Distance Function X X X on R Metric Space is Continuous

Basic Representation Theory of Groups and Algebras

In Pure and Applied Mathematics, 1988

Proof

Let B denote the Banach space of continuous functions from S into X ₀ under the natural linear operations and norm ||f|| = sup{||f(s)||:a ∈A}; and consider the Banach representation T of A on B given by (T_af)(s) = af(s)for a ∈ A,s ∈ S, f ∈ B. If {e_α } is a left approximate unit in A, then e_αx → x for each x ∈ X ₀, and since {e_α } is bounded, e_αx → x uniformly on compact subsets of X ₀. If f ∈ B, then f(S) is compact; so e_αf(s) → f(s) uniformly on S. Hence, B ₀ = B. Let i ∈ B be the identity, i.e., i(s) = s for all s∈ S. Then by 9.2 there exists a∈ A and g∈ B such that i = ag and, in particular, S = ag(S). Setting T = g(S), the corollary follows.

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Riesz Spaces and Ideals of Measurable Functions

Martin Väth , in Handbook of Measure Theory, 2002

3.4 Bibliographical remarks

Spaces of measurable functions are together with spaces of continuous functions the most natural examples of Riesz spaces. Using the axiom of choice, standard results state that any Riesz space X satisfying certain natural assumptions is Riesz isomorphic to (a subspace of) C(K) where K is the set of all maximal (order) ideals, equipped with a certain topology (analogously to the Gel'fand representation theorem for Banach algebras). Similar results establish a Riesz isomorphism to $L_{\infty }\left(S\right)$ For such representation theorems, we refer to Luxemburg and Zaanen (1971), Semadeni (1971), Schaefer (1974), and Zaanen (1983),

In particular, one may also establish Riesz isomorphisms of ideals of $\mathfrak{M}\left(S\right)$ to C(K). However, these homomorphisms do not immediately imply, e.g., Theorem 3. Moreover, ideals in $\mathfrak{M}\left(S\right)$ are among the best understood classes of Riesz spaces, and so it makes not much sense to represent them in a less known form.

The precise connection of Example 3 with the duality map is revealed in Theorem 2.1.1 of Väth (1997). Lemma 1 is from Din and Väth (2000).

Proposition 14 is apparently new, although it was known already in the special case of preideal spaces (Lemma 2.2.1 in Väth (1997)). The notion of entire topologies is related with the so-called support of the topology, see, e.g., Feledziak and Nowak (1997). For the latter, see also Aliprantis and Burkinshaw (1977, 1980a, 1980b). In these references, the reader will also find other conditions than metrizability which ensure that a Hausdorff topology is entire (on σ-finite measure spaces, the Fatou property is such a condition, for example, in view of the remarks following Definition 16 and by the results of Aliprantis and Burkinshaw (1977)). Theorem 10 was proved for scalar functions by Wnuk (1986); the case of vector functions follows from this straightforwardly in view of Proposition 12 (see Feledziak and Nowak (1997) for details). In the context of preideal spaces, the important Corollary 6 was obtained first independently by Zabreiíko (1974) and by Luxemburg and Zaanen (1963d). (The arguments of Luxemburg and Zaanen are completely different.)

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Handbook of the Geometry of Banach Spaces

Robert Deville , Nassif Ghoussoub , in Handbook of the Geometry of Banach Spaces, 2001

2.2 First examples of admissible classes of perturbations

One can easily recover Ekeland's result as well as the theorem of Borwein and Preiss [8] from Theorem 2.1. For that, we may consider the cones $\mathcal{A}$ ₁ (respectively $\mathcal{A}$ ₂) consisting of those functions f on (X, d) of the form

$f\left(x\right)=-{\displaystyle \sum _n{\lambda }_{n}d\left(x,x_n\right)}\left(\text{respectively}f\left(x\right)=-{\displaystyle \sum _n{\lambda }_n{d}^2\left(x,x_n\right)}\right)$

with λ _n , ≥ 0. We assume here, without loss of generality, that d is a bounded metric on X. Let us check that they are admissible cones.

Indeed, suppose F is a closed subset of $\tilde{X}=X\times \text{R}$ such that $\tilde{h}=\left(h,-1\right)\in {\tilde{\mathcal{A}}}_1$ is bounded above on F. For small τ > 0, consider a point (x ₀, s ₀) ∈ F such that $h\left(x_0\right)-s_0>{\sup }_F\tilde{h}-{\tau }^2$ . Let $\tilde{k}$ be the functional in ${\tilde{\mathcal{A}}}_1$ defined by $\tilde{k}=\left(h-\tau d\left(,x_0\right),-1\right)$ and let S be the slice of F given by

$S=\left\{\left(y,s\right)\in F;\tilde{k}\left(y,s\right)>\tilde{k}\left(x_0,s_0\right)-{\tau }^2\right\}.$

It is easy to see that the d-diameter of S is less than τ, which means that $\tilde{X}$ is ${\tilde{\mathcal{A}}}_1$ -uniformly dentable. A similar proof works for $\mathcal{A}$ ₂.

Note that Ekeland's result would then follow from Theorem 2.2 and the triangular inequality. Note also that we could have used the space $\mathcal{A}$ = Lip(X) of Lipschitz functions on X as an admissible space.

We shall now investigate the possibility of having other classes of functions as perturbation spaces. For simplicity, we shall only deal, in the sequel, with the case where X is a Banach space.

A function b : X → R is said to be a bump function on X if it has a bounded and nonempty support.

Proposition 2.1

Let $\mathcal{A}$ be a Banach space of continuous functions on a Banach space X satisfying the following properties:

(i)

For each $g\in A,{||g||}_{\mathcal{A}}\ge {||g||}_{\infty }=\sup \left\{\left|g\left(x\right)\right|;x\in X\right\}$ .

(ii)

$\mathcal{A}$ is translation invariant, i.e., if g ∈ $\mathcal{A}$ and x ∈ X, then τ_xg: X → R given by ${\tau }_{x}g\left(t\right)=g\left(x+t\right)$ is in $\mathcal{A}$ and ${||{\tau }_{x}g||}_{\mathcal{A}}={||g||}_{\mathcal{A}}$ .

(iii)

$\mathcal{A}$ is dilation invariant, i.e., if g ∈ $\mathcal{A}$ and α ∈ R then g^α : X → R given by g^α (t) = g(αt) is in $\mathcal{A}$ .

(iv)

There exists a bump function b in $\mathcal{A}$ .

Then $\mathcal{A}$ is an admissible space of perturbations for the space X.

Proof. According to (ii) and (iv), we can find a bump function b in $\mathcal{A}$ such that b(0) ≠ 0. Using (iii) and replacing b(x) by α₁ b(α₂ x) with suitable coefficients α₁, α₂ ∈ R, we can assume that $b\left(0\right)>0,{||b||}_A<\epsilon$ and b(x) = 0 whenever $||x||\ge \epsilon$ .

Let now $\tilde{g}=\left(g,-1\right)\in \tilde{\mathcal{A}}$ be bounded above on a closed subset F of $\tilde{X}$ and let (x ₀, s ₀) ∈ F be such that $g\left(x_0\right)-s_0>{\sup }_F\tilde{g}-b\left(0\right)$ and consider the function h(x) = b(x – x ₀) and $\tilde{k}=\left(g+h,-1\right)$ .

