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For a scaling refinement equation $\varphi ={\sum }_n{a}_n\varphi (2\cdot -n)$, set $m(t)=\frac{1}{2}{\sum }_n{a}_n{e}^{-int}$. The quadrature mirror filter identity makes the two rows $(m(t),m(t+\pi ))$ and $(q(t),q(t+\pi ))$ a unitary matrix, where $q(t)=-e^{-it}\overline{m(t+\pi )}$. The complementary coefficients $b_n=(-1{)}^n\overline{a_{1-n}}$ yield $\psi ={\sum }_n{b}_n\varphi (2\cdot -n)$. In a multiresolution analysis, coefficient-space completeness shows its translates form an orthonormal basis of $V_1\ominus V_0$, hence its dilations and translates form an orthonormal wavelet basis of the whole space.

Every bounded sequence in a Hilbert space has a weakly convergent subsequence. The closed span of a sequence is separable; diagonal convergence of coordinates along an orthonormal basis gives a bounded limiting linear functional, represented by a vector of the Hilbert space. Applied to a bounded family of mollifications, this proves that their strong Lebesgue limit inherits the weak derivative bound.

Generalized Gaussian white noise over a real Hilbert space $H=L^2$ is an isonormal Gaussian process $W(h)$ with covariance $⟨h,g{⟩}_H$. Its law $P_0$ is carried on a larger observation space; the identity is not the covariance of an infinite-dimensional $H$-valued Gaussian measure because it is not a trace-class operator. A deterministic signal $h\in H$ defines a translated observation law $P_h$ with Radon-Nikodym derivativeIn an orthonormal basis, $W_m(h)={\sum }_j{m}_j{h}_j$ is a stochastic series with total variance $\Vert h{\Vert }_H^2$, not an inner product of two $H$-valued observations. The formula follows from the Cameron-Martin theorem for a Gaussian measure in its generalized white-noise version, or from finite-dimensional Gaussian likelihood ratios. On ${\mathbb{R}}^2$, generalized noise can be realized on tempered distributions; an unweighted global negative Sobolev space is not automatically a suitable almost-sure support on an unbounded domain. The stochastic series avoids imposing that unsupported regularity.

Normalizing the tableau lines in a compatible real phase convention gives the orthonormal basis formAdmissible swaps have $|d|\ge 2$. Nonadmissible swaps act by $+1$ within a row and $-1$ within a column. The symmetric matrix is an orthogonal involution and is useful for making the unitary representation and its phases explicit.