A natural regularization of the adsorption integral equation with Langmuir-kernel

Abstract

Adsorption integral equations are used to compute adsorption energy distributions by measured total isotherms. Since these types of equations are Fredholm integral equations of the first kind on bounded domains, they are unstable or ill-posed, respectively. Hence, there is a need for regularization. In this work, we present general regularizations based on the fourier transform for a special kernel, the Langmuir kernel. The regularization parameters are chosen as zeros or minimizers of simple functions depending on the mean absolute error of a transformed total isotherm. In difference to many other solutions proposed, an explicit error and convergence analysis is made. Additionally, we consider adsorption energy distributions with sharp peaks or for ideal adsorbents. Here, we construct a regularization for computing averages of the adsorption energy distribution and the maximal approximation error is estimated uniformly.

1 Introduction

Adsorption processes on solid surfaces depend strongly on the type and arrangement of the various adsorption sites. The quantitative description of the energetical heterogeneity of these surfaces is therefore important to characterize adsorbents. To this end one tries to compute the cumulative adsorption energy distribution (CAED) $\chi =\chi (u)$ or rather its density, i.e. the (differential) adsorption energy distribution (AED) F by means of the experimentally measured total adsorption isotherms $\Theta _t=\Theta _t(p,T)$. Here $\chi (u)$ denotes the percentage of adsorption sites releasing molar energies less or equal then u by adsorption with respect to the total number of adsorption sites. Hence $\chi (u_b)-\chi (u_a)$, $u_a<u_b$ is the percentage of adsorption sites releasing molar energies $u_a<u\le u_b$. $\Theta _t(p,T)$ represents the total coverage of the surface at pressure p and temperature T.

It is supposed that $\Theta _t$ is a superposition of the local isotherms $\Theta _l=\Theta _l(p,T,u)$ representing the local coverage on adsorption sites at the same molar energy u [1, 2]. Under the assumption that every energy intervall contributes to the superposition by the percentage of its adsorption sites and because of $\chi (0)=0$, the total isotherm is given by a Stieltjes integral:

$$\begin{aligned} \Theta _t(p,T) = \int \limits _0^\infty \Theta _l(p,T,u)\,d\chi (u). \end{aligned}$$

(1)

If $\chi$ has an integrable density F, that is

$$\begin{aligned} \chi (u)=\int \limits _0^u F(u^\prime )\,du^\prime ,\ 0\le u, \end{aligned}$$

(2)

then (1) reads as

$$\begin{aligned} \Theta _t(p,T) = \int \limits _{0}^{\infty } \Theta _l(p,T,u) F(u)\,du. \end{aligned}$$

(3)

(3) is known as adsorption integral equation (AIE) and F as AED. Since the AED gives a better resolution of the energetic inhomogeneities than the CAED, the computation of the AED is in the centre of interest. For that reason, let us first justify that it is reasonable to assume the existence of an AED.

We consider an ideal adsorbent. Here, there are finitely many types of adsorption sites with well-defined molar adsorption energies $0<u_1<u_2\cdots <u_n,$ where the adsorption takes place. The corresponding CAED is a step function

$$\begin{aligned} \chi (u) =\sum \limits _{j=1}^n a_j H(u-u_j), \ 0\le u \end{aligned}$$

(4)

with

$$\begin{aligned} a_j \ge 0\text { for }j=1,...,n\text { and } \sum \limits _{j=1}^n a_j=1, \end{aligned}$$

(5)

where $a_j$ is the percentage of adsorption sites releasing the molar energy $u_j$ and H denotes the Heaviside function

$$\begin{aligned} H(u) = {\left\{ \begin{array}{ll} 1,\ u \ge 0,\\ 0,\ u<0. \end{array}\right. } \end{aligned}$$

(6)

(1) then reduces to a finite sum

$$\begin{aligned} \Theta _t(p,T) =\sum \limits _{j=1}^n a_j \Theta _l(p,T,u_j). \end{aligned}$$

(7)

If a real adsorbent is considered, the interaction of an adsorption site with an adsorptive particle is influenced by adsorbed particles in the neighborhood of the site. This neighborhood can differ for different particles. So there is rather a continuum of adsorption energies than one specified energy that contributes to the adsorption on an adsorption site. These smear effects lead to CAED’s with smoothed steps. Therefore, eq, (2) is a reasonable assumption, i.e. the existence of F is justified.

Although (4) is very unlikely, we will also consider ideal adsorbents as limiting cases of real adsorbents with more and more localized adsorption energies. As we will see, this approach is useful for estimating the reconstruction quality of AED’s with sharp peaks.

Based on the assumption that $\chi$ is smooth, F is often supposed to be a continuous function [1, 2]. At least we can assume that F is bounded and integrable.

Furthermore, the molar adsorption energies are bounded by minimal and maximal values $0\le u_{{\textrm{min}}}<u_{{\textrm{max}}}<\infty$ ([2, 3]) implying $F(u)=0$ for all $u\notin \left[ u_{{\textrm{min}}},u_{{\textrm{max}}}\right]$. With this condition, (3) reads as

$$\begin{aligned} \Theta _t(p,T) = \int \limits _{u_{{\textrm{min}}}}^{u_{{\textrm{max}}}} \Theta _l(p,T,u) F(u)\,du. \end{aligned}$$

(8)

By definition of $\chi$, F meets additionally the constraints

$$\begin{aligned} F\ge 0&\ \ \text { and } \int \limits _{u_{{\textrm{min}}}}^{u_{{\textrm{max}}}} F(u)\,du=1. \end{aligned}$$

(9)

Equation (3) as well as (8), where the temperature T is kept fixed, are Fredholm integral equations of the first kind with kernel $\Theta _l (\cdot ,T,\cdot )$, given data $\Theta _t(\cdot ,T,\cdot )$ and unknown F. In general, these equations are ill-posed, in particular unstable with respect to the usual settings [4, 5].

