Linear Subspace of Continuous Functions is Closed

Closed Subspace

A closed subspace Y of X either embeds into L1(μ) for some measure μ or contains a normalized basic sequence which is equivalent to (even equal to a small perturbation of) a disjoint sequence.

From: Handbook of the Geometry of Banach Spaces , 2001

Foundations of Complex Analysis in Non Locally Convex Spaces

Aboubakr Bayoumi , in North-Holland Mathematics Studies, 2003

10.4 Levi Problem (Quotient Map Approach)

In this Part we solve the Levi problem in some separable p-Banach spaces, (0   < p ≤  1). That is, in a complete locally bounded space with p-homogeneous norm. (The case p  =   1 corresponds to Banach spaces).

Since every separable Banach space Eis isomorphic to a quotient space of

$l_0={\cap }_{\mathrm{p>}}{l}_p\text{,}$

for a closed subspace Mof E,see Stiles [203], and since Eis a Levi space, see Corollary 20, p.240, we can obtain the following interesting partial result for the Levi problem.

Theorem 120

(Levi problem in certain separable Banach space) [14]

Let U be a pseudoconvex domain in a Banach space E. Suppose that E is isomorphic to a quotient space of l₀, that is,

(10.79) $E\simeq l_0/M\text{,}$

and l₀/Mhas the bounded approximation property.Then U is a domain of holomorphy.

Proof

Since l ₀is a Fréchet space, its quotient space l ₀ /Mis a Fréchet space. Now the space l ₀ /Mis assumed to have the b.a.p.Hence a direct application of Corollary 20, p.242, will imply the required result.This completes the proof of the theorem. ■

Let E_p be a p-Banach space. The fact that every separable locally bounded space is isomorphic to a quotient space of l_p (0   < p  ≤   1) will help us to obtain the following consequence which is of special interest.

Theorem 121

(Levi problem in separable p-Banach space) [14]

Let E_p be a separable p-Banach space which is isomorphic to a quotient space of l_p , that is

(10.80) $E\simeq l_p/M$

and l_p/M has the bounded approximation property. Then every pseudoconvex domain in E_p is a domain of holomorphy.

Proof

E_p is isomorphic to l_p/M for some closed subspace M of l_p. l_p/M is a Fréchet space, and by assumption it has the b.a.p. Then by applying Corollary 20, p.242, we achieve the solution and the proof is established. ■

Remark 44

The assumption that l_p/M has the b.a.p. cannot be dropped out of the above theorem. We note that L_p [0,1], (0   < p <  1) does not admit holomorphic functions other than0, and so each domain (pseudoconvex or not) is not a domain of holomorphy.

Problem 2

What is the class of separable p-Banach spaces each of its elements is isomorphic to a quotient space l_p/Mwith the b.a.p.?

Could Hardy space H_p (0   < p <   1) be an element of this class?

We note that l_p (0 <   p <  1) is an element of this class since it has a basis and hence this class is nonempty.

We give now more examples of non-locally convex spaces which have either a Schauder basis or the bounded approximation property. In fact, by the results of this chapter, all pseudoconvex domains in these spaces are domains of holomorphy, that is,they are Levi spaces.

Example 49

The Hardy spaces Hp(l >   p >  0)

The Hardy space H_p of all analytic functions f on the unit disc of $\mathbb{C}$ is separable, locally bounded, non locally convex space with respect to the p-norm

$\Vert f\Vert =\underset{r\to 1}{lim}{\displaystyle \underset{0}{\overset{2\pi }{\int }}}|f\left(r{e}^{i\theta }\right)|{}^{p}d\theta ,f\in H_p\left(\mathbb{C}\right)\text{.}$

The Banach envelope of H_p is isomorphic to l ₁,that is

(10.81) $H_p^{~}\simeq l_1$

by Kalton [130]. This implies that H _p ^~ has the b.a.p.

Moreover, H_p contains a non locally convex closed subspace M_p of E isomorphic to l_p (0   < p <  1), that is

(10.82) $M_p\simeq l_p\text{,}$

see Shapiro [189].

Now every pseudoconvex domain in this complemented subspaces Mof H_p,in H _p ^~ is a domain of holomorphy, by corollary 20, p.242. That is, H _p ^~ andM_p are Levi spaces. ▲

Remark 45 (Important)

The previous results of this chapter can be generalized to non-schlicht domains over a suitable space E. Let us first recall the following concepts which are analogues of those considered by Schottenloher[197] for locally convex spaces:

A Riemann domain spread overa metric vector spaceE is a pair

(10.83) $\left(\Omega ,q\right)$

where Ω is a connected Hausdorff space and

(10.84) $q:\Omega \to E$

is a local homeomorphism. That is, for every x ∈ Ω there exists a neighborhood w of x such that

(10.85) $q_w:w\to E$

is a homeomorphism of w ontoq(w)

If q is injective, the domain(Ω, q)is called schlicht domain, and can be identified, viaq, with a domain inE.

The boundary distance function d _Ω , on a fixed domain (Ω, q)over E is defined by:

(10.86) $d_{\Omega }\left(x\right)=sup(r;\mathit{there}\mathit{exists}a\mathit{neighborhood}U\mathit{of}xs.t\text{.}$

(10.87) $q|{}_U:U\to B\left(q\left(x\right),r\right)\mathit{is}a\mathit{homeomorphism}\mathit{)},x\in \Omega$

The ball B(x,r)for x ∈  Ω, r   >   d _Ω(x)is just the component of q ^-1(B(q(x),r))which contains x. The plurisubharmonic, the holomorphic or for that matter any locally defined class of functions, can now be defined on(Ω,q)using restrictions q|_w of the projection q, see the author[12],[13],and[14].

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Tensor Norms and Operator Ideals

In North-Holland Mathematics Studies, 1993

Ex 3.17.

If G is a closed subspace of a Banach space E with $\mathfrak{N}\left(F,G\right)=F\prime {\tilde{\otimes }}_{\pi }G$ for some F and F′ ⊗_π G ↪F′ ⊗_π E is not an isomorphic embedding, then there is a non–nuclear T ∈

(ℓ _p (F), ℓ _p (G)) which is nuclear as an operator ℓ _p (F) → ℓ _p (E). Hint: The operator F → G associated with z ∈ F′ ⊗ G has a norm ≤ π(z; F′, E).