By (ii), h ∈ $\mathcal{A}$ and ${||h||}_{\mathcal{A}}={||b||}_{\mathcal{A}}<\epsilon$ , which implies that ${||\tilde{g}-\tilde{k}||}_{\mathcal{A}}<\epsilon$ On the other hand, consider the following slice of F,

$S=\left\{\left(x,s\right)\in F;\tilde{k}\left(x,s\right)>\underset{F}{\sup }\tilde{g}\right\}.$

It is non-empty since it contains (x ₀, s ₀). On the other hand, if (x, s) ∈ F and $||x-x_0||\ge \epsilon$ , then b(x – x ₀) = 0 and (x, s) cannot belong to S. It follows that the d-diameter of S is less than 2ε. Consequently, $\mathcal{A}$ is an admissible family of perturbations.

Corollary 2.1

(Localization). Assume $\mathcal{A}$ is a Banach space of bounded continuous functions on X satisfying conditions (i)–(iv) above. Then, for some constant a > 0, depending only on X and $\mathcal{A}$ , the following holds:

If $\varphi :X\to \text{R}\cup \left\{+\infty \right\}$ is lower semi-continuous and bounded below with $D\left(\varphi \right)\ne ø$ and if y ₀ ∈ X satisfies $\varphi \left(y_0\right)<{\inf }_X\varphi +a{\epsilon }^2$ for some ε ∈ (0, 1), then there exist g ∈ $\mathcal{A}$ and x ₀ ∈ X such that

(i)

$||x_0-y_0||\le \epsilon$ ,

(ii)

${||g||}_A\le \epsilon ,$

(iii)

φ + g attains its minimum at x ₀.

Proof. We can clearly assume that there exists a bump function b in $\mathcal{A}$ with b(0) = 1, 0 ≤ b ≤ 1 and such that the support of b is contained in the unit ball of X. Hypothesis (i) implies that $M≔{||b||}_A\ge {||b||}_{\infty }=1$ . Let a = 1/4M and suppose that ε and y ₀ are given. Define the function

$\begin{array}{l}\tilde{\varphi }\left(x\right)=\varphi \left(x\right)-2a{\epsilon }^{2}b\left(\frac{x-y_0}{\epsilon }\right).\end{array}$

We have $\tilde{\varphi }\left(y_0\right)<{\inf }_X\varphi -a{\epsilon }^2$ and $\tilde{\varphi }\left(y\right)\ge {\inf }_X\varphi$ whenever $||y-y_0||\ge \epsilon .$

From Proposition 2.1 and Theorem 2.1, we can find x ₀ ∈ X and k ∈ $\mathcal{A}$ such that ${||k||}_A\le \min \left\{\epsilon /2,a{\epsilon }^2/2\right\}$ and $\tilde{\varphi }+k$ attains its minimum at x ₀. The above conditions imply that $||x_0-y_0||<\epsilon$ Thus, the function g ∈ $\mathcal{A}$ defined by $g\left(x\right)=-2a{\epsilon }^{2}b\left(\frac{x-x_0}{\epsilon }\right)+k\left(x\right)$ satisfies claims (i), (ii) and (iii) of the corollary.

Remark 2.1.

Again, we can recover Ekeland's minimization principle on Banach spaces from the (easily verifiable) fact that the space $\mathcal{A}$ ₁ of all bounded Lipschitz functions on X equipped with the norm

${||f||}_{A_1}=\sup \left\{\left|f\left(x\right)\right|;x\in X\right\}+\sup \left\{\frac{\left|f\left(x\right)-f\left(y\right)\right|}{||x-y||};x\ne y\right\}$

satisfies the conditions of Proposition 2.1 and hence it is an admissible space of perturbations. Note that, as an additional bonus, we get that the perturbation is also small in the uniform norm as well as in the Lipschitz norm.

To derive an analogue of the Borwein–Preiss Theorem, we can consider the space $\mathcal{A}$ ₂ of all bounded Lipschitz functions f on X that also verify the following second order condition

${||f||}_{A_1}=\sup \left\{\frac{\left|f\left(x+2h\right)-2f\left(x+h\right)+f\left(x\right)\right|}{h^2};x,h\in x\right\}<\infty .$

The space $\mathcal{A}$ ₂ equipped with the norm ${||f||}_{A_2}={||f||}_{A_1}+||f||$ is also an admissible space of perturbations.

Clearly, the above norms will correspond to the C ¹ and C ²-norms whenever the functions are differentiable. But since X is in general infinite dimensional, we have to deal with two types of difficulties: firstly, the appearance of various different and generally non-equivalent types of differentiability and secondly, the problem of admissibility of these spaces of differentiable functions which requires extra assumptions on the Banach spaces involved. We deal with some of these problems in Section 3.

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Asymptotically Faster Translation Methods

NAIL A. GUMEROV , RAMANI DURAISWAMI , in Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, 2004

8.3.3.1 Integral representation of spherical filter

Let us return to representation of translation operators in the space of continuous functions. As we found, it is sufficient to perform translation with the regular or singular translation kernel of bandwidth p + p′ − 1,where p′ and p are the bandwidths of the initial and translated functions determined by the error bounds. Unfiltered translation of the signature function ${\Psi }^{\left(p^{\prime }\right)}\left(\text{s}\right)$ can then be represented as

(8.3.33) ${\widehat{\Psi }}^{\left(2p^{\prime }+p-2\right)}\left(\text{s}\right)={\Lambda }^{\left(p^{\prime }+p-2\right)}\left(\text{t}:{\text{s}}^{\prime }\right){\Psi }^{\left(p^{\prime }\right)}\left({\text{s}}^{\prime }\right).$

Using integral representations of translation coefficients (e.g. Eq. (7.1.58)) and signature function (7.1.28) we can show that the translation with truncated reexpansion matrix is equivalent to the following operation in the space of signature functions:

(8.3.34) $\begin{array}{ll}{\widehat{\Psi }}^{\left(p\right)}\left(\text{s}\right)& ={\displaystyle \sum _{n=0}^{p-1}{\displaystyle \sum _{m=-n}^n{\text{i}}^{-n}}}{\displaystyle \sum _{n^{\prime }=0}^{p^{\prime }-1}{\displaystyle \sum _{m^{\prime }=-n^{\prime }}^{n^{\prime }}{\left(E/F\right)}_{n{n}^{\prime }}^{m{m}^{\prime }}\left(\text{t}\right)}{C}_{n^{\prime }}^{m^{\prime }}{\Upsilon }_n^m\left(\text{s}\right)}\\ & {\displaystyle ={\int }_{S_u}{\Lambda }^{\left(p+p^{\prime }-1\right)}\left(\text{t};{\text{s}}^{\prime }\right)\left[{\displaystyle \sum _{n^{\prime }=0}^{p^{\prime }-1}{\displaystyle \sum _{m^{\prime }=-n^{\prime }}^{n^{\prime }}{\text{i}}^{-n^{\prime }}}{C}_{n^{\prime }}^{m^{\prime }}{\Upsilon }_{n^{\prime }}^{m^{\prime }}\left({\text{s}}^{\prime }\right)}\right]}\\ & \times \left[{\displaystyle \sum _{n=0}^{p-1}{\displaystyle \sum _{m=-n}^n{\Upsilon }_n^{-m}\left({\text{s}}^{\prime }\right){\Upsilon }_n^m\left(\text{s}\right)}}\right]\text{d}S\left({\text{s}}^{\prime }\right)\\ & ={\displaystyle {\int }_{S_u}{\Lambda }^{\left(p+p^{\prime }-1\right)}\left(\text{t};{\text{s}}^{\prime }\right){\Psi }^{\left(p^{\prime }\right)}\left({\text{s}}^{\prime }\right){\delta }^{\left(p\right)}\left({\text{s}}^{\prime };s\right)\text{d}S\left({\text{s}}^{\prime }\right)}\\ & {\displaystyle ={\int }_{S_u}{\widehat{\Psi }}^{\left(2p^{\prime }+p-2\right)}\left({\text{s}}^{\prime }\right){\delta }^{\left(p\right)}\left({\text{s}}^{\prime };\text{s}\right)\text{d}\text{S}\left({\text{s}}^{\prime }\right)},\end{array}$