There are numerous general methods for a stable solution of ill-posed problems. Each of these so-called regularizations has it’s limitations [5,6,7]. The reliability of regularizations depends on properties of the given data as well as on properties of the (integral) operator, see Sect. 2.

Since the properties of an integral operator are determined by the properties of its kernel and since there is a variety of possible kernels, i.e. local isotherms $\Theta _l$ ( [8]), it cannot be expected that there is only one method that fits all possible AIEs in the same way. In most cases, a general method is best practice for solving the AIE. Here often the problem occurs that the criteria estimating the approximation quality are unhandy or even impractical in the special case. As a result of this argument, in our opinion, whenever it is possible for given local isotherm $\Theta _l$, one should construct a taylormade regularization of the corresponding AIE to achieve a proper error analysis.

A widely used statistical-thermodynamical model for the local isotherms $\Theta _l$ describing type-I total isotherms $\Theta _t$ is the Langmuir model

$$\begin{aligned} \Theta _l(p,T,u)&= \frac{K_L (T,u) p}{1+K_L(T,u)p}, \end{aligned}$$

(10)

$$\begin{aligned} K_L(T,u)&= K_0(T) \exp \left( \frac{u}{RT}\right) . \end{aligned}$$

(11)

Here, R denotes the ideal gas constant and $K_0(T)$ is a known temperature-dependent constant [9]. For the sake of brevity, we call the AIE (8) with local isotherms $\Theta _l$ as defined by (10) and (11) the Langmuir AIE.

The Langmuir AIE is uniquely solvable and for continuous F there is a theoretical formula for a pointwise computation of F by means of $\Theta _t$ [4]. However, because the formula is impractical, different solution methods were presented. While older methods neglected the instability of the Langmuir AIE, newer ones make no statements about the approximation quality or, as mentioned above, the implicit given criteria estimating this quality are of little use [10]- [17].

This fact motivated the search for a taylormade solution procedure, where at least its limitations should be made clear. A first step into this direction was made in [4]. Here, we used the change of variables

$$\begin{aligned} \xi :=-RT\ln \left( K_0(T)p\right) ,&\ \varphi (\xi ,T) := \Theta _t (p,T) \end{aligned}$$

(12)

in order to transform the Langmuir AIE into

$$\begin{aligned} \varphi (\xi ,T)&= \int \limits _{u_{{\textrm{min}}}}^{u_{{\textrm{max}}}} \frac{F(u)}{1+\exp \left( \frac{\xi -u}{RT}\right) }\,du,\ \xi \in \mathbb {R}. \end{aligned}$$

(13)

By means of an inversion formula for the transformed total (adsorption) isotherm $\varphi$, a regularization scheme on the basis of fourier series was established including quantitative statements about the error amplification. In [18] the scheme was extended to fourier transform and a heuristic graphical criterion for the right choice of the approximate solution in the case of unknown error level (“error free case”) was given.

In this paper, we present a general and complete theory based on fourier transform to solve the AIE with Langmuir kernel by means of regularization. The advantages of this theory are:

• 1. For a wide class of AED’s, the approximation error is estimated in terms of magnitude. Here, the error caused by the amplification of measurement errors is explicitely estimated. If only reconstructions of an averaged AED are considered, then the approximation error itself is explicitely estimated in dependence of the measurement error level.

• 2. The “optimal” regularization parameter is easily computed.

• 3. The theory is easy to apply, i.e. it yields a blueprint constructing concrete regularizations by means of simple “damping functions".

In the following, we outline the contents of the paper. In Sect. 2, we explain the method of regularization for ill-posed problems and adapt it to our needs. According to our definition of regularization, we establish in Sect. 3 settings for the approximate solution of eq. (13) with constraints (9) in the case of erroneous data. In Sect. 4, we construct suitable approximate solution operators by means of fourier transform, while in Sect. 5 strategies for the right choice of one of these operators in dependence on the measurement error level are presented. The reconstruction of averaged AED’s is in the focus of Sect. 6. Here, we deal with AED’s with sharp peaks, or with ideal adsorbents, respectively. Finally, we summarize and discuss the results in Sect. 7.

2 General solution strategy: regularization

Usually, integral equations are considered as operator equations $\textbf{A}x=y$ between normed (function) spaces $\left( X,\Vert \cdot \Vert _X\right)$ and $\left( Y,\Vert \cdot \Vert _Y\right)$. For integral equations of the first kind, the linear integral operator $\textbf{A}$ is often one-to-one, while the inverse operator $\textbf{A}^{-1}$ is not continuous (unbounded) on $\textbf{A}(X)$. In applications, the right-hand side y is usually erroneous with a known error level $\delta >0$. More precisely, instead of y we have to deal with a pertubed right-hand side $y_\varepsilon =y+\varepsilon$, where $\Vert \varepsilon \Vert _Y\le \delta$ applies to the pertubation $\varepsilon$. The challenge is now to find a reasonable approximation $x_\varepsilon$ to the exact solution x that depends continuously on the data $y_\varepsilon$ (“stability condition”) [5].

