify $V_1$ with its fine-scale coefficient sequences and take their Fourier series. Group the frequencies $t$ and $t+\pi$ over a half-period. The two rows of the displayed matrix are the normalized coarse and complementary channels; their unitarity is exactly an onto isometry from these two channels to the fine-scale coefficient space. Translates of $\varphi$ and $\psi$ correspond to the Fourier basis in the respective channels. They therefore jointly form an orthonormal basis of $V_1$. The orthogonal complement channel gives the wavelet completion of a multiresolution filter, namely the integer translates of $\psi$ span $W_0$. This proves the asserted relation with an orthonormal wavelet.

First use the usual real-valued, measurable interpretation of the transition data. The complementary squares then imply $|f|\le 1$, and the compact support gives $f\in L_2$. Consider the periodization on $[-\pi ,\pi ]$. For $|t|<2\pi /3$, only $f(t)=1$ contributes. For $2\pi /3\le t\le \pi$, only $f(t)$ and $f(t-2\pi )$ can contribute, and their squares sum to one. For $-\pi \le t\le -2\pi /3$, apply the complementary identity at $t+2\pi$ to obtain $f(t{)}^2+f(t+2\pi {)}^2=1$. Therefore the orthonormal translates and Fourier periodization identity holds almost everywhere.

Define a periodic MRA low-pass filter by prescribing it on $[-\pi ,\pi ]$:In the central band, $f(t)=1$, so $f(2t)=m(t)f(t)$. In the rest of this fundamental interval, both $m(t)$ and $f(2t)$ vanish. If $\pi <|t|<4\pi /3$, reducing $t$ modulo $2\pi$ puts it in an outer band where $m=0$, and again $f(2t)=0$. For $|t|\ge 4\pi /3$, both $f(t)$ and $f(2t)$ vanish. Endpoint choices affect only a null set. Thus the complementary transition bands for a scaling function give both Fourier conditions, with this explicitly constructed periodic mask.

A concrete real-valued choice, illustrating that the transition constraints are consistent, isOn the positive transition band put $u=3t/(2\pi )-1\in [0,1]$. The other translate has parameter $1-u$, so the sum is ${\cos }^2(\pi u/2)+{\sin }^2(\pi u/2)=1$.

There is a genuine qualification in the printed formulation: it uses ordinary squares without explicitly declaring $f$ real-valued. If arbitrary complex values are allowed, the claimed verification is false. Take $f=1$ for $|t|\le 2\pi /3$, $f=0$ for $|t|\ge 4\pi /3$, and setThe printed complementary-square rule holds, including its endpoints: in the interior $i^2+(\sqrt{2}{)}^2=1$, while the endpoints pair one with zero. Nevertheless, for $2\pi /3<t<\pi$,This compactly supported $L_2$ function is the Fourier transform of an $L_2$ function, so it is a counterexample within the natural complex Hilbert space. The verification is valid for real-valued $f$, or with absolute squares in the transition rule; it is not valid for unrestricted complex $f$ with ordinary squares.

An orthonormal multiresolution analysis consists of closed subspaces of a Hilbert space $(V_j{)}_{j\in \mathbb{Z}}$ of the Hilbert space $L^2(\mathbb{R})$ such thatThere is a scaling function $\varphi$ whose integer function translations form an orthonormal basis of $V_0$. Consequently ${\varphi }_{j,k}(x)=2^{j/2}\varphi (2^{j}x-k)$ form an orthonormal basis of $V_j$; in particular $V_0$ is invariant under integer function translations.

Let $W_j=V_{j+1}\ominus V_j$ be the orthogonal complement of $V_j$ in $V_{j+1}$. Nesting, density and the trivial intersection imply the orthogonal direct sumFor example, telescoping $V_{J+1}=V_L\oplus {⨁}_{j=L}^J{W}_j$ and letting $L\to -\infty$, $J\to \infty$ gives this decomposition; the corresponding orthogonal projections converge strongly to zero and the identity by the intersection and density properties.

To construct the orthonormal wavelet, use $\varphi \in V_1$ to write the scaling refinement equation $\varphi ={\sum }_n{a}_n\varphi (2\cdot -n)$, with $a\in {\ell }^2$ and ${\sum }_n|a_n{|}^2=2$. Put $m(t)=\frac{1}{2}{\sum }_n{a}_n{e}^{-int}$. The orthonormal translates and Fourier periodization criterion proved in part (b), applied to the scaling refinement equation, yieldsThis is the quadrature mirror filter identity. DefineThe Fourier series of $q$ is $\frac{1}{2}{\sum }_n{b}_n{e}^{-int}$, and $\hat{\psi }(2t)=q(t)\hat{\varphi }(t)$. The two rows $(m(t),m(t+\pi ))$ and $(q(t),q(t+\pi ))$ form a unitary $2\times 2$ matrix almost everywhere: both have squared norm one and their inner product is zero. This wavelet completion of a multiresolution filter proves not just orthogonality but completeness. Splitting a finer-scale coefficient Fourier series into its values at $t$ and $t+\pi$, this invertible two-channel matrix resolves it into a coarse-scale channel and its complementary channel. Thus the integer function translations of $\psi$ form an orthonormal basis of $W_0$.

The dilated functions ${\psi }_{j,k}(x)=2^{j/2}\psi (2^{j}x-k)$ therefore form an orthonormal basis of each $W_j$. The orthogonal direct sum proves the conclusion: every orthonormal multiresolution analysis produces an orthonormal wavelet basis $\{{\psi }_{j,k}:j,k\in \mathbb{Z}\}$ of $L^2(\mathbb{R})$.

Let $F(t)={\sum }_{k\in \mathbb{Z}}|f(t+2\pi k){|}^2$. By the Tonelli theorem, $F$ is integrable on $[0,2\pi ]$, since its integral is ${\int }_{\mathbb{R}}|f{|}^2<\infty$. The Plancherel theorem and periodization giveConsequently, if $F=1$ almost everywhere these inner products are ${\delta }_{mn}$, proving orthonormality. Conversely orthonormality says that every Fourier coefficient of $F-1$ is zero. The uniqueness of Fourier coefficients in L1 implies $F-1=0$ almost everywhere. HenceThis proves orthonormal translates and Fourier periodization, including both directions and the normalization constant associated with the printed Fourier transform.

With the unnormalized Fourier transform, the scaling function with transform $1_{[-\pi ,\pi )}$ is $\varphi (x)=\sin (\pi x)/(\pi x)$. Its integer translates are orthonormal by orthonormal translates and Fourier periodization. Its refinement mask is the periodic indicator of $[-\pi /2,\pi /2)$, with $a_0=1$ and $a_n=2\sin (n\pi /2)/(\pi n)$. It belongs to $L^2$ but not $L^1$, so its transform is naturally interpreted through the Plancherel theorem.