where we used definitions (8.2.36) and (8.3.33). In fact, we could obtain the final result immediately, if we recall Eq. (8.2.37), and interpretation of the band-limited function δ^(p)(s′; s) as a kernel of a low-pass spherical filter. Nevertheless, it is useful to have this derivation which proves the fact that we obtained by some discrete procedures from a different viewpoint and provides a different representation of the filtering matrix, via kernel δ^(p)(s′; s).

Indeed, using a cubature exact for functions of bandwidth 2p′ + 2p − 3 or, to simplify, for $N_b^{\prime }=2\left(p^{\prime }+p-1\right),$ , with $N_c^{\prime }$ nodes, we can represent the latter equation in the form

(8.3.35) $\begin{array}{cc}{\widehat{\Psi }}^{\left(p\right)}\left({\text{s}}_j\right)={\displaystyle \sum _{j^{\prime }=0}^{N_c^{″}-1}{w}_{j^{\prime }}{\widehat{\Psi }}^{\left(2p^{\prime }+p-2\right)}\left({{\text{s}}^{\prime }}_{j^{\prime }}\right){\delta }^{\left(p\right)}\left({{\text{s}}^{\prime }}_{j^{\prime }}{\text{;s}}_j\right)},& j=0,\dots ,N_c-1,\end{array}$

where s _j are the nodes of the grid at which function ${\widehat{\Psi }}^{\left(p\right)}\left(\text{s}\right)$ should be sampled.

Function δ^(p)(s′; s) has different representations. We proceed with the following decomposition based on the Christoffel–Darboux formula for the normalized associated Legendre functions (7.1.63):

(8.3.36) $\left({\mu }^{\prime }-\mu \right){\displaystyle \sum _{n=\left|m\right|}^{p-1}{\underset{\_}{P}}_n^m}\left({\mu }^{\prime }\right){\underset{\_}{P}}_n^m(\mu )=a_{p-1}^m\left[{\underset{\_}{P}}_p^m\left({\mu }^{\prime }\right){\underset{\_}{P}}_{p-1}^m(\mu )-{\underset{\_}{P}}_{p-1}^m\left({\mu }^{\prime }\right){\underset{\_}{P}}_p^m(\mu )\right],$

where $a_{p-1}^m$ are the differentiation coefficients (2.2.8). With this equation the truncated surface delta-function (8.2.36) can be represented as

(8.3.37) $\begin{array}{cc}{\delta }^{\left(p\right)}\left({\text{s}}^{\prime };\text{s}\right)& ={\displaystyle \sum _{m=-\left(p-1\right)}^{p-1}{\displaystyle \sum _{n=\left|m\right|}^{p-1}{\Upsilon }_n^{-m}\left({\text{s}}^{\prime }\right){\Upsilon }_n^m\left(\text{s}\right)}}\hfill \\ & =\frac{1}{2\pi }{\displaystyle \sum _{m=-\left(p-1\right)}^{p-1}{\text{e}}^{im\left(\phi ={\phi }^{\prime }\right)}{a}_{p-1}^m\left[\frac{{\underset{\_}{P}}_p^m\left({\mu }^{\prime }\right){\underset{\_}{P}}_{p-1}^m(\mu )}{{\mu }^{\prime }-\mu }-\frac{{\underset{\_}{P}}_{p-1}^m\left({\mu }^{\prime }\right){\underset{\_}{P}}_p^m(\mu )}{{\mu }^{\prime }-\mu }\right]},\end{array}$

where (θ, φ) and (θ′, φ′) are the spherical polar angles of surface points s and s′, respectively, and μ = cos θ, μ′ = cos θ′.

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Assessing structural relationships between distributions - a quantile process approach based on Mallows distance

G. Freitag , ... M. Vogt , in Recent Advances and Trends in Nonparametric Statistics, 2003

2.2 Location model

In order to make ideas more transparent we treat the location model separately. Observe that

${\mathrm{\Gamma }}_{L,\beta }^2\left(F,G\right)=\frac{1}{1-2\beta }{\displaystyle {\int }_{\beta }^{1-\beta }{\left(F^{-1}\left(t\right)-G^{-1}\left(t\right)-{\delta }_{\beta }\left(F,G\right)\right)}^2}dt\text{,}$

where

(8) ${\delta }_{\beta }={\delta }_{\beta }\left(F,G\right)=\frac{1}{1-2\beta }\left({\displaystyle {\int }_{F^{-1}\left(\beta \right)}^{F^{-1}\left(1-\beta \right)}xdF\left(x\right)-}{\displaystyle {\int }_{G^{-1}\left(\beta \right)}^{G^{-1}\left(1-\beta \right)}xdG\left(x\right)}\right)\text{.}$

Let C [0, 1] be the metric space of continuous functions on [0, 1] equipped with the sup-norm and the Borel σ-field. A Brownian Bridge will be denoted as $\mathrm{ℬ}$ , i.e. a centered Gaussian process in C[0, 1] with $cov\left[\mathrm{ℬ}\left(s\right),\mathrm{ℬ}\left(t\right)\right]=s\wedge t-st$ , where s ∧ t:= min{s, t}. The following theorem treats the case of paired data.

Theorem 2.1

Let $H\in C^2\left[{\overline{\mathrm{ℝ}}}^2\right]$ with marginals F, G and $Z_1,\dots ,Z_n\stackrel{i.i.d.}{\sim }H\left(\cdot ,\cdot \right)$ . Then under the assumptions (7) we have for Γ² _L,β >   0, as n  →   ∞,

$n^{\frac{1}{2}}\left\{{\mathrm{\Gamma }}_{L,\beta }^2\left(F_n,G_n\right)-{\mathrm{\Gamma }}_{L,\beta }^2\left(F,G\right)\right\}\stackrel{d}{\to X_{L1},}$

where X _L1 is given by

$\begin{array}{l}{X}_{L1}=\frac{2}{1-2\beta }\left\{{\displaystyle {\int }_0^{1-\beta }{\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{F^{-1}\left(t\right)-G^{-1}\left(t\right){\delta }_{\beta }}{{\displaystyle \int \left(F^{-1}\left(t\right)\right)}}}}\right.dtd{\mathrm{ℬ}}_F\left(s\right)\\ \left.+{\displaystyle {\int }_0^{1-\beta }{\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{G^{-1}\left(t\right)-F^{-1}\left(t\right){\delta }_{\beta }}{g\left(G^{-1}\left(t\right)\right)}dtd{\mathrm{ℬ}}_G\left(s\right)}}\right\}\text{.}\end{array}$