Since $y_\varepsilon$, in general, doesn’t belong to the range of $\textbf{A}$, the inverse operator $\textbf{A}^{-1}: \textbf{A}(X)\rightarrow X$ has to be approximated by a continuous (bounded) linear operator $\textbf{R}: Y\rightarrow X$. Furthermore, $x_\varepsilon$ should converge to the exact solution x if the error level $\delta$ tends to 0. Hence, we need a family of continuous linear operators $\textbf{R}_{\sigma }: Y\rightarrow X$, $\sigma >0$ that converges pointwise to $\textbf{A}^{-1}$ on $\textbf{A}(X)$ as $\sigma \rightarrow 0$. Now, each

$$\begin{aligned} x_{\varepsilon ,\sigma } := \textbf{R}_{\sigma }\left( y_\varepsilon \right) =\textbf{R}_{\sigma }y+\textbf{R}_{\sigma }\varepsilon \end{aligned}$$

(14)

can be regarded as an approximation of x. The approximation error is estimated by

$$\begin{aligned} \Vert x_{\varepsilon ,\sigma } -x\Vert _X&\le \left\| \textbf{R}_{\sigma }y - \textbf{A}^{-1} y \right\| _X + \left\| \textbf{R}_{\sigma }\varepsilon \right\| _{X} \nonumber \\&\le \left\| \textbf{R}_{\sigma }y - \textbf{A}^{-1} y \right\| _X + \left\| \textbf{R}_{\sigma }\right\| _{Y\rightarrow X} \delta . \end{aligned}$$

(15)

It remains to find a strategy to choose $\sigma$ dependent on the error level $\delta$ or more general dependent on $\delta$ and the given data $y_\varepsilon$ to assure $x_{\varepsilon ,\sigma }\rightarrow x$, $\delta \rightarrow 0$. Such a strategy has to balance the needs for accuracy and for stability of the approximate solution to achieve an acceptable approximation error [5].

On the one hand, a small $\sigma$ is required for accuracy, since $\textbf{R}_{\sigma }y\rightarrow \textbf{A}^{-1} y=x$, $\sigma \rightarrow 0$. On the other hand, a larger $\sigma$ is required for stability. This results from the following argument:

By (14) it holds for two perturbations $\varepsilon$ and ${\bar{\varepsilon }}$ with $\Vert \varepsilon \Vert _Y\le \delta$ and $\Vert {\bar{\varepsilon }}\Vert _Y\le \delta$

$$\begin{aligned} \Vert x_{\varepsilon ,\sigma } -x_{{\bar{\varepsilon }},\sigma }\Vert _X\le 2\left\| \textbf{R}_{\sigma }\right\| _{Y\rightarrow X} \delta . \end{aligned}$$

(16)

As a consequence of the unboundedness of $\textbf{A}^{-1}$ we get $\left\| \textbf{R}_{\sigma }\right\| _{Y\rightarrow X}\rightarrow \infty$, $\sigma \rightarrow 0$. Therefore, the approximation error (15) becomes large for small $\sigma$. An optimal strategy would try and make the right-hand side of (15) minimal.

The method outlined above constructing an approximate stable solution to an ill-posed (unstable) problem is called regularization. Mostly, generalized regularizations are specialized to Hilbert space settings, i.e. X and Y are Hilbert spaces and $\textbf{A}: X\rightarrow Y$ is a bounded linear operator [4]- [7]. In this paper, we go another way and consider settings for metric spaces. The intention is to include the (nonlinear) constraints (9) into the setting. This means, we are only interested in reconstructing functions belonging to a subset $X_a$ of X. $X_a$ is called the set of admissible functions. X can then be considered as a space of approximations for $X_a$. Similarly, the linear space Y can be replaced by a metric subspace. An advantage of this approach is that $\textbf{A}^{-1}$ needs only to be approximated on $\textbf{A}(X_a)$. So we have more freedom to choose the $\textbf{R}_{\sigma }$.

These considerations lead to the following definition of regularization motivated by [5]:

Definition 1

Let $\textbf{A}$ be an operator and $X_a$ be a subset of the domain of $\textbf{A}$ such that $\textbf{A}$ is one-to-one on $X_a$. Furthermore, let $\left( X,d_X\right)$ and $\left( Y,d_Y\right)$ be metric spaces with $X_a\subset X$ and $\textbf{A}(X_a)\subset Y$.

A regularization scheme for the equation $\textbf{A}x =y$ with respect to the setting $X_a$, $\left( X,d_X\right)$, $\left( Y,d_Y\right)$ is a family of continuous operators $\textbf{R}_{\sigma }: Y\rightarrow X$, $\sigma >0$ with the property of pointwise convergence:

$$\begin{aligned} \lim \limits _{\sigma \rightarrow 0} d_X\left( \textbf{R}_{\sigma }y, \textbf{A}^{-1}y\right) = 0 \text { for all } y\in \textbf{A}\left( X_a\right) . \end{aligned}$$

(17)

¹

The parameter $\sigma$ is called the regularization parameter.

A strategy of choosing $\sigma =\sigma (\delta )$ in dependence of the error level $\delta >0$ is called regular if it satisfies for all $y\in \textbf{A}\left( X_a\right)$ the condition:

For all families of $y_\delta \in Y$, $\delta >0$ with $d_Y\left( y_\delta ,y\right) \le \delta$ there holds

$$\begin{aligned} \lim \limits _{\delta \rightarrow 0} d_X\left( \textbf{R}_{\sigma (\delta )} y_\delta , \textbf{A}^{-1} y\right) =0. \end{aligned}$$

(18)

A regularization scheme for $\textbf{A}x=y$ with respect to the setting $X_a$, $(X,d_X)$, $(Y,d_Y)$ including a regular strategy is called a regularization for $\textbf{A}x=y$ with respect to the setting $X_a$, $(X,d_X)$, $(Y,d_Y)$.