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Numerical Analysis of Wavelet Methods

Albert Cohen , in Studies in Mathematics and Its Applications, 2003

Definition 2.2.1

A multiresolution analysis is a sequence of closed subspaces of L ²(ℝ), such that the following properties are satisfied:

1.

The sequence is nested, i.e. for all j,

(2.2.1) $V_j\subset V_{j+1}.$

2.

The spaces are related to each other by dyadic scaling, i.e.

(2.2.2) $f\in V_j\iff f\left(2\cdot \right)\in V_{j+1}\iff f\left(2^{-j}\cdot \right)\in V_0.$

3.

The union of the spaces is dense, i.e. for all f in L ²(ℝ)

(2.2.3) $\underset{j\to +\infty }{\lim }{\Vert f-P_j^{o}f\Vert }_{L^2}=0,$

where $P_j^o$ is the orthonormal projection onto V_j .

4.

The intersection of the spaces is reduced to the null function, i.e.

(2.2.4) $\underset{j\to -\infty }{\lim }{\Vert P_j^{o}f\Vert }_{L^2}=0.$

5.

There exists a function φ ∈ V ₀ such that the family

(2.2.5) $\phi \left(\cdot -k\right),k\in \mathbb{Z},$

is a Riesz basis of V ₀.

By definition, a family {e_k } _k∈Z is a Riesz basis of a Hilbert space H, if and only if it spans H, i.e. the finite linear combinations of the e_k are dense in H, and if there exist 0 < C ₁ ≤ C ₂ such that for all finitely supported sequences (x_k ), we have

(2.2.6) $C_1{{\displaystyle \sum _k{\left|x_k\right|}^2\le \Vert {\displaystyle \sum _k{x}_k{e}_k}\Vert }}_H^2\le C_2{\displaystyle \sum _k{\left|x_k\right|}^2}.$

This property expresses the "stability" of the expansion in this basis with respect to the coordinates. It also means that the application

(2.2.7) $T:{\left(c_k\right)}_{k\in \mathbb{Z}}\mapsto {\displaystyle \sum _{k\in \mathbb{Z}}{c}_k{e}_k},$

defines an isomorphism from ℓ ²(ℤ) to H.

The following three facts concerning Riesz bases are easy to check: firstly, the series ${\displaystyle {\sum }_{k\in \mathbb{Z}}{x}_k{e}_k}$ converges unconditionally in L ² (i.e. its terms can be permutated without affecting the convergence) if and only if ${\displaystyle {\sum }_{k\in \mathbb{Z}}{\left|x_k\right|}^2}$ is finite. Secondly, any x ∈ H can be decomposed in a unique way according to $x={\displaystyle {\sum }_{k\in \mathbb{Z}}{x}_k{e}_k}$ with ${\left(x_k\right)}_{k\in \mathbb{Z}}$ in ℓ ²(ℤ) and the equivalence in (2.2.6) also holds for such infinite linear combinations. Finally, there exists a unique biorthogonal Riesz basis ${\left\{{\tilde{e}}_k\right\}}_{k\in \mathbb{Z}}$ , (defined by ${\tilde{e}}_k={\left(T{T}^{*}\right)}^{-1}{e}_k)$ such that $〈e_k,{\tilde{e}}_l〉={\delta }_{k,l}$ and the coordinates of x in the basis are given by $x_k=〈x,{\tilde{e}}_k〉$ . We therefore have for all x ∈ H

(2.2.8) $x={\displaystyle \sum _{k\in \mathbb{Z}}〈x,{\tilde{e}}_k〉e_k={\displaystyle \sum _{k\in \mathbb{Z}}〈x,e_k〉}}{\tilde{e}}_k.$

In this sense, Riesz bases are very close to orthonormal bases, which appear as a particular case for which $e_k={\tilde{e}}_k$ and C ₁  = C ₂  =   1.

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Recent Progress in Functional Analysis

Jean Schmets , in North-Holland Mathematics Studies, 2001

3.6 Direct sum decomposition of spaces

In [119] and [128], M. Valdivia continues previous investigations on the direct sum decompositions of locally convex spaces.

The main result of [119] deals with two closed subspaces Y and Z of a Banach space X. If Y ≠ {0}, X = Y + Z and Z is weakly countably determined, then there is a continuous linear projection P on X such that ∥P∥ = 1, PX ⊃ Y, ker(P) ⊂ Z and dens(TX) = dens(Y). It leads to the fact that every Banach space is a topological direct sum X = X ₁ ⊕ X ₂ with X ₁ reflexive and $\text{dens}\left(X_2^{**}\right)=\text{dens}\left(X^{**}/X\right)$ , a result that he had already established under the assumption that X**/X is separable [54].

In [128], M. Valdivia considers the case of a Fréchet space E such that (E′, μ(E′, E″))is barrelled. He then proves that E is the direct sum of two closed subspaces F and G such that G is reflexive and dens(F″, β(F″, F′)) ≤ dens(E′, β(E′,E″)/E), generalizing the result of [119] mentioned above, as well as the one of [110].

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Spaces of Analytic Functions with Integral Norm

P. Wojtaszczyk , in Handbook of the Geometry of Banach Spaces, 2003

Subspaces and complemented subspaces

Let us first discuss subspaces of H ₁. The next proposition shows that many subspaces of L ₁ are also subspaces of H ₁.

Proposition 37

Let $X\subset L_1\left(\mathbb{T}\right)$ be a closed subspace. Suppose that X either has an unconditional basis or X is reflexive. Then X is isomorphic to a subspace of $H_1\left(\mathbb{T}\right)$ .