Here ${\mathrm{ℬ}}_F$ , ${\mathrm{ℬ}}_G$ are Brownian bridges with cov $\left[{\mathrm{ℬ}}_F\left(s\right),{\mathrm{ℬ}}_G\left(t\right)\right]=H\left(F^{-1}\left(s\right),G^{-1}\left(t\right)\right)-st$ . Thus, X _L1 is a centered normally distributed random variable with variance

$\begin{array}{l}{\sigma }_{L1,\beta }^2=\frac{4}{{\left(1-2\beta \right)}^2}\left[{\displaystyle {\int }_0^{1-\beta }{\left({\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{F^{-1}\left(t\right)-G^{-1}\left(t\right)-{\delta }_{\beta }}{f\left(F^{-1}\left(t\right)\right)}dt}\right)}^{2}ds-{\left({\displaystyle {\int }_0^{1-\beta }{\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{F^{-1}\left(t\right)-G^{-1}\left(t\right)-{\delta }_{\beta }}{f\left(F^{-1}\left(t\right)\right)}\mathit{dtds}}}\right)}^2}\right.\\ +{\displaystyle {\int }_0^{1-\beta }{\left({\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{G^{-1}\left(t\right)-F^{-1}\left(t\right)+{\delta }_{\beta }}{g\left(G^{-1}\left(t\right)\right)}dt}\right)}^{2}ds-{\left({\displaystyle {\int }_0^{1-\beta }{\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{G^{-1}\left(t\right)-F^{-1}\left(t\right)+{\delta }_{\beta }}{g\left(G^{-1}\left(t\right)\right)}\mathit{dtds}}}\right)}^2}\\ +2{\displaystyle {\int }_0^{1-\beta }{\displaystyle {\int }_0^{1-\beta }\left({\displaystyle {\int }_{\beta \mathrm{V}r}^{1-\beta }\frac{F^{-1}\left(t\right)-G^{-1}\left(t\right)-{\delta }_{\beta }}{f\left(F^{-1}\left(t\right)\right)}dt}\right)\left({\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{G^{-1}\left(t\right)-F^{-1}\left(t\right)+{\delta }_{\beta }}{g\left(G^{-1}\left(t\right)\right)}dt}\right)}\frac{\partial H\left(F^{-1}\left(r\right),G^{-1}\left(s\right)\right)}{\partial r\partial s}}\mathit{drds}\\ \left.-2\left({\displaystyle {\int }_0^{1-\beta }{\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{F^{-1}\left(t\right)-G^{-1}\left(t\right)-{\delta }_{\beta }}{f\left(F^{-1}\left(t\right)\right)}\mathit{dtds}}}\right)\left({\displaystyle {\int }_0^{1-\beta }{\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{G^{-1}\left(t\right)-F^{-1}\left(t\right)+{\delta }_{\beta }}{g\left(G^{-1}\left(t\right)\right)}\mathit{dtds}}}\right)\right]\text{.}\end{array}$

Now we deal with the case of two independent samples as in the Example 1.

Theorem 2.2

Assume that $X_1,\dots ,X_m\stackrel{i.i.d.}{\sim }F,Y_1,\dots ,Y_n\stackrel{i.i.d.}{\sim }G$ are independent samples, and F, G satisfy the conditions in (7) . If n/(n  + m)   → λ ∈ (0, 1) as m ∧ n  →   ∞, then we obtain

${\left(\frac{\mathit{nm}}{n+m}\right)}^{\frac{1}{2}}\left\{{\mathrm{\Gamma }}_{L,\beta }^2\left(F_m,G_n\right)-{\mathrm{\Gamma }}_{L,\beta }^2\left(F,G\right)\right\}\stackrel{d}{\to }{X}_{L2}\text{,}$

Where X _L2 is given by

$\begin{array}{lll}{X}_{L2}\hfill & :=\hfill & \frac{1}{1-2\beta }\left\{\sqrt{\lambda }\right.{\displaystyle {\int }_0^{1-\beta }{\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{F^{-1}\left(t\right)-G^{-1}\left(t\right)-{\delta }_{\beta }}{f\left(F^{-1}\left(t\right)\right)}}dtd{B}_F\left(s\right)}\hfill \\ \hfill & \hfill & \left.+\sqrt{1-\lambda }{\displaystyle {\int }_0^{1-\beta }{\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{G^{-1}\left(t\right)-F^{-1}\left(t\right)-{\delta }_{\beta }}{g\left(G^{-1}\left(t\right)\right)}}dtd{B}_G\left(s\right)}\right\}\text{,}\hfill \end{array}$

with two independent Brownian bridges ${\mathrm{ℬ}}_F$ and ${\mathrm{ℬ}}_G$ Furthermore, X _L2 is a centered normally distributed random variable with variance

$\begin{array}{lll}{\sigma }_{L2,\beta }^2\hfill & =\hfill & \frac{4}{{\left(1-2\beta \right)}^2}\left[\lambda \left({\displaystyle {\int }_0^{1-\beta }{\left({\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{F^{-1}\left(t\right)-G^{-1}\left(t\right)-{\delta }_{\beta }}{f\left(F^{-1}\left(t\right)\right)}dt}\right)}^{2}ds-{\left({\displaystyle {\int }_0^{1-\beta }{\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{F^{-1}\left(t\right)-G^{-1}\left(t\right)-{\delta }_{\beta }}{f\left(F^{-1}\left(t\right)\right)}\mathit{dtds}}}\right)}^2}\right)\right.\hfill \\ \hfill & \hfill & +\left(1=\lambda \right)\left.\left({\displaystyle {\int }_0^{1-\beta }{\left({\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{G^{-1}\left(t\right)-F^{-1}\left(t\right)-{\delta }_{\beta }}{g\left(G^{-1}\left(t\right)\right)}dt}\right)}^{2}ds-{\left({\displaystyle {\int }_0^{1-\beta }{\displaystyle {\int }_{\beta \mathrm{V}s}^{1-\beta }\frac{G^{-1}\left(t\right)-F^{-1}\left(t\right)-{\delta }_{\beta }}{g\left(G^{-1}\left(t\right)\right)}\mathit{dtds}}}\right)}^2}\right)\right]\text{.}\hfill \end{array}$

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Computational Theory of Iterative Methods

In Studies in Computational Mathematics, 2007

Example 5.1.4

Let X=Y=C[0, 1]the space of continuous functions on[0, 1]equipped with the sup-norm. Consider the integral equation on Ū[x ₀,r/2), (r∈ [0,R]) given by

(5.1.116) $x\left(t\right)={\displaystyle {\int }_0^{1}k\left(t,s,x\left(s\right)\right)}ds,$

where the kernel k (t, s, x (s)) with (t, s) ∈ [0, 1] × [0, 1]is a nondifferentiable operator on Ū (x ₀,r/2). Define operators F, G on Ū (x ₀,r/2) by

(5.1.117) $F\left(x\right)\left(t\right)=Ix\left(t\right)\left(\mathrm{I\hspace{0.17em}the\hspace{0.17em}identity\hspace{0.17em}operator}\right)$

(5.1.118) $G\left(x\right)\left(t\right)=-{\displaystyle {\int }_0^{1}k}\left(t,s,x\left(s\right)\right)ds.$

Choose x ₀= 0,and assume there exists a constant k ₀∈ [0, 1),a real function k ₁(t, s) such that