To emphasize the importance of the setting, we paraphrase [5], p. 222: The instability of the equation $\textbf{A}x = y$ is a property of the setting, since the continuity of $\textbf{A}^{-1} : \textbf{A}(X_a)\rightarrow X$ depends on $X_a$ and the metrics $d_X$ and $d_Y$. Therefore one could try and restore stability by changing the setting. But because of practical needs, in general, this approach is inadequate. In particular, the data space Y and its metric $d_Y$ must be suitable to describe the measured data, moreover the deviation of the reconstructed solution of the true solution ought to be measured by $d_X$ in a reasonable way.

For the sake of simplicity, we end this paragraph by setting the following conventions:

Convention 1

From now on if not otherwise specified all functions are complex valued and defined on the whole real axis. For a function f denotes $\Re f$ its real part and $\Im f$ its imaginary part. The letter i indicates the imagnary unit. A function f is called real-valued if $\Im f$vanishes. For real-valued functions f and g we still write $f\lhd g$ instead of $\Re f\lhd \Re g$ for $\lhd = <,\le ,>,\ge$. Furthermore we identify two functions f and g if it holds

$$\begin{aligned} \int \limits _{-\infty }^{\infty } \vert f(x)-g(x)\vert \,dx=0. \end{aligned}$$

(19)

We call a function f essentially bounded if there is $c\ge 0$ such that there holds

$$\begin{aligned} \int \limits _{-\infty }^{\infty } H(\vert f(x)\vert -c)\,dx= 0. \end{aligned}$$

(20)

If for two functions f and g the integral $\int \limits _{-\infty }^{\infty } f(x-y)g(y)\,dy$ does exist for every real x, then the convolution $f*g$ of f and g is defined as the function

$$\begin{aligned} f*g (x) := \int \limits _{-\infty }^\infty f(x-y)g(y)\,dy. \end{aligned}$$

(21)

By $\textrm{supp}f$ we denote the support of the function f, which is the closure of the set of all real points x with $f(x)\ne 0$.

3 Settings

We want to establish reasonable settings for the treatment of the Langmuir AIE in the version of equation (13).

Let $\textbf{A}$ be the operator that assigns to a function f the function

$$\begin{aligned} \textbf{A}f&:=k*f \ \ \text { with }\ k(x) := \frac{1}{1+\exp \left( \frac{x}{RT}\right) }, \ x\in \mathbb {R}, \end{aligned}$$

(22)

whenever $k*f$ exists. Then (13) can be formulated as

$$\begin{aligned} \varphi =\textbf{A}F \end{aligned}$$

(23)

with the constraint that F vanishes outside of $\left[ u_{{\textrm{min}}},u_{{\textrm{max}}}\right]$, i.e. it holds

$$\begin{aligned} \textrm{supp}F \subset \left[ u_{{\textrm{min}}},u_{{\textrm{max}}}\right] . \end{aligned}$$

(24)

We include (24) as well as the constraints (9) and the considerations about the regularity of the AED into the definition of the admissible functions, which suggests a reasonable approximation space X:

Definition 2

A function F is called admissible if it is real-valued, essentially bounded, integrable and satisfies the constraints (9) and (24).

Let $X_a$ be the set of all admissible functions and $C_{a}$ be the set of all continuous admissible functions.

Furthermore, let X be the space of all essentially bounded and absolutely integrable functions and $C_0$ be the space of all continuous functions f with $\lim \limits _{x\rightarrow \pm \infty } f(x) =0.$

Since k is bounded and continuous, $\textbf{A}f$ is well-defined for every absolutely integrable function f. According to [4], $\textbf{A}$ is one-to-one on $X_a$ and thus on $C_a$, too. Obviously, it holds $X_a\subset X$ and $C_a\subset C_0$.

Next we are looking for a suitable data space Y describing erroneous (measured) transformed total isotherms. The following observation is helpful.

Proposition 1

For $\varphi \in \textbf{A}(X_a)$ holds that $\varphi$ is bounded, continuous and that $\varphi -k$ is absolutely integrable.

Proof

Let $F\in X_a$ with $\varphi = \textbf{A}F$, then (9) and the fact that k is strictly monotonically decreasing and satisfies $0<k<1$ yields for $\xi \in \mathbb {R}$:

$$\begin{aligned} 0<k(\xi -u_{{\textrm{min}}})\le \int \limits _{u_{{\textrm{min}}}}^{u_{{\textrm{max}}}} k(\xi -u) F(u)\,du =\varphi (\xi )\le k(\xi -u_{{\textrm{max}}})<1, \end{aligned}$$

(25)

which proves boundedness.

Continuity results from $\vert k^\prime (x) \vert = \frac{1}{RT}\frac{\exp \left( \frac{x}{RT}\right) }{\left( 1+\exp \left( \frac{x}{RT}\right) \right) ^2}\le \frac{1}{4RT}$, $x\in \mathbb {R}$, which implies for $\xi _1,\xi _2\in \mathbb {R}$:

$$\begin{aligned} \left| \varphi (\xi _2)-\varphi (\xi _1)\right| \le \int \limits _{u_{{\textrm{min}}}}^{u_{{\textrm{max}}}} \vert k(\xi _2-u)-k(\xi _1-u)\vert F(u)\,du \le \frac{\vert \xi _2-\xi _1\vert }{4RT}. \end{aligned}$$

Finally by (25), we get $0<k(\xi -u_{{\textrm{min}}})-k(\xi )\le \varphi (\xi )-k(\xi )\le k(\xi -u_{{\textrm{max}}})-k(\xi )$ and thus

$$\begin{aligned} 0\le \varphi (\xi )-k(\xi )&\le \frac{\exp \left( \frac{\xi }{RT}\right) \left( 1-\exp \left( \frac{-u_{{\textrm{max}}}}{RT}\right) \right) }{\left( 1+\exp \left( \frac{\xi }{RT}\right) \right) \left( 1+\exp \left( \frac{\xi -u_{{\textrm{max}}}}{RT}\right) \right) } \nonumber \\&< \exp \left( \frac{u_{{\textrm{max}}}-\vert \xi \vert }{RT}\right) . \end{aligned}$$

(26)

Hence $\vert \varphi -k\vert = \varphi -k$ is bounded by an interable function and consequently, the continuous function $\varphi -k$ is absolutely integrable. $\square$

The above result motivates:

Definition 3

Let Y be the set of all essentially bounded functions $\psi$ for which $\psi -k$ is absolutely integrable.