To see the first part fix an unconditional basis (x_n ) in X and assume (perturbing it slightly) that each x_n (t) is a trigonometric polynomial and fix a sequence k_n such that for some strictly increasing sequence of integers l_n we will have Λ ^{2l _n} (e^{ik _n t} x_n ) = e^{ik _n t} x_n , where Λ^s are multipliers discussed after Theorem 18. Then using the unconditionality of (x_n ) and Theorem 18 we get

$\begin{array}{ll}{\Vert {\displaystyle \sum _n{a}_n{x}_n}\Vert }_1\hfill & ~{\displaystyle {\int }_{\mathbb{T}}{\left({\displaystyle \sum _n{\left|a_n{x}_n\left(t\right)\right|}^2}\right)}^{1/2}\text{d}t}~{\displaystyle {\int }_{\mathbb{T}}{\left({\displaystyle \sum _n{\left|a_n{\text{e}}^{\text{i}{k}_{n}t}{x}_n\left(t\right)\right|}^2}\right)}^{1/2}\text{d}t}\hfill \\ \hfill & ~\Vert {\displaystyle \sum _n{a}_n{\text{e}}^{\text{i}{k}_{n}t}{x}_n}\Vert \hfill \end{array}$

which shows that e^ik_nt x_n is a basic sequence in $H_1\left(\mathbb{T}\right)$ equivalent to (x_n ).

For the second part recall that by Rosenthal's theorem a reflexive subspace of L ₁ is a subspace of L_p for some p > 1, that such L_p is a subspace of L ₁ [40], and that L_p for p > 1 has an unconditional basis (cf., e.g., [15]).

As a corollary of the above we get that a reflexive complemented subspace of H ₁ is isomorphic to ℓ ₂ (cf. [27, Corollary 2.1]). It was also shown in [27] that every Hilbertian subspace of H ₁ contains an infinite-dimensional complemented subspace. To see it take ${\left(f_n\right)}_{n=1}^{\infty }$ , a sequence of functions in $H_1(\mathbb{D})$ equivalent to the unit vector basis in ℓ ₂ and let $V:{\ell }_2\to H_1(\mathbb{D})$ be defined as V(e_n ) = ƒ _n . Think that $H_1(\mathbb{D})=X^{*}$ where X ⊂ K(ℓ ₂) the space of all compact operators on ℓ₂. We have V = U* where U : X → ℓ ₂ is onto. Take x_n ∈ X such that ∥x_n ∥ ⩽ C and U(x_n ) = e_n . Then for some subsequence z_k = x _{n 2k} − x _{n 2k+1} is weakly null. Now we work in K(ℓ ₂) and see that z_k either contains a subsequence z _{k s} equivalent to the unit vector basis in c ₀ (what in our case leads to a contradiction) or to ℓ ₂ This gives that P(f) = ∑ _s 〈f, z_ks 〉 f _{n 2k s} is tne desired projection.

Now let us consider subspaces of $H_1(\mathbb{D})$ which are invariant under rotation. Such a subspace is described by a subset Λ ⊂ ℕ and equals span{zⁿ : n ∈ Λ}. Also, by invariance, if such a subspace is complemented, it is complemented by a multiplier 1 _Λ . We have seen examples of such multipliers in Paley's theorem. Other easily checked examples are arithmetic progressions intersected with ℕ. All the other examples are build from the above ones.

Theorem 38

(Klemes [25]) A subspace span $\left\{z^n:n\in \Lambda \right\}\subset H_1(\mathbb{D})$ is complemented in $H_1(\mathbb{D})$ iff Λ is a finite Boolean combination of lacunary sets, finite sets and arithmetical progressions intersected with ℕ.

The version (much more difficult) of Klemes' theorem for $H_1(ℝ)$ , i.e., a characterisation of translation invariant complemented subspaces of $H_1(ℝ)$ was given by Alspach [1]. No extension of those results to several variables are known.

Clearly the existence of unconditional basis gives many projections. In particular the following result easily follows from the form of Haar basis in H ₁(δ).

Theorem 39

The space H ₁ is isomorphic to ${\left({\displaystyle \sum {}_{n=1}^{\infty }{H}_1}\right)}_1$ its infinite ℓ ₁ sum. Actually the Haar basis in H ₁(δ) and the Haar basis in H ₁[0, 1] are permutatively equivalent and each of them is permutatively equivalent to its infinite ℓ ₁ sum.

Proof

Let us identify the normalised in H ₁(δ) Haar function with its support I and denote it by h ₁. Let $\mathcal{O}=\left\{h_I:I\subset \left[0,1\right]\right\}$ and let ${\mathcal{D}}_n=\{h_I:I\subset [1-2^{-n+1},1-2^{-n}]\}$ with n = 1,2,…Clearly the basis ${\mathcal{D}}_n$ is isometrically equivalent to the Haar basis in H ₁ [0, 1] and ${\displaystyle {\cup }_{n=1}^{\infty }{\mathcal{D}}_n}$ isometrically equivalent to ${\left({\displaystyle {\sum }_{n=1}^{\infty }{\mathcal{D}}_n}\right)}_1$ . One easily observes that $\mathcal{O}\backslash {\displaystyle {\cup }_{n=1}^{\infty }{\mathcal{D}}_n}$ is a basis equivalent to the unit vector basis ℓ ₁. Since each ${\mathcal{D}}_n$ contains a subsequence equivalent to the unit vector basis in ℓ ₁ we infer that $\mathcal{O}$ is permutatively equivalent to ${\left({\displaystyle \sum \mathcal{O}}\right)}_1$ . To treat the case of H ₁(δ) we define ${\mathcal{O}}^{\prime }=\left\{h_I:I\subset ℝ\right\}$ and define ${\mathcal{D}}_n^{\prime }=\{h_I:I\subset [2^{n-1},2^n]{\cup }^{\text{undefined}}[-2^n,-2^{n-1}]\}$ where n = 1, 2,… and ${\mathcal{D}}_0^{\prime }=\{h_I:I\subset [-1,1]\}$ . The argument now is analogous.