(5.1.119) $\left|\right|k\left(t,s,x\right)-k\left(t,s,y\right)\left|\right|\le k_1\left(t,s\right)\left|\right|x-y\left|\right|$

and

(5.1.120) $\underset{t\in \left[0,1\right]}{\sup }{\displaystyle {\int }_0^1{k}_1\left(t,s\right)ds\le k_0}$

for all t, s∈ [0, 1], x, y∈Ū(x ₀,r ₂). Moreover choose: r ₀= 0,y ₀=y ₋₁,A (x) =I (x),I the identity operator on X, v ₀(r) =r, a=b= 0,v= 0,and v ₁(r) =k ₀ for all x, y∈Ū(x ₀,r/2),r, s∈ [0, 1]. It can easily be seen that the conditions of Theorem 5.1.6 hold if

(5.1.121) $t^{*}=\frac{\eta }{1-k_0}\le \frac{r}{2}.$

In order to cover the local case, let us assume x ^* is a zero of Equation (5.1.1),A (x ^*,x ^*)⁻¹ exists and for any x, y∈Ū(x ^*,r) ⊆Ū(x ^*,R),r∈ [0,R], t∈ [0, 1]:

(5.1.122) $\left|\right|A{\left(x^{*},x^{*}\right)}^{-1}\left[A\left(x,y\right)-A\left(x^{*},x^{*}\right)\right]\left|\right|\le {\overline{\theta }}_0\left(\left|\right|x-x^{*}\left|\right|,\left|\right|y-x^{*}\left|\right|\right),$

and

(5.1.123) $\begin{array}{l}\left|\right|A{\left(x^{*},x^{*}\right)}^{-1}[F^{\prime }\left(x^{*}+t\left(y-x^{*}\right)\right)\left(y-x^{*}\right)\\ -A\left(x,y\right)\left(y-x^{*}\right)+G\left(y\right)-G\left(x^{*}\right)]||\\ \le \overline{\theta }\left(\left|\right|y-r^{*}\left|\right|,\left|\right|x-x^{*}\left|\right|,t\right)\left|\right|y-x^{*}\left|\right|,\end{array}$

where, $\overline{\theta }$ ₀, $\overline{\theta }$ ₁ are as θ₀, θ (in three variables), respectively. In order for us to compare our results with earlier ones, we only consider the case r ₀= 0,x ₋₁=v, x ₀=w in (5.5.1), and call the corresponding sequence {x _n } instead of {y _n }. Then as in the identity (5.1.36) and estimate (5.1.37) with y ^*,v, w, replaced by x ^*,x ^*,x ^* respectively but using (5.1.22), (5.1.123), instead of (5.1.4), (5.1.5) we can show along the lines of the proof of the uniqueness part of Theorem 5.1.2 the following local result for method (5.7.2).

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Newton's method

Á. Alberto Magreñán , Ioannis K. Argyros , in A Contemporary Study of Iterative Methods, 2018

3.3 Numerical examples

Example 3.1

Let $\mathcal{X}=\mathcal{Y}=\mathcal{C}[0,1]$ , the space of continuous functions defined on $[0,1]$ , be equipped with the max-norm. Let $D^{⁎}=\{x\in \mathcal{C}[0,1];\Vert x\Vert \le R\}$ for $R>0$ and consider F defined on $D^{⁎}$ by

$F(x)(s)=x(s)-f(s)-\xi {\int }_0^{1}G(s,t)x{(t)}^{3}dt,$

where

$x\in C[0,1]$

and

$s\in [0,1],$

where $f\in \mathcal{C}[0,1]$ is a given function, ξ is a real constant, and the kernel G is the Green's function

$G(s,t)=\{\begin{array}{c}(1-s)t,t\le s,\hfill \\ s(1-t),s\le t.\hfill \end{array}$

In this case, for each $x\in D^{⁎}$ , $F^{\prime }(x)$ is a linear operator defined on $D^{⁎}$ by the following expression:

$[F^{\prime }(x)(v)](s)=v(s)-3\xi {\int }_0^{1}G(s,t)x{(t)}^{2}v(t)dt,$

where

$v\in C[0,1]$

and

$s\in [0,1].$

If we choose

$x_0(s)=1$

and

$f(s)=1,$

it follows that

$\Vert I-F^{\prime }(x_0)\Vert \le \frac{3|\xi |}{8}.$

Hence, if

$|\xi |<\frac{8}{3},$

$F^{\prime }{(x_0)}^{-1}$ is defined,

$\Vert F^{\prime }{(x_0)}^{-1}\Vert \le \frac{8}{8-3|\xi |},$

$\Vert F(x_0)\Vert ⩽\frac{|\xi |}{8},$

and

$\eta =\Vert F^{\prime }{(x_0)}^{-1}F(x_0)\Vert \le \frac{|\xi |}{8-3|\xi |}.$

Choosing $\xi =1.00$ and $\lambda =3$ , we have:

$\eta =0.2,$

$L=3.8,$

$L_0=2.6,$

$K=2.28,$

$H=1.28$

and

$L_1=1.38154\dots$

Using this values, we obtain that conditions (3.1.4)–(3.1.7) are not satisfied, since

$h_K=1.52>1,$

$h_1=1.28>1,$

$h_2=1.19343\dots >1,$

$h_3>1.07704\dots >1,$

but conditions (3.1.8) and condition (3.1.11) are satisfied, since

$h_4=0.985779\dots <1\text{and}{h}_5=0.97017\dots <1.$

Hence, we can ensure the convergence of the Newton's method by Theorem 3.2.1.

Example 3.2

Let

$\mathcal{X}=\mathcal{Y}=\mathbb{R},$

$x_0=1,$

$p\in [0,0.5),$

$D=\overline{U}(x_0,1-p),$

and define function F on D by

(3.3.1) $F(x)=x^3-p.$

Then we have

$L_0=3-p$

and

$L=2(2-p).$

Condition (3.1.4) is not satisfied, since $h_K>1$ for each $p\in (0,0.5)$ . Conditions (3.1.5), (3.1.6), and (3.1.7) (see Fig. 3.1) are satisfied respectively for

$p\in [0.494816242,0.5),$

$p\in [0.450339002,0.5),$

and

$p\in [0.4271907643,0.5).$

Figure 3.1. Condition (3.1.5) in black and dashed, condition (3.1.6) in black and dot-dashed and condition (3.1.7) in black and dotted.

We are now going to consider such an initial point that previous conditions cannot be satisfied but our new criteria are satisfied. Hence, we obtain an improvement with our new weaker criteria.

Moreover, we get that

$H=\frac{5+p}{3},$

$K=2,$

and

$L_1=\frac{2}{3(3-p)}(-2p^2+5p+6).$

Using these values, we obtain that condition (1.8) is satisfied for

$p\in [0.0984119,0.5)$

and condition of Theorem 3.2.1 is satisfied for $p\in [0,0.5)$ , so there exist several values of p for which the previous conditions cannot guarantee the convergence but our new ones can.

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The majorization method in the Kantorovich theory

Á. Alberto Magreñán , Ioannis K. Argyros , in A Contemporary Study of Iterative Methods, 2018

1.5 Examples

First, we provide two examples where ${\nu }_{\star }<\nu$ .