By definition 3, we deal only with erroneous transformed total isotherms $\varphi _\varepsilon = \varphi +\varepsilon$ where the functions of errors $\varepsilon$ is not only essentially bounded but also absolutely integrable. Practically this is no real restriction. On the one hand, we measure only finitely many points of the total isotherm $\Theta _t$ between $\Theta _t\left( p_{{\textrm{min}}},T\right)$ and $\Theta _t\left( p_{{\textrm{max}}},T\right)$, $0<p_{{\textrm{min}}}<p_{{\textrm{max}}}<\infty$. So we can always find a $\varphi _\varepsilon$ as an approximation of the transformed total isotherm $\varphi$ that suits definition 3.

On the other hand, consider the following heuristic argument about the behaviour of the error:

Let $V=V(p)$ be the volume of the adsorbed gas. The total isotherm $\Theta _t$ is then given by $\Theta _t =\frac{V}{V_{\infty }}$ with $V_{\infty } := \lim \limits _{p\rightarrow \infty } V(p)$. For the measured volume $V_\epsilon =V+\epsilon$ let $\epsilon =\epsilon (p)$ be the function of the measurement errors. $V_\epsilon$ tends to a limit volume $V_{\epsilon ,\infty }$ for large pressures which is assumed to be larger than 0, otherwise the measurement has to be rejected. It is natural to assume that $\epsilon$ is controlled by V while the oscillations $\epsilon _{\infty }-\epsilon$ around the limit error $\epsilon _{\infty } := V_{\epsilon ,\infty } - V_{\infty } =\lim \limits _{p\rightarrow \infty } \epsilon (p)$ are controlled by $V_{\infty } -V$.² More precisely, we postulate

$$\begin{aligned} \epsilon \sim V\ \text { and }\ \epsilon _{\infty } -\epsilon \sim V_{\infty } - V. \end{aligned}$$

(27)

³ (27) yields together with $V\sim \Theta _t$ and $V_{\infty } -V\sim 1-\Theta _t$ for $\epsilon _\Theta := \Theta _{t\epsilon }-\Theta _t$ (the error with respect to $\Theta _t$):

$$\begin{aligned} \epsilon _\Theta = \frac{V+\epsilon }{V_{\infty }+\epsilon _{\infty }} -\frac{V}{V_{\infty }} =\frac{\epsilon (V_{\infty }-V)+ V(\epsilon -\epsilon _{\infty })}{(V_{\infty }+\epsilon _{\infty })V_{\infty }} \sim \Theta _t(1-\Theta _t). \end{aligned}$$

This implies for $\varepsilon =\varphi _\varepsilon -\varphi$ (the error of the transformed total isotherm $\varphi$):

$$\begin{aligned} \varepsilon \sim \varphi (1-\varphi )\overset{(25)}{\sim } k(\cdot -u_{{\textrm{max}}})(1-k(\cdot -u_{{\textrm{min}}})). \end{aligned}$$

(28)

The right-hand side of (28) is a positive and integrable function. This justifies to assume that $\varphi _\varepsilon$ belongs to Y as defined in definition 3.

It leaves to choose the metrics $d_X$ and $d_Y$. For numerical reasons, the metrics should allow the approximation of the elements of X and Y by continuous or even smooth functions. Last but not least, the metrics should be compatible with integral equations. It is therefore natural to chose metrics assigned to integral norms:

Definition 4

Let for functions f, g and $1\le q<\infty$

$$\begin{aligned} \Vert f\Vert _q&:=\left( \int \limits _{-\infty }^\infty \vert f(x)\vert ^q\,dx\right) ^{\frac{1}{q}}, \end{aligned}$$

(29)

$$\begin{aligned} \Vert f \Vert _{\infty }&:= \inf \left\{ c\ge 0\left| \int \limits _{-\infty }^\infty H(\vert f(x)\vert -c)\,dx=0\right. \right\} , \end{aligned}$$

(30)

$$\begin{aligned} d_q(f,g)&:=\Vert f-g\Vert _q \text { and } d_{\infty }(f,g):=\Vert f-g\Vert _{\infty } \end{aligned}$$

(31)

whenever the defining right-hand sides exists. For $1\le r\le \infty$ we denote by $L_r$ the set of all functions f for which $\Vert f\Vert _r$ is well-defined.

Since $X=L_1\cap L_{\infty }$ and it holds $\varphi _1-\varphi _2\in L_1\cap L_{\infty }$ for all $\varphi _1,\varphi _2\in Y$, $d_r$ defines a metric on X as well as on Y for every $1\le r \le \infty$. On $C_0$, $d_{\infty }$ defines a metric and it holds for all $f,g\in C_0$

$$\begin{aligned} d_{\infty }(f,g) =\Vert f-g\Vert _{\infty } = \mathop {{\textrm{max}}}\limits _{x\in \mathbb {R}} \vert f(x)-g(x)\vert . \end{aligned}$$

(32)

This motivates to consider the settings

$$\begin{aligned} X_a, \left( X,d_q\right) , \left( Y,d_r \right)&\ \text { and }\ C_a, \left( C_0,d_{\infty }\right) , \left( Y,d_s\right) \end{aligned}$$

(33)

for (23), where $1\le q,r,s\le \infty$ have to be further specified to assure the continuity of the used operators $\textbf{R}_{\sigma }$.