The main unsolved problem about infinite-dimensional complemented subspaces of $H_1(\mathbb{D})$ is whether there are infinitely many isomorphic types of them. The easy ones are obtained from the above theorem, Paley's theorem which implies that ℓ ₂ is complemented in $H_1(\mathbb{D})$ and finite-dimensional spaces $H_1^n$ spanned by increasing subsets of unconditional basis. A routine argument yields 10 non-isomorphic types of them as follows ℓ ₁, ℓ ₂, ℓ ₁ ⊕ ℓ ₂, ${\left({\displaystyle \sum {\ell }_n^n}\right)}_1,{\left({\displaystyle \sum {\ell }_2^n}\right)}_1,\oplus {\ell }_2{\left({\displaystyle \sum {\ell }_2}\right)}_1,{\left({\displaystyle \sum H_1^n}\right)}_1,{\left({\displaystyle \sum H_1^n}\right)}_1\oplus {\ell }_2,{\left({\displaystyle \sum H_1^n}\right)}_1\oplus {\left({\displaystyle \sum {\ell }_2}\right)}_1,H_1$ . Two essentially new examples were obtained by a more refined martingale techniques by Müller and Schechtman [37]. One of them, called Y ₁ in [37], can isomorphically be described as $\left\{\left({\alpha }_n\right):{\displaystyle \sum {}_{n=1}^{\infty }\min {\left\{\left|{\alpha }_n\right|,{\left|{\alpha }_n\right|}^{2}n\right\}}^{1/2}<\infty }\right\}$ . This space is not isomorphic to H ₁ but contains subspaces isomorphic to all ℓ_p with 1 ⩽ p ⩽ 2. The other one is a sum of independent copies of $H_1^n$ and can be described as span ${\left\{G_n\right\}}_{n=1}^{\infty }$ where each G_n is an isometric copy of $H_1^n$ but different G_n 's consist of statistically independent functions. To be more explicite let ${\left\{r_n\right\}}_{n=1}^{\infty }$ be the sequence of Rademacher functions and let G_n be the span in H ₁ [0, 1] of all Walsh functions of the form $r_{k_1}^{n_1}·····r_{k_j}^{n_j}$ where n_i = 0, 1 and n ² ⩽ k_i ⩽ (n + 1)². Clearly one can use those two new spaces to form direct sums with old ones to get a more extensive (but still finite) list of all known complemented subspaces of H ₁.

One should also note that the Haar basis in H ₁(δ) gives only three isomorphic types

Theorem 40

([34])Let Λ ⊂ ℤ · ℤ be an infinite subset. Then the subspace span {h_λ : λ ∈ Λ} ⊂ H ₁ (δ) is isomorphic to ℓ ₁ or to ${\left({\displaystyle \sum {}_{n=1}^{\infty }{H}_1^n}\right)}_1$ or to H ₁(δ).

The proof of this theorem is quite complicated and technical so we will only indicate how different possibilities arise. In order to make things more transparent let us consider H ₁[0, 1]. If we take any subset $\mathcal{B}$ of Haar functions whose supports are in [0,1] we define the set $\sigma \left(\mathcal{B}\right)$ as

$\left\{t:t\in \text{supp}{h}_1\text{for infinitely many}{h}_I\in \mathcal{B}\right\}.$

If $\left|\sigma \left(\mathcal{B}\right)\right|>0$ then span $\left\{h_I\in \mathcal{B}\right\}\sim H_1$ . The proof builds a block basic sequence of $\left\{h_I\in \mathcal{B}\right\}$ which is very close in the H ₁ norm and distribution to the original Haar system. This gives that our space contains complemented H ₁ so by decomposition we get the claim.

If $\left|\sigma \left(\mathcal{B}\right)\right|=0$ but sup $\left\{{\left|I\right|}^{-1}{\displaystyle \sum {{}_{hJ\in }}_{\mathcal{B},J\subset I}}\left|J\right|:h_1\in \mathcal{B}\right\}=\infty$ then we have span $\left\{h_I\in \mathcal{B}\right\}\sim {\left({\displaystyle \sum H_1^n}\right)}_1$ . We show that in this case our space contains complemented ${\left({\displaystyle \sum H_1^n}\right)}_1$ and is contained as a complemented subspace in one.

If $\left|\sigma \left(\mathcal{B}\right)\right|=0$ but sup $\left\{{\left|I\right|}^{-1}{\displaystyle \sum {{}_{hJ\in }}_{\mathcal{B},J\subset I}}\left|J\right|:h_1\in \mathcal{B}\right\}<\infty$ then a direct calculation shows that the basis $\mathcal{B}$ is equivalent to the unit vector basis in ℓ ₁.

The condition distinguishing cases in the situation when $\left|\sigma \left(\mathcal{B}\right)\right|=0$ is a martingale version of the Carleson condition (cf. [26]). Note also that there is a close similarity between conditions and conclusion of this theorem and Theorem 33. This is not accidental and actually the methods of proof of Theorem 33 are an outgrowth and elaboration of the methods used to prove Theorem 40.

There are also some general results about complemented subspaces of H ₁. Let us formulate some of them as one theorem.

Theorem 41

(a)

([34]) Let X ⊂ H ₁ be isomorphic to H ₁ . Then there exists Y ⊂ X complemented in H ₁ and isomorphic to H ₁.

(b)

([34]) The space H ₁ is primary, i.e., whenever H ₁ = X ⊕ Y then either X or Y is isomorphic to H ₁.

(c)

([37]) A complemented subspace X of H ₁ either contains ℓ ₂ or is isomorphic to a complemented subspace of ${\left({\displaystyle \sum {}_{n=1}^{\infty }{H}_1^n}\right)}_1$ .

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Functional Analysis and its Applications

Anatolij Plichko , in North-Holland Mathematics Studies, 2004

Definition 5. [15]

An operator T : X → Y is called strictly cosingular if for every closed subspace E ⊂ Y of infinite codimension, the map QT (where Q : Y → Y/E is a quotient map) has non-closed range.

It is easy to see ([15]) that if T* is strictly singular, then T is strictly cosingular and if T* is strictly cosingular then T is strictly singular.

The following quantitative characteristic of the operator T was introduced in [14]:

$a_n\left(T\right)=\underset{E^n}{\sup }\inf \left\{{\Vert \widehat{Tx}\Vert }_{Y/E^n}:x\in X,{\Vert \widehat{x}\Vert }_{X/T^{-1}{E}^n}=1\right\},$

where supremum is taken over all closed subspaces Eⁿ ⊂ Y of codimension n and caps denote the corresponding quotient classes.

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Advanced Theory

In Pure and Applied Mathematics, 1986

10.1.15 Theorem.

Suppose that $\mathfrak{A}$ is aC ^*-algebra acting on a Hilbert space $\mathcal{H}$ .

(i)

The ultraweakly continuous linear functionals on $\mathfrak{A}$ form a norm-closed subspace $\mathfrak{A}$ ^u of the Banach dual space $\mathfrak{A}$ ^#.

(ii)

The singular elements of $\mathfrak{A}$ ^#, together with 0, form a norm-closed subspace $\mathfrak{A}$ ^s of $\mathfrak{A}$ ^#.