Example 1.5.1

Let $\mathcal{X}=\mathcal{Y}=\mathcal{C}[0,1]$ , the space of continuous functions defined on $[0,1]$ , be equipped with the max norm and $\mathcal{D}=\bar{U}(0,1)$ . Define function F on $\mathcal{D}$ by

(1.5.1) $F(h)(x)=h(x)-5{\int }_0^{1}x\theta h{(\theta )}^{3}d\theta .$

Then we have

$F^{\prime }(h[u])(x)=u(x)-15{\int }_0^{1}x\theta h{(\theta )}^{2}u(\theta )d\theta \mathit{for\; all}u\in \mathcal{D}.$

Using (1.5.1) we see that hypotheses of Theorem 1.3.1 hold for $x^{\star }(x)=0$ , where

$x\in [0,1],$

$\nu (t)=15t,$

and

${\nu }_{\star }(t)=7.5t.$

So we can ensure the convergence by Theorem 1.3.1 .

Example 1.5.2

Let $\mathcal{X}=\mathcal{Y}=\mathbb{R}$ . Define function F on $\mathcal{D}=[-1,1]$ by

(1.5.2) $F(x)=e^x-1.$

Then, using (1.5.2) for $x^{\star }=0$ , we get that

$F(x^{\star })=0$

and

$F^{\prime }(x^{\star })=e^0=1.$

Moreover, hypotheses of Theorem 1.3.1 hold for

$\nu (t)=et$

and

${\nu }_{\star }(t)=(e-1)t.$

Note that ${\nu }_{\star }<\nu$ . So we can ensure the convergence by Theorem 1.3.1 .

Example 1.5.3

Let $\mathcal{X}=\mathcal{Y}=\mathcal{C}[0,1]$ , equipped with the max-norm. Let $\theta \in [0,1]$ be a given parameter. Consider the "cubic" integral equation

(1.5.3) $u(s)=u^3(s)+\lambda u(s){\int }_0^1\mathcal{G}(s,t)u(t)dt+y(s)-\theta .$

Nonlinear integral equations of the form (1.5.3) are considered Chandrasekhar-type equations [10] , [13] , [18] , and they arise in the theories of radiative transfer, neutron transport, and in the kinetic theory of gasses. Here the kernel $\mathcal{G}(s,t)$ is a continuous function of two variables $(s,t)\in [0,1]\times [0,1]$ satisfying

(i)

$0<\mathcal{G}(s,t)<1$ ,

(ii)

$\mathcal{G}(s,t)+\mathcal{G}(t,s)=1$ .

The parameter λ is a real number called the "albedo" for scattering; $y(s)$ is a given continuous function defined on $[0,1]$ , and $x(s)$ is the unknown function sought in $\mathcal{C}[0,1]$ . For simplicity, we choose

$u_0(s)=y(s)=1$

and

$\mathcal{G}(s,t)=\frac{s}{s+t}$

for all $(s,t)\in [0,1]\times [0,1](s+t\ne 0)$ .

Let $\mathcal{D}=U(u_0,1-\theta )$ and define the operator F on $\mathcal{D}$ by

(1.5.4) $\begin{array}{c}F(x)(s)=x^3(s)-x(s)+\lambda x(s){\int }_0^1\mathcal{G}(s,t)x(t)dt+y(s)-\theta \hfill \\ \mathit{for\; all}s\in [0,1].\hfill \end{array}$

Then every zero of F satisfies equation (1.5.3) . Therefore, the operator $F^{\prime }$ satisfies conditions of Theorem 1.2.8 , with

$\eta =\frac{|\lambda |\ln 2+1-\theta }{2(1+|\lambda |\ln 2)},$

$\omega (t):=Lt=\left(\frac{|\lambda |\ln 2+3(2-\theta )}{1+|\lambda |\ln 2}\right)t,$

and

${\omega }_0(t):=L_{0}t=\left(\frac{2|\lambda |\ln 2+3(3-\theta )}{2(1+|\lambda |\ln 2)}\right)t.$

It follows from our main results that if one condition in Application 1.4.1 holds, then problem (1.5.3) has a unique solution near $u_0$ . This assumption is weaker than the one given before using the Newton–Kantorovich hypothesis. Note also that $L_0<L$ for all $\theta \in [0,1]$ and ${\omega }_0<\omega$ . Next we pick some values of λ and θ such that all hypotheses are satisfied, so we can compare the "h" conditions (see Table 1.1 ).

Table 1.1. Different values of the parameters involved.

λ	θ	h ⋆	h 1	h 2	h 3
0.97548	0.954585	0.4895734208	0.4851994045	0.4837355633	0.4815518345
0.8457858	0.999987	0.4177974405	0.4177963046	0.4177959260	0.4177953579
0.3245894	0.815456854	0.5156159025	0.4967293568	0.4903278739	0.4809439506
0.3569994	0.8198589998	0.5204140737	0.5018519741	0.4955632842	0.4863389899
0.3789994	0.8198589998	0.5281518448	0.5093892893	0.5030331107	0.4937089648
0.458785	0.5489756	1.033941504	0.9590659445	0.9332478337	0.8962891928

Example 1.5.4

Let

$\mathcal{X}=\mathcal{Y}=\mathbb{R},$

$x_0=1,$

$\mathcal{D}=[\xi ,2-\xi ],$

$\xi \in [0,0.5).$

Define function F on $\mathcal{D}$ by

(1.5.5) $F(x)=x^3-\xi .$

(a)

Using (1.1.4) , (1.1.7) , we get that

$\begin{array}{c}\hfill \eta =\frac{1}{3}(1-\xi ),{\omega }_0(t):=L_{0}t=(3-\xi )t,\mathit{and}\hfill \\ \hfill \omega (t):=Lt=2(2-\xi )t.\hfill \end{array}$

Using Application 1.4.1 (a), we have that

$h_{\star }=\frac{2}{3}(1-\xi )(2-\xi )>0.5\mathit{for\; all}\xi \in (0,0.5).$

Hence, there is no guarantee that Newton's method (1.1.2) starting at $x_0=1$ converges to $x^{\star }$ . However, one can easily see that if, for example, $\xi =0.49$ , Newton's method (1.1.2) converges to $x^{\star }=\sqrt[3]{0.49}$ .

(b)

Consider our "h" conditions given in Application 1.4.1 . Then we obtain that

$\begin{array}{c}\hfill h_1=\frac{1}{6}(7-3\xi )(1-\xi )⩽0.5\mathit{for\; all}\xi \in [0.4648162415,0.5),\hfill \\ \hfill \begin{array}{c}{h}_2=\frac{1}{12}(8-3\xi +{(5{\xi }^2-24\xi +28)}^{1/2})(1-\xi )⩽0.5\hfill \\ \mathit{for\; all}\xi \in [0.450339002,0.5)\hfill \end{array}\hfill \end{array}$

and

$\begin{array}{c}{h}_3=\frac{1}{24}(1-\xi )(12-4\xi +{(84-58\xi +10{\xi }^2)}^{1/2}\hfill \\ +{(12-10\xi +2{\xi }^2)}^{1/2})⩽0.5\mathit{for\; all}\xi \in [0.4271907643,0.5).\hfill \end{array}$

Next we pick some values of ξ such that all hypotheses are satisfied, so we can compare the "h" conditions. Now, we can compare our "h" conditions (see Table 1.2 ).

Table 1.2. Different values of the parameters involved.