4 Regularization schemes by means of Fourier transform

Starting point for the construction of the regularization operators $\textbf{R}_{\sigma }$ is the following heuristical argument: Applying formally the convolution theorem of fourier transform ( [19]) to (23) yields

$$\begin{aligned} {\hat{\varphi }} = \widehat{\textbf{A}F}={\hat{k}}{\hat{F}}, \end{aligned}$$

(34)

where ${\hat{F}}$ denotes the spectral function of the function F, see definition 5 below.

Again formally, we get by (34)

$$\begin{aligned} {\hat{F}} = {\hat{k}}^{-1} {\hat{\varphi }}. \end{aligned}$$

(35)

Inserting an erroneous $\varphi _\varepsilon =\varphi +\varepsilon$ into the right-hand side of (35) provides an approximation of ${\hat{F}}$:

$$\begin{aligned} {\hat{F}}_\varepsilon := {\hat{k}}^{-1} {\hat{\varphi _\varepsilon }} = {\hat{F}} + {\hat{k}}^{-1} {\hat{\varepsilon }}. \end{aligned}$$

(36)

Because of the instability of (13), ${\hat{k}}^{-1}$ and consequently ${\hat{F}}_\varepsilon$ are unbounded. A stable solution can be found replacing ${\hat{k}}^{-1}$ by bounded functions $\kappa _{\sigma }$ with $\kappa _{\sigma } \rightarrow {\hat{k}}^{-1}$, $\sigma \rightarrow 0$. This suggests to choose the approximation operators $\textbf{R}_{\sigma }$ defined by $\textbf{R}_{\sigma }f := {\varvec{{\mathcal {F}}}}^{-1} \left( \kappa _{\sigma } {\hat{f}}\right)$ where ${\varvec{{\mathcal {F}}}}^{-1}$ denotes the inverse fourier transform.

We will follow this approach with slight modifications. For this purpose and for the sake of convenience, we first summarize the properties of the fourier transformation that are relevant for us, see [19] and [20].⁴

Definition 5

Let for a function f and $\rho \ge 0$ the functions ${\hat{f}}$, ${\check{f}}$, ${\hat{f}}_\rho$ and ${\check{f}}_\rho$ defined by

$$\begin{aligned} {\hat{f}} (\omega ):= \int \limits _{-\infty }^\infty f(u) \exp (-i\omega u)\,du, \ {\check{f}} (\omega ):= \frac{1}{2\pi }{\hat{f}}(-\omega ), \end{aligned}$$

(37)

$$\begin{aligned} {\hat{f}}_\rho (\omega ):= \int \limits _{-\rho }^\rho f(u) \exp (-i\omega u)\,du, \ {\check{f}}_\rho (\omega ):= \frac{1}{2\pi }{\hat{f}}_\rho (-\omega ) \end{aligned}$$

(38)

whenever the right-hand sides exists for every real $\omega$. Moreover, let the operators ${\varvec{{\mathcal {F}}}}$, ${\varvec{{\mathcal {F}}}}^*$, ${\varvec{{\mathcal {F}}}}_\rho$ and ${\varvec{{\mathcal {F}}}}^*_\rho$ be defined by

$$\begin{aligned} {\varvec{{\mathcal {F}}}}(f):= {\hat{f}},\ {\varvec{{\mathcal {F}}}}^*(f)={\check{f}},\ {\varvec{{\mathcal {F}}}}_\rho (f)={\hat{f}}_\rho \ \text { and }\ {\varvec{{\mathcal {F}}}}^*_\rho (f)={\check{f}}_\rho . \end{aligned}$$

(39)

${\hat{f}}$ is called the spectral function of the function f and the operator ${\varvec{{\mathcal {F}}}}$ is called the fourier transform.

Lemma 2

The following statements about the fourier transform or spectral functions, respectively, are true:

• 1. For $f\in L_1$, there hold

  $$\begin{aligned} {\hat{f}}, {\check{f}}\in C_0,\ \Vert {\hat{f}}\Vert _{\infty } \le \Vert f\Vert _1\ \text { and } \ \Vert {\check{f}}\Vert _{\infty } \le \frac{1}{2\pi }\Vert f\Vert _1. \end{aligned}$$

  (40)

• 2. For $f,g\in L_1$, it holds

  $$\begin{aligned} \widehat{f*g}= {\hat{f}} {\hat{g}}. \end{aligned}$$

  (41)

• 3. For continuous differentiable $f\in L_1$ with $f^\prime \in L_1$, it holds

  $$\begin{aligned} \widehat{f^\prime }(\omega )= i\omega {\hat{f}}(\omega ),\ \omega \in {\textbf{R}}. \end{aligned}$$

  (42)

• 4. For $f\in L_2$, there are (uniquely determined) ${\bar{f}},{\bar{f}}^*\in L_2$ with

  $$\begin{aligned} \lim \limits _{\rho \rightarrow \infty }\left\| {\hat{f}}_\rho -{\bar{f}}\right\| _2 = \lim \limits _{\rho \rightarrow \infty }\left\| {\check{f}}_\rho -{\bar{f}}^*\right\| _2 = 0. \end{aligned}$$

  (43)