(iii)

Each ρ in $\mathfrak{A}$ ^# can be expressed, uniquely, in the form ρ = ρ_u + ρ_#, with ρ_u in $\mathfrak{A}$ ^u and ρ_# in $\mathfrak{A}$ ^s. Moreover, ||ρ|| = ||ρ_u|| + ||ρ_s||; and if ρ is positive, or hermitian, the same is true of ρ_u and ρ_s.

(iv)

If φ is the universal representation of $\mathfrak{A}$ , and S ∈ φ( $\mathfrak{A}$ )⁻, the mappings ρ → S ρ, ρ → ρ S: $\mathfrak{A}$ ^#→ $\mathfrak{A}$ ^# leave $\mathfrak{A}$ ^u and $\mathfrak{A}$ ^s invariant.

Proof. We continue to use the notation established in Proposition 10.1.14 and the discussion that precedes it. We have already noted that $\mathfrak{A}$ ^u and $\mathfrak{A}$ ^s are complementary closed subspaces of $\mathfrak{A}$ ^#, and that the corresponding projection from $\mathfrak{A}$ ^# onto $\mathfrak{A}$ ^u is the mapping ρ → P ρ. This proves (i), (ii), and the first assertion in (iii), with

${\rho }_u=P\rho ,{\rho }_s=\left(I-P\right)\rho .$

Since P and I − P are positive elements of the center of φ( $\mathfrak{A}$ )⁻, ρ_u and ρ_s are positive, or hermitian, when ρ has this property. Given A ₁ and A ₂ in the unit ball ( $\mathfrak{A}$ )₁, φ(A ₁)P + φ(A ₂)(I— P) lies in (φ( $\mathfrak{A}$ )⁻)₁, so

$\begin{array}{c}\left|\right|\rho \left|\right|=\left|\right|\tilde{\rho }\left|\right|\ge \left|\tilde{\rho }\left(\phi \left(A_1\right)P+\phi \left(A_2\right)\left(I-P\right)\right)\right|\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=|\left(P\rho \right)\left(A_1\right)+\left(\rho -P\rho \right)\left(A_2\right)|=|{\rho }_u\left(A_1\right)+{\rho }_s\left(A_2\right)|.\end{array}$

By taking the upper bound of the right-hand side, as A ₁ and A ₂ vary in ( $\mathfrak{A}$ )₁, we obtain

$\left|\right|\rho \left|\right|\ge \left|\right|{\rho }_u\left|\right|+\left|\right|{\rho }_s\left|\right|.$

Since the reverse inequality is apparent, this completes the proof of (iii). With ρ in $\mathfrak{A}$ ^# and S in φ( $\mathfrak{A}$ )⁻,

$P\left(S\rho \right)=\left(PS\right)\rho =\left(SP\right)\rho =S\left(P\rho \right),P\left(\rho S\right)=\left(P\rho \right)S.$

Accordingly, the mappings ρ → S ρ, ρ → ρ S: $\mathfrak{A}$ ^#→ $\mathfrak{A}$ ^# both commute with the projection ρ → p ρ from $\mathfrak{A}$ ^# onto $\mathfrak{A}$ ^u, and hence leave invariant both its range space $\mathfrak{A}$ ^u and its null space $\mathfrak{A}$ ^s.

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Introduction to Fibrewise Homotopy Theory

I.M. James , in Handbook of Algebraic Topology, 1995

PROPOSITION 5.1

Let X be a fibrewise pointed space over B and let A be a closed subspace of X. The inclusion A → X is a fibrewise cofibration if and only if ({0} × X) ∪ (I × A) is a fibrewise retract of I × X.

This characterization enables us to see that the associated bundle functor P _# transforms cofibrations in the equivariant sense into cofibrations in the fibrewise sense. Specifically, let P be a principal G-bundle over B. Let X be a pointed G-space and let A be a closed invariant subspace of X. Suppose that the inclusion A → X is a G-fibration. Then the inclusion P _# A → P _# X is a fibrewise cofibration.

Unsurprisingly there is a fibrewise version of the well-known Puppe sequence. This concerns sequences

$X_1\stackrel{f_1}{\to }{X}_2\stackrel{f_2}{\to }{X}_3\to \cdots$

of fibrewise pointed spaces and fibrewise pointed maps, over the given base space B. We describe such a sequence as exact, in this context, if the induced sequence

${\text{π}}_B^B\left(X_1,E\right)\stackrel{f_1^{*}}{\leftarrow }{\text{π}}_B^B\left(X_2,E\right)\stackrel{f_2^{*}}{\leftarrow }{\text{π}}_B^B\left(X_3,E\right)\leftarrow \cdots$

is exact for all fibrewise pointed spaces E.

Given a fibrewise pointed map ϕ : X → Y, where X and Y are fibrewise pointed spaces over B, the fibrewise mapping-cone ${\Gamma }_B^B(\varphi )$ of ϕ is defined to be the push-out of the cotriad

${\Gamma }_B^B\left(X\right)\supset X\stackrel{\varphi }{\to }Y.$

Now ${\Gamma }_B^B(\varphi )$ comes equipped with a fibrewise embedding

$\varphi \prime :Y\to {\Gamma }_B^B(\varphi ),$

and we easily see that the sequence

${\text{π}}_B^B(X,E)\stackrel{{\varphi }^{*}}{\leftarrow }{\text{π}}_B^B(Y,E)\stackrel{{\varphi }^{\prime *}}{\leftarrow }{\text{π}}_B^B({\Gamma }_B^B(\varphi ),E)$

is exact, for all fibrewise pointed spaces E. Obviously the procedure can be iterated so as to obtain exact sequences of unlimited length, but that in itself is not of great interest.

To understand the situation better consider the case of a fibrewise cofibration u : A → X, where A and X are fibrewise pointed spaces over B. Then the natural projection

$\rho :{\Gamma }_B^B(u)\to {\Gamma }_B^B(u){/}_B{\Gamma }_B^B(A)=X{/}_{B}A$

is a fibrewise pointed homotopy equivalence. Moreover if u′ is derived from u in the way that ϕ′ above is derived from ϕ then ρ ○ u′ : X → X/ _B A is just the fibrewise collapse.