ξ	x ⋆	h ⋆	h 1	h 2	h 3
0.486967	0.6978302086	0.5174905727	0.4736234295	0.4584042632	0.4368027442
0.5245685	0.7242710128	0.4676444075	0.4299718890	0.4169293786	0.3983631448
0.452658	0.6727986326	0.5646168433	0.5146862992	0.4973315343	0.4727617854
0.435247	0.6597325216	0.5891326340	0.5359749755	0.5174817371	0.4913332192
0.425947	0.6526461522	0.6023932312	0.5474704233	0.5283539600	0.5013421729
0.7548589	0.8688261621	0.2034901726	0.1934744795	0.1900595014	0.1850943832

(c)

Consider the case $\xi =0.7548589$ where our "h" conditions given in Application 1.4.1 are satisfied. Let

$L_{-1}=2$

and

$L_{-2}=\frac{\xi +5}{3}=1.918286300.$

Hence, we have that

$\alpha =0.5173648648,$

${\alpha }_1=0.01192474572,$

and

${\alpha }_2=0.7542728992.$

Then conditions (1.4.1) – (1.4.3) hold. We also have that

${\beta }_1=9.613132975,$

${\beta }_2=-0.5073095684,$

${\beta }_3=-2.069459459,$

${\gamma }_1=-0.02751250012,$

${\gamma }_2=4.097708498,$

${\gamma }_3=-0.965270270,$

${\lambda }_1=0.4911124649,$

${\mathrm{\Delta }}_2=16.68498694,$

and

${\lambda }_2=148.7039440.$

Since ${\gamma }_1<0$ , ${\mathrm{\Delta }}_2>0$ and using (1.4.9) , we get in turn that

$\frac{1}{2L_4}=\min \{{\lambda }_1,{\lambda }_2\}=0.4911124649.$

Then, we deduce that condition (1.4.10) holds, provided that

$L_4=1.018096741$

and

$h_4=0.08319245167<0.5.$

If we consider $L_{\star }=1.836572600$ in Application 1.4.1 , we get that

$h_5=0.07234026489<h_4.$

Finally, we pick the same values of ξ as in Table 1.2 , so we can compare the $h_4$ and $h_5$ conditions (see Table 1.3 ).

Table 1.3. Different values of the parameters involved.

ξ	h 4	h 5
0.486967	0.1767312629	0.1377052696
0.5245685	0.1634740591	0.1290677624
0.452658	0.1888584478	0.1454742725
0.435247	0.1950234005	0.1493795529
0.425947	0.1983192281	0.1514558980
0.7548589	0.08319245167	0.07234026489

Example 1.5.5

Let $\mathcal{X}$ and $\mathcal{Y}$ as in Example 1.5.3 . Consider the following nonlinear boundary value problem [10] :

$\{\begin{array}{c}\hfill u^{″}=-u^3-\gamma u^2,\hfill \\ \hfill u(0)=0,\hfill \\ \hfill u(1)=1.\hfill \end{array}$

It is well known that this problem can be formulated as the integral equation

(1.5.6) $u(s)=s+{\int }_0^1\mathcal{Q}(s,t)(u^3(t)+\gamma u^2(t))dt$

where $\mathcal{Q}$ is the Green function given by

$\mathcal{Q}(s,t)=\{\begin{array}{c}\hfill t(1-s),t⩽s,\hfill \\ \hfill s(1-t),s<t.\hfill \end{array}$

Then problem (1.5.6) is in the form (1.1.1) , where $F:\mathcal{D}⟶\mathcal{Y}$ is defined as

$[F(x)](s)=x(s)-s-{\int }_0^1\mathcal{Q}(s,t)(x^3(t)+\gamma x^2(t))dt.$

Set $u_0(s)=s$ and $\mathcal{D}=U(u_0,R_0)$ . It is easy to verify that $U(u_0,R_0)\subset U(0,R_0+1)$ since $\parallel u_0\parallel =1$ . If $2\gamma <5$ , the operator $F^{\prime }$ satisfies conditions of Theorem 1.2.8 with

$\begin{array}{c}\hfill \eta =\frac{1+\gamma }{5-2\gamma },\hfill \\ \hfill \omega (t):=Lt=\frac{\gamma +6R_0+3}{4}t,\hfill \end{array}$

and

${\omega }_0(t):=L_{0}t=\frac{2\gamma +3R_0+6}{8}t.$

Note that ${\omega }_0<\omega$ . Next we pick some values of γ and $R_0$ such that all hypotheses are satisfied, so we can compare the "h" conditions (see Table 1.4 ).

Table 1.4. Different values of the parameters involved.

γ	R 0	h ⋆	h 1	h 2	h 3
0.00025	1	0.4501700201	3376306412	0.2946446274	0.2413108547
0.25	0.986587	0.6367723612	0.4826181423	0.4240511567	0.3508368298
0.358979	0.986587	0.7361726023	0.5600481163	0.4932612622	0.4095478068
0.358979	1.5698564	1.013838328	0.7335891949	0.6245310288	0.4927174588
0.341378	1.7698764	1.084400750	0.7750792917	0.6539239239	0.5088183074

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Complex Potentials and R-linear problem

In Computational Analysis of Structured Media, 2018

1.2 Equations in functional spaces

Let L be a finite union of simple closed curves on $\widehat{\mathbb{C}}$ . Functions continuous in L form the Banach space of continuous functions $\mathcal{C}\left(L\right)$ endowed with the norm

${‖\phi ‖}_{\mathcal{C}}≔\underset{z\in L}{max}\left|\phi \left(z\right)\right|.$

Analogous space is introduced for functions continuous in D  ∪   ∂  D with the norm

(2.1.6) ${‖\phi ‖}_{\mathcal{C}}≔\underset{z\in D\cup \partial D}{max}\left|\phi \left(z\right)\right|.$

A function φ is called the Hölder continuous function on L (we write $\phi \in {\mathcal{H}}^{\alpha }\left(L\right)$ ) with the power 0 < α ≤ 1 if there exists such a constant C > 0 that |  φ(x)   − φ(y)|   < C  |  x  − y  | ^α for all x, y ∈ L. ${\mathcal{H}}^{\alpha }\left(L\right)$ is a subspace of the continuous functions. It is a Banach space with the norm

(2.1.7) ${‖\phi ‖}_{\alpha }≔{‖\phi ‖}_{\mathcal{C}}+\underset{x,y\in L,x\ne y}{sup}\frac{\left|\phi \left(x\right)-\phi \left(y\right)\right|}{{\left|x-y\right|}^{\alpha }}.$

Let L be a smooth curve consisting of not more than finite collection of connected component. Let L divide the complex plane onto two domains D ⁺ and D ⁻. The set of all continuous functions on L contains two subsets of functions analytically extended into D ⁺ and D ⁻. These subsets will be denoted $\mathcal{C}\left(D^{\pm }\right)$ .

Analogous designations ${\mathrm{\mathscr{H}}}^{\alpha }\left(D^{\pm }\right)\equiv \mathrm{\mathscr{H}}\left(D^{\pm }\right)$ are used for Hölder continuous functions. It follows from Maximum Modulus Principle for analytic functions that convergence in the space $\mathrm{\mathscr{H}}\left(D^{\pm }\right)$ implies the uniform convergence in the closure of D ^±.