  For the operators ${\varvec{{\mathcal {F}}}}_2$ and ${\varvec{{\mathcal {F}}}}_2^*$ defined for all $g\in L_2$ by ${\varvec{{\mathcal {F}}}}_2 (g):={\bar{g}}$ and ${\varvec{{\mathcal {F}}}}_2^* (g)={\bar{g}}^*$ and for all $f\in L_2$, there hold

  $$\begin{aligned} {\varvec{{\mathcal {F}}}}^*_2\left( {\bar{f}}\right) =f, \ {\varvec{{\mathcal {F}}}}_2\left( {\bar{f}}^*\right) =f,\ \left\| {\bar{f}}\right\| _2 =\sqrt{2\pi }\Vert f\Vert _2,\ \left\| {\bar{f}}^*\right\| _2 =\frac{1}{\sqrt{2\pi }}\Vert f\Vert _2. \end{aligned}$$

  (44)

• 5. For $f,g\in L_2$, it holds

  $$\begin{aligned} \overline{f*g}= {\bar{f}} {\bar{g}}. \end{aligned}$$

  (45)

• 6. For $f\in L_1\cap L_2$, there hold

  $$\begin{aligned} {\bar{f}} ={\hat{f}} \ \text { and } \ {\bar{f}}^*={\check{f}}. \end{aligned}$$

  (46)

Since k and thus $\varphi$ are not integrable, they also have no spectral function.⁵ This is why we cannot follow the argument based on (34) directly. A way out is to differentiate $\varphi =k*F$. Since $k^\prime$ is continuous and absolutely integrable, we get for absolutely integrable F

$$\begin{aligned} \varphi ^\prime = k^\prime *F. \end{aligned}$$

(47)

with continuous and absolutely integrable $\varphi ^\prime$.⁶ (40) can therefore be applied to (47). This yields

$$\begin{aligned} \widehat{\varphi ^\prime } = \widehat{k^\prime }{\hat{F}}. \end{aligned}$$

(48)

For $F\in X_a$⁷, (48) can be inverted by means of an inversion formula proved in [4]. It holds

$$\begin{aligned} \int \limits _{-\infty }^\infty F(u) \zeta (u)\,du = \int \limits _{-\infty }^\infty \varphi ^\prime (\xi )\frac{Z(\xi -i\pi RT)-Z(\xi +i\pi RT)}{2i\pi RT}\,d\xi \end{aligned}$$

(49)

for all globally analytical functions $\zeta$ defined on $\mathbb {C}$, whose derivatives grow polynomially at most on the real axis, where Z satisfies $Z^\prime =\zeta$.

Inserting the functions $\zeta _\omega =\exp (-i\omega \cdot )$, $\omega \in \mathbb {R}$ into (49) yields together with

$$\begin{aligned} \frac{Z_\omega (\xi -i\pi RT)-Z_\omega (\xi +i\pi RT)}{2i\pi RT}= -\textrm{sinhc}(\pi RT\omega )\exp (-i\omega \xi ),\ \xi \in \mathbb {R}\ \end{aligned}$$

(50)

the equation

$$\begin{aligned} {\hat{F}}(\omega ) = -\textrm{sinhc}(\pi RT\omega )\widehat{\varphi ^\prime }(\omega ),\ \omega \in \mathbb {R}, \end{aligned}$$

(51)

where $\textrm{sinhc}$ is the function defined by

$$\begin{aligned} \textrm{sinhc} (x):= {\left\{ \begin{array}{ll} \frac{\sinh (x)}{x},\ x\ne 0,\\ \ 1,\ \qquad x=0.\end{array}\right. } \end{aligned}$$

(52)

In applications, we have to deal with erroneous $\varphi _\varepsilon$. Since computing the derivative of an erroneous function itself is unstable, we have to get rid of the derivative in (51). To this end we fix some

$$\begin{aligned} F_0\in C_a \ \text { and }\ \varphi _0:=k*F_0 \end{aligned}$$

(53)

and set for arbitrary $G\in X$ and $\psi \in Y$

$$\begin{aligned} \Delta G:= G-F_0 \ \text { and } \ \Delta \psi :=\psi -\varphi _0. \end{aligned}$$

(54)

(51) then implies $\widehat{\Delta F} = -\textrm{sinhc}(\pi RT\cdot )\widehat{(\Delta \varphi )^\prime }$ for $F\in X_a$. As $\widehat{(\Delta \varphi )}$ and $\widehat{(\Delta \varphi )^\prime }$ are continuous and absolutely integrable, (42) can be applied. Thus we get

$$\begin{aligned} {\hat{F}} (\omega ) = {\hat{F}}_0(\omega )+ a(\omega )\widehat{\Delta \varphi }(\omega ) \text { where } a(\omega ):= \frac{\sinh (\pi RT\omega )}{i\pi RT}, \ \omega \in \mathbb {R}. \end{aligned}$$

(55)

Instead of the inappropriate eq. (35), we use (55) to construct the approximation operators $\textbf{R}_{\sigma }$.

Lemma 3

Let b be a function such that b and ba are bounded and absolutely integrable. Let $\textbf{R}$ be the operator defined on Y by

$$\begin{aligned} \textbf{R}(\psi ):= {\varvec{{\mathcal {F}}}}^*\left( b{\hat{F}}_0 +ba\widehat{\Delta \psi }\right) , \end{aligned}$$

(56)

then there hold for all $\psi ,\psi _1,\psi _2\in Y$

$$\begin{aligned} \textbf{R}(\psi )&\in C_0\cap L_2, \end{aligned}$$

(57)

$$\begin{aligned} \left\| \textbf{R}\left( \psi _1\right) -\textbf{R}\left( \psi _2\right) \right\| _{\infty }&\le \frac{1}{2\pi } \Vert ba\Vert _1\left\| \psi _1-\psi _2\right\| _1, \end{aligned}$$

(58)

$$\begin{aligned} \left\| \textbf{R}\left( \psi _1\right) -\textbf{R}\left( \psi _2\right) \right\| _2&\le \frac{1}{\sqrt{2\pi }} \Vert ba\Vert _2\left\| \psi _1-\psi _2\right\| _1. \end{aligned}$$

(59)

Proof

(57) results from (40), (46) and the fact that $b{\hat{F}}_0 +ba\widehat{\Delta \psi }$ is a bounded, absolutely integrable and thus also square-integrable function.