Returning to the general case, where ϕ : X → Y, we observe that the embedding ${\varphi }^{\prime }:Y\to {\Gamma }_B^B(\varphi )$ is a fibrewise cofibration. In fact the embedding $X\to {\Gamma }_B^B\left(X\right)$ is a fibrewise cofibration, from first principles, and so the conclusion follows from the observation that the push-out of a fibrewise cofibration is again a fibrewise cofibration.

By combining these last two results we see that the fibrewise mapping-cone ${\Gamma }_B^B\left(\varphi \prime \right)$ is equivalent to the fibrewise suspension

${\sum }_B^B\left(X\right)={\Gamma }_B^B(\varphi ){/}_{B}Y={\Gamma }_B^B\left(\varphi \prime \right){/}_B{\Gamma }_B^B(\varphi ),$

up to fibrewise pointed homotopy equivalence. In the process, moreover, (ϕ′)′ is transformed into the fibrewise pointed map

$\varphi ″:{\sum }_B^B(\varphi )\to {\sum }_B^B\left(X\right).$

Repeating the procedure we find that Γ _B (ϕ′)′ is equivalent to the fibrewise suspension ${\displaystyle \sum {}_B^B\left(Y\right)}$ , in the same sense. In the process, moreover, ((ϕ′)′)′ is transformed into the fibrewise suspension

${\sum }_B^B(\varphi ):{\sum }_B^B\left(X\right)\to {\sum }_B^B\left(Y\right)$

of ϕ, precomposed with the fibrewise reflection in which (t, x) ↦ (1 – t, x). This last does not affect the exactness property and so we arrive at an exact sequence of the form

$X\stackrel{\varphi }{\to }Y\to {\Gamma }_B^B(\varphi )\to {\sum }_B^B\left(X\right)\stackrel{{\sum }_B^B(\varphi )}{\to }{\sum }_B^B\left(Y\right)\to \cdots$

When the given fibrewise pointed map ϕ is varied by a fibrewise pointed homotopy the exact sequence varies similarly. In particular, if ϕ is fibrewise pointed nulhomotopic the sequence has the same fibrewise pointed homotopy type (in an obvious sense) as in the case of the fibrewise constant, where the sequence reduces to the form

$X\to Y\to Y{\vee }_B{\sum }_B^B\left(X\right)\to {\sum }_B^B\left(X\right)\to \cdots$

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SOME UNSOLVED PROBLEMS IN QUANTUM LOGICS

S. Gudder , in Mathematical Foundations of Quantum Theory, 1978

Example 2

(Hilbert Logic). Let L(H) be the lattice of all closed subspaces of a complex Hilbert space H (dim H ≥ 3) with the usual order and orthocomplementation. If M is the set of states on L(H), we call (L(H), M) a Hilbert logic. Observables are represented by self-adjoint operators and pure states have the form m(a) = 〈φ, P_a φ〉 where φ is a unit vector and P_a is the orthogonal projection onto a ∈ L(H). If A and B are bounded self-adjoint operators such that 〈φ, Aφ〉 = 〈φ,Bφ〉 for every unit vector φ ∈ H, then it is well known that A = B. Hence, uniqueness holds on Hilbert logics.

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Wavelets, Advanced

Su-Yun Huang , Zhidong Bai , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III Multiresolution Analysis

Wavelets can be used as an analysis tool to describe mathematically the increment in information needed to go from a coarser approximation to a higher resolution approximation. This insight was put into the framework of multiresolution analysis by Mallat (1989).

A multiresolution analysis of L ₂(R ) consists of a nested sequence of closed subspaces $\cdots {\text{V}}_{\text{j}-1}\subset {\text{V}}_{\text{j}}\subset {\text{V}}_{\text{j}+1}\cdots$ for approximating L ₂(R) functions. The multiresolution analysis satisfies the following conditions:

1.

${\cup }_{\text{j}=-\infty }^{\infty }{\text{V}}_{\text{j}}$ is dense in L ₂(R).

2.

${\cap }_{\text{j}=-\infty }^{\infty }{\text{V}}_{\text{j}}=\{0\}$ .

3.

f(x)   ∈ V _j if and only if f(2x)   ∈ V _{j  +   1}, ∀ _j   ∈ Z.

4.

There exists a function f(x) such that {f(x  − k): k  ∈ Z forms an unconditional basis (also known as Riesz basis) for V ₀.

Notice that {f(2 ^j x  − k): k  ∈ Z} is an unconditional basis for V _j .

Given this translation-invariant unconditional basis, we are seeking whether a translation-invariant orthonormal basis exists. The answer is affirmative. There exists a function ϕ(x) such that ${\int }_{-\infty }^{\infty }\phi (\text{x})\text{d}\text{x}=1$ and that ${\{\phi (\text{x}-\text{k})\}}_{\text{k}\in \text{Z}}$ forms an orthonormal basis for V ₀. The function ϕ(x) is called a wavelet scaling function or the father wavelet.

The following is a brief description of construction of ϕ(x) from the unconditional basis {f(x  − k)} _{k  ∈ Z} based on the Fourier method.

The family of integral translates of one single function ${\phi }^{(\text{o}.\text{n}.)}(\text{x})\in {\text{L}}_1(\text{R})\cap {\text{L}}_2(\text{R})$ is an orthonormal set if and only if

(3) ${\displaystyle \sum _{\text{k}\in \text{Z}}}|{\hat{\phi }}^{(\text{o}.\text{n}.)}(\omega +2\pi \text{k}){|}^2=\frac{1}{2\pi }$

for all ω ε R. Given the unconditional basis {f(x  − k): k  ∈ Z}, let

$\text{M}(\omega )={\left(2\pi {\displaystyle \sum _{\text{k}=-\infty }^{\infty }}|\hat{\text{f}}(\omega +2\text{k}\pi ){|}^2\right)}^{-1/2}$

and let ϕ(x) be given by

$\hat{\phi }(\omega )=\text{M}(\omega )\hat{\text{f}}(\omega )$

It can be easily checked that ϕ(x) satisfies (3). Thus {ϕ(x  − k)} _{k  ∈ Z} forms an orthonormal basis for V ₀. Notice that $\{{\phi }_{\text{j},\text{k}}(\text{x})=\sqrt{2^{\text{j}}}\phi (2^{\text{j}}\text{x}-\text{k}):\text{k}\in \text{Z}\}$ is then an orthonormal basis for V _j . Since ${\text{V}}_0\subset {\text{V}}_1$ , the scaling function ϕ(x) can be represented as a linear combination of functions ϕ_1,k (x):