The Lebesgue space ${\mathrm{ℒ}}_p\left(L\right)$ consists of functions f having the finite integral ${‖\phi ‖}_p≔{\left({\int }_L{\left|\phi \left(t\right)\right|}^p\mathit{ds}\right)}^{\frac{1}{p}}$ . The space ${\mathrm{ℒ}}_p\left(L\right)$ for p ≥ 1 is Banach. The Hardy space ${\mathrm{\mathscr{H}}}_p\left(\mathbb{U}\right)$ for p > 0 is defined for any p > 0 as the space of all analytic functions in the unit disk $\mathbb{U}=\left\{z\in \mathbb{C}::\left|z\right|<1\right\}$ satisfying the following condition

$\underset{0<r<1}{sup}{\int }_0^{2\pi }{\left|f\left({\mathit{re}}^{\mathit{i\theta }}\right)\right|}^p\mathit{d\theta }<\infty .$

${\mathrm{\mathscr{H}}}_p\left(\mathbb{U}\right)$ are Banach spaces for any p, 1 ≤ p ≤ ∞. Such spaces can be defined also for harmonic setting. For instance harmonic Hardy spaces $h_p\left(\mathbb{U}\right)$ , 0 < p < ∞, are the spaces of all harmonic functions u, satisfying the following condition

$\underset{0<r<1}{sup}{\int }_0^{2\pi }{\left|u\left({\mathit{re}}^{\mathit{i\theta }}\right)\right|}^p\mathit{d\theta }<\infty .$

$h_{\infty }\left(\mathbb{U}\right)$ is the space of all bounded harmonic functions on $\mathbb{U}$ .

Consider an operator equation

(2.1.8) $x=\mathbf{A}x+b,$

where A is a linear operator on a Banach space $\mathcal{B}$ . Widely used method of the solution of the equation (2.1.8) is the method of successive approximation. Taking an arbitrary element $x_0\in \mathcal{B}$ (called an initial point of approximation) one can determine the sequence of approximations

(2.1.9) $x_{n+1}=\mathbf{A}{x}_n+b,n=0,1,\dots$

The convergence of this sequence is connected with the convergence of the operator series

(2.1.10) $\mathbf{I}+\mathbf{A}+{\mathbf{A}}^2+\cdots +{\mathbf{A}}^n+\cdots$

Usually, the absolute convergence of the series (2.1.10) is considered when the following number series converges

(2.1.11) $1+‖\mathbf{A}‖+{‖\mathbf{A}‖}^2+\cdots +{‖\mathbf{A}‖}^n+\cdots$

It is not always the case of our study. We shall use another type of convergence associated to the space $\mathcal{B}$ . Let $\mathcal{B}$ be the space of continuous functions $\mathcal{C}$ on D  ∪   ∂  D endowed with the norm (2.1.6). The convergence in $\mathcal{C}$ means the uniform convergence on D  ∪   ∂  D which differs from the absolute convergence. It will be seen below that the method of successive approximations for the $\mathbb{R}$ -linear problem converges uniformly for any location of inclusions and may become divergent when separation condition (2.3.106) is not satisfied. Therefore, the uniform convergence is preferable for our investigations. In order to study such a convergence we consider an equation dependent on a spectral parameter.

Theorem 1

([21, p. 75]). Let A be a linear bounded operator in a Banach space $\mathcal{B}$ . If for any element $b\in \mathcal{B}$ and for any complex number ν satisfying the inequality |  ν  |   ≤   1 equation

(2.1.12) $x=\nu \mathbf{A}x+b$

has a unique solution, then the unique solution of the equation

(2.1.13) $x=\mathbf{A}x+b$

can be found by method of successive approximations. The approximations converge in $\mathcal{B}$ to the solution

(2.1.14) $x=\sum _{k=0}^{\infty }{\mathbf{A}}^{k}b.$

Corollary 1

Let A be a compact operator in $\mathcal{B}$ . If the equation

(2.1.15) $x=\nu \mathbf{A}x$

has only trivial solution for all $v\in \mathbb{C}$ , |  ν  |   ≤   1, then equation (2.1.12) has a unique solution for all $b\in \mathcal{B}$ , and for all |  ν  |   ≤   1. This solution can be found by the method of successive approximations convergent in $\mathcal{B}$ .

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Convex Analysis and Duality Methods

G. Bouchitté , in Encyclopedia of Mathematical Physics, 2006

Entropy

$f(x)=\{\begin{array}{cc}x\text{log}x& \text{if}x\in {\mathbb{R}}_{+}\\ +\infty & \text{otherwise}\end{array},f*(y)=\text{exp}(y-1)$

Example 3

Let C⊂X be convex, and let $f={\chi }_C$ . Then,

$\begin{array}{l}f*(x*)={\sigma }_C(c*)=\underset{x\in C}{\text{sup}}〈x|x*〉\\ (\text{support}\text{function of}C)\end{array}$

Notice that if M is a subspace of X, then $({\chi }_M)*={\chi }_{M^{\perp }}$ . We specify now a particular case of interest.

Let Ω be a bounded open subset of ${\mathbb{R}}^n$ . Take $X=C_0({\displaystyle \overline{\text{Ω}}};{\mathbb{R}}^d)$ to be the Banach space of continuous functions on the compact ${\displaystyle \overline{\text{Ω}}}$ ) with values in ${\mathbb{R}}^d$ . As usual, we identify the dual X* with the space ${\mathcal{M}}_b({\displaystyle \overline{\text{Ω}}};{\mathbb{R}}^d)$ of ${\mathbb{R}}^d$ -valued Borel measures on ${\displaystyle \overline{\text{Ω}}}$ with finite total variation. Let K be a closed convex of R^d such that 0∈K. Then ${\rho }_K^0(\xi ):=\text{sup}\{(\xi |z):z\in K\}$ is a non-negative convex l.s.c. and positively 1-homogeneous function on ${\mathbb{R}}^d$ (e.g., ρ_K is the Euclidean norm if K is the unit ball of ${\mathbb{R}}^d$ ). Let us define $C:=\{\phi \in X:\phi (x)\in K,\forall x\in \text{Ω}\}$ . Then, we have

[1] $\begin{array}{c}(\chi C)*(\lambda )={\int }_{\text{Ω}}{\rho }_K^0(\lambda )\\ :={\int }_{\text{Ω}}{\rho }_K^0(\frac{\text{d}\lambda }{\text{d}\theta })\theta (\text{d}x)\end{array}$

where θ is any non-negative Radon measure such that $\lambda \ll \theta$ (the choice of θ is indifferent). In the case where K is the unit ball, we recover the total variation of λ.

Example 4

(Integral functionals). Given $1\le p<+\infty$ a measured space and $\phi :\text{Ω}\times {\mathbb{R}}^d\to [0,+\infty ]$ a $\mathcal{T}\otimes B_{{\mathbb{R}}^d}$ -measurable integrand. Then the partial conjugate $\phi *(x,z*):=\text{sup}\{〈z|z*〉-\phi (x,z):z\in {\mathbb{R}}^d\}$ is a convex measurable integrand. Let us define

$I_{\phi }:u\in {(L_{\mu }^{\rho })}^d\to {\int }_{\text{Ω}}\phi (x,u(x))\text{d}\mu \in \mathbb{R}\cup \{+\infty \}$

and assume that I_ϕ is proper. Then there holds $(I_{\phi })*=I_{\phi *}$ , where

$(I_{\phi })*:\nu \in {(L_{\mu }^{p\prime })}^d\to {\int }_{\text{Ω}}\phi *(x,\nu (x))\text{d}\mu$

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