(4) $\phi (\text{x})={\displaystyle \sum _{\text{k}\in \text{Z}}}{\text{h}}_{\text{k}}{\phi }_{1,\text{k}}(\text{x})$

for some coefficients h _k , k  ∈ Z. The preceding equation is called the two-scale equation or the scaling equation. The sequence of coefficients h _k :k  ∈ Z has the properties

$\begin{array}{lll}\hfill \sum _{\text{K}\in \text{Z}}\text{h}\text{k}=\sqrt{2}& \hfill \text{and}\hfill & \sum _{\text{k}\in \text{Z}}{\text{h}}_{\text{k}}^2=1\hfill \end{array}$

The two-scale equation (4) can be rewritten in terms of Fourier transform as

(5) $\hat{\phi }(\omega )={\displaystyle \sum _{\text{k}\in \text{Z}}}{\text{h}}_{\text{k}}{\hat{\phi }}_{1,\text{k}}(\omega )={\text{m}}_0(\omega /2)\hat{\phi }(\omega /2)$

where

${\text{m}}_0(\omega )=\frac{1}{\sqrt{2}}{\displaystyle \sum _{\text{k}\in \text{Z}}}{\text{h}}_{\text{k}}{\text{e}}^{-\text{i}\text{k}\omega }$

Orthogonality of these ϕ(x  − k) implies that

$|{\text{m}}_0(\omega ){|}^2+|{\text{m}}_0|(\omega +\pi ){|}^2=1$

By iterating (5), we have

$\hat{\phi }(\omega )=\hat{\phi }(0){\displaystyle \prod _{\text{j}=1}^{\infty }}{\text{m}}_0\left(\frac{\omega }{2^{\text{j}}}\right)=\frac{1}{\sqrt{2\pi }}{\displaystyle \prod _{\text{j}=1}^{\infty }}{\text{m}}_0\left(\frac{\omega }{2^{\text{j}}}\right)$

Define the orthogonal complementary subspaces W _j   = V _{j  +   1}  − V _j so that V _{j  +   1}  = V _j ⊕ W _j , where ⊕ is an orthogonal direct sum. Then the space L ₂(R) can be decomposed as

${\text{L}}_2(\text{R})={\displaystyle \underset{\text{j}\in \text{Z}}{\oplus }}{\text{W}}_{\text{j}}$

For an arbitrary multiresolution analysis of L ₂(R), there exists a function ψ(x) such that $\{{\psi }_{\text{j},\text{k}}(\text{x})=\sqrt{2^{\text{j}}}\psi (2^{\text{j}}\text{x}-\text{k}):\text{k}\in \text{Z})$ forms an orthonormal basis of the difference space W _j . Thus, {ψ _j,k (x):j, k  ∈ Z} forms an orthonormal basis for L ₂(R). The function ψ(x) is called a wavelet function or the mother wavelet. Since $\psi (\text{x})\in {\text{W}}_0\subset {\text{V}}_1$ , it can be represented as

(6) $\psi (\text{x})={\displaystyle \sum _{\text{k}\in \text{Z}}}{\text{g}}_{\text{k}}{\phi }_{1,\text{k}}(\text{x})$

for some coefficients g _k , k  ∈ Z. One possibility for the choice of these coefficients may be set to relate to h _k by

${\text{g}}_{\text{k}}={(-1)}^{\text{k}}{\overline{\text{h}}}_{1-\text{k}}$

That is, essentially only one two-scale sequence h governs the multiresolution analysis and its wavelet decomposition. The two-scale equation (6) can be rewritten in terms of Fourier transform as

(7) $\hat{\psi }(\omega )={\text{m}}_1(\omega /2)\hat{\phi }(\omega /2)$

where

${\text{m}}_1(\omega )=\frac{1}{\sqrt{2}}{\displaystyle \sum _{\text{k}\in \text{Z}}}{\text{g}}_{\text{k}}{\text{e}}^{-\text{i}\text{k}\omega }=-{\text{e}}^{-\text{i}\omega }\overline{{\text{m}}_0(\omega +\pi )}$

The sequence of coefficients {g _k : k  ∈ Z} has the properties

$\begin{array}{lll}\hfill {\displaystyle \sum _{\text{k}\in \text{Z}}}{\text{g}}_{\text{k}}=0& \hfill \text{and}\hfill & {\displaystyle \sum _{\text{k}\in \text{Z}}}{\text{g}}_{\text{k}}^2=1\hfill \end{array}$

A multiresolution analysis {V _j : j  ∈ Z} of L ₂(R) is called r-regular, if the scaling function ϕ(x) (or the wavelet ψ(x)) can be so chosen to satisfy the condition

$|{\phi }^{(\alpha )}(\text{x})|\le {\text{C}}_{\text{m}}{(1+|\text{x}|)}^{-\text{m}}$

for each integer m  ∈ N and for every α   =   0,1, …, r.

For the rest of this section, the multiresolution analysis {V _j : j  ∈ Z} is assumed r-regular, unless otherwise specified.

Let

$\begin{array}{ll}\mathcal{V}(\text{x},\text{y})\hfill & ={\displaystyle \sum _{\text{k}\in \text{Z}}}\phi (\text{x}-\text{k})\overline{\phi (\text{y}-\text{k})}\hfill \\ {\mathcal{V}}_{\text{j}}(\text{x},\text{y})\hfill & ={\displaystyle \sum _{\text{k}\in \text{Z}}}{2}^{\text{j}}\phi (2^{\text{j}}\text{x}-\text{k})\overline{\phi (2^{\text{j}}\text{y}-\text{k})}\hfill \\ \mathcal{W}(\text{x},\text{y})\hfill & ={\displaystyle \sum _{\text{k}\in \text{Z}}}\psi (\text{x}-\text{k})\overline{\psi (\text{y}-\text{k})}\hfill \\ {\mathcal{W}}_{\text{j}}(\text{x},\text{y})\hfill & ={\displaystyle \sum _{\text{k}\in \text{Z}}}{2}^{\text{j}}\psi (2^{\text{j}}\text{x}-\text{k})\overline{\psi (2^{\text{j}}\text{y}-\text{k})}\hfill \end{array}$

By the r-regularity property, we immediately deduce that

$|{\partial }_{\text{x}}^{\alpha }{\partial }_{\text{y}}^{\beta }\mathcal{V}(\text{x},\text{y})|\le {\text{C}}_{\text{m}}{(1+|\text{x}-\text{y}|)}^{-\text{m}}$

for each integer m  ∈ N and for every α   =   0,1, …, r, and β   =   0,1, …, r.

The wavelet subspaces V _j and W _j are reproducing kernel Hilbert spaces with the reproducing kernels $\mathcal{V}$ _j (x, y) and $\mathcal{W}$ _j (x, y), respectively. The orthogonal projection of f  ∈ L ₂(R) onto the subspaces V _j and W _j can be written respectively as

$\mathcal{P}{\text{v}}_{\text{j}}\text{f}(\text{x})={\int }_{-\infty }^{\infty }{\mathcal{V}}_{\text{j}}(\text{x},\text{y})\text{f}(\text{y})\text{<mi mathvariant="italic" is="true">dy</mi>}$

and

$\mathcal{P}{\text{w}}_{\text{j}}\text{f}(\text{x})={\int }_{-\infty }^{\infty }{\mathcal{W}}_{\text{j}}(\text{x},\text{y})\text{f}(\text{y})\text{d}\text{y}$

For an r-regular multiresolution analysis, we have

${\int }_{-\infty }^{\infty }{\mathcal{V}}_{\text{j}}(\text{x},\text{y}){\text{y}}^{\text{k}}\text{d}\text{y}={\text{x}}^{\text{k}}$

and

${\int }_{-\infty }^{\infty }{\mathcal{W}}_{\text{j}}(\text{x},\text{y}){\text{y}}^{\text{k}}\text{d}\text{y}=0$

for every j  ∈ Z and k  =   0,1, …, r.

A kernel $\mathcal{V}$ (x, y) is said to be of order m if and only if it satisfies the moment conditions

${\int }_{-\infty }^{\infty }\mathcal{V}(\text{x},\text{y}){\text{y}}^{\text{k}}\text{d}\text{y}=\{\begin{array}{ll}1,\hfill & \text{k}=0\hfill \\ {\text{x}}^{\text{k}},\hfill & \text{k}=1,\dots ,\text{m}-1\hfill \\ \alpha (\text{x})\ne {\text{x}}^{\text{m}},\hfill & \text{k}=\text{m}\hfill \end{array}$

A projection kerne $\mathcal{V}$ (x, y) arising from a multiresolution analysis is of order m if and only if the associated wavelet ψ(x) has vanishing moments of order m. Usually the order of vanishing moments is much higher than the degree of regularity.

For a multiresolution analysis, one may be interested in knowing the wellness of the projection approximation ${\mathcal{P}}_{{\text{V}}_{\text{j}}}\text{f}$ to f. The answer depends on the regularity of f and the regularity of functions in V _j . The higher these regularities are, the better the projection approximation is.

Theorem 1

Suppose that the multiresolution analysis has regularity r. Then f (x) is in the Sobolev space W ₂ ^s (R), 0   ≤ s  ≤ r, if and only if

$\underset{\text{j}\to \infty }{limsup}{2}^{\text{j}\text{s}}{||\text{f}-\mathcal{P}{\text{v}}_{\text{j}}\text{f}||}_2<\infty$

This theorem reveals the approximation order of the projection ${\mathcal{P}}_{{\text{V}}_{\text{j}}}$ in L ₂-norm. It also characterizes the function space of W ^s ₂(R).

To look further into the pointwise behavior of the projection ${\mathcal{P}}_{{\text{v}}_{\text{j}}}$ , assume that f(x) belongs to C ^m,α (R) ∩ L ₂(R) for some 0   <   α   ≤   1, where

$\begin{array}{ll}{\text{C}}^{\text{m},\alpha }(\text{R})=\hfill & \{\text{f}\in {\text{C}}^{\text{m}}(\text{R}):|{\text{f}}^{(\text{m})}(\text{x})-{\text{f}}^{(\text{m})}(\text{y})|\hfill \\ \hfill & \le \text{A}|\text{x}-\text{y}{|}^{\alpha },\text{A}>0\}.\hfill \end{array}$

Suppose that $\mathcal{V}$ (x, y) is of order m. Define

${\text{b}}_{\text{m}}(\text{x})={\text{x}}^{\text{m}}-{\int }_{-\infty }^{\infty }\mathcal{V}(\text{x},\text{y}){\text{y}}^{\text{m}}\text{d}\text{y}$

Notice that the function b _m (x) is a continuous periodic function with period one. Assume that the kernel $\mathcal{V}$ (x, y), or ϕ(x) or ψ(x), is sufficiently localized so that

${\int }_{-\infty }^{\infty }|\mathcal{V}(\text{x},\text{y})(\text{y}-\text{x}{)}^{\text{m}+\alpha }|\text{d}\text{y}<\infty$

Then

(8) $\text{f}(\text{x})-\mathcal{P}{\text{v}}_{\text{j}}\text{f}(\text{x})=\frac{1}{2^{\text{j}\text{m}}\text{m}!}{\text{f}}^{(\text{m})}(\text{x}){\text{b}}_{\text{m}}(2^{\text{j}}\text{x})+\text{O}(2^{-\text{j}(\text{m}+\alpha )})$

A multiresolution analysis {V _j : j  ∈ Z} is said to be symmetric if the projection kernel satisfies the condition $\mathcal{V}$ (−x, y)   = $\mathcal{V}$ (x,−y). Notice that $\mathcal{V}$ (−x, y) is a time-reversed kernel, and ${\int }_{-\infty }^{\infty }\mathcal{V}(\text{x},-\text{y})\text{f}(\text{y})\text{d}\text{y}$ corresponds to the transform of time-reversed signal f. If the multiresolution analysis is symmetric, then b _m (−x)   =   (−1) ^m b _m (x). When m is even, b _m (x) is symmetric about zero. Since b _m (x) is periodic with period one, b _m (x) is also symmetric about all the points x  = k/2, k  ∈ Z. When m is odd b _m (x) is antisymmetric about zero and hence antisymmetric about all the points x  = k/2, k  ∈ Z.

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