Let now $r=1$ for $q=\infty$ and $r=2$ for $q=2$ and let $\chi :=\psi _1-\psi _2$, then we get by (40), (44) and (46)

$$\begin{aligned} \left\| \textbf{R}\left( \psi _1\right) -\textbf{R}\left( \psi _2\right) \right\| _q \le \frac{1}{(2\pi )^{\frac{1}{r}}}\left\| ba{\hat{\chi }}\right\| _{r} \le \frac{1}{(2\pi )^{\frac{1}{r}}}\Vert ba\Vert _{r}\left\| {\hat{\chi }}\right\| _{\infty } \le \frac{1}{(2\pi )^{\frac{1}{r}}}\Vert ba\Vert _r\left\| \chi \right\| _{1}. \end{aligned}$$

$\square$

Lemma 3 allows constructing regularization schemes for the settings $X_a$, $(X,d_2)$, $(Y,d_1)$ and $C_a$, $(C_0,d_{\infty })$, $(Y,d_1)$. In the first case - the general case - the restriction of boundedness for functions belonging to $X_a$ as well as X can be relaxed, cf. footnote 7. Instead of the boundedness only square-integrability of the functions in question is needed.

Definition 6

Let $X_{a,2}$ be the set of all real-valued square-integrable functions satisfying (9) and (24).

As a final preparation for the proof of the main statement of this section and for the later treatment of ideal adsorbents, we introduce the notion of a (generalized) dirac sequence.

Definition 7

A family $\left( \gamma _\tau \right) _{\tau >0}$ of real-valued functions is called a (generalized) dirac-sequence if it satisfies

$$\begin{aligned} \gamma _\tau \ge 0,\ \Vert \gamma _\tau \Vert _1=1\text { for all }\tau>0\ \text { and }\ \lim \limits _{\tau \rightarrow 0} \int \limits _{\vert x\vert \ge \rho } \gamma _\tau (x)\,dx= 0 \text { for all } \rho >0. \end{aligned}$$

We need the following statements:

Lemma 4

Let $\left( \gamma _\tau \right) _{\tau >0}$ be a dirac-sequence and $f\in C_0$, then there hold $f*\gamma _\tau \in C_0$ and

$$\begin{aligned} \lim \limits _{\tau \rightarrow 0}\left\| f -f*\gamma _\tau \right\| _{\infty } =0. \end{aligned}$$

(60)

Proof

As $f\in C_0$, there hold for $\tau >0$:

$$\begin{aligned} \lim \limits _{x\rightarrow x_0} f(x-y)\gamma _\tau (y)&= f(x_0-y)\gamma _\tau (y)\text { for all }x_0, y\in \mathbb {R}, \end{aligned}$$

(61)

$$\begin{aligned} \lim \limits _{\vert x\vert \rightarrow \infty } f(x-y)\gamma _\tau (y)&= 0 \text { for all }x_0, y\in \mathbb {R}, \end{aligned}$$

(62)

$$\begin{aligned} \left| f(x-y)\gamma _\tau (y)\right|&\le \Vert f\Vert _{\infty } \gamma _\tau (y) \text { for all } x,y\in \mathbb {R}. \end{aligned}$$

(63)

$f*\gamma _\tau \in C_0$ is then a consequence of the dominated convergence theorem [22].

Let now $\epsilon >0$. Due to the uniform continuity of $f\in C_0$, there is a $\rho _\epsilon >0$ such that

$$\begin{aligned} \vert f(x-y)-f(x)\vert \le \epsilon \text { for all } x\in \mathbb {R} \ \text { and }\ \vert y\vert \le \rho _\epsilon . \end{aligned}$$

Hence, it holds

$$\begin{aligned} \left| f(x) -f*\gamma _\tau (x)\right|&\le \int \limits _{-\rho _\epsilon }^{\rho _\epsilon } \vert f(x)-f(x-y)\vert \gamma _\tau (y)\,dy +\int \limits _{\vert y\vert \ge \rho _\epsilon } \vert f(x)-f(x-y)\vert \gamma _\tau (y)\,dy \nonumber \\&\le \epsilon + 2\Vert f\Vert _{\infty } \int \limits _{\vert y\vert \ge \rho _\epsilon } \gamma _\tau (y)\,dy \text { for all } x\in \mathbb {R} \text { and }\tau >0. \end{aligned}$$

(64)

Choose now $\tau _\epsilon$ such that $\int \limits _{\vert y\vert \ge \rho _\epsilon } \gamma _\tau (y)\,dy \le \epsilon$ for all $0<\tau \le \tau _\epsilon$. It follows by (64)

$$\begin{aligned} \left| f(x) -f*\gamma _\tau (x)\right| _{\infty }\le \left( 1+2\Vert f\Vert _{\infty }\right) \epsilon \text { for all } x\in \mathbb {R} \text { and } 0<\tau \le \tau _\epsilon \end{aligned}$$

and thus $\left\| f -f*\gamma _\tau \right\| _{\infty }\le \left( 1+2\Vert f\Vert _{\infty }\right) \epsilon$ for all $0<\tau \le \tau _\epsilon$ , which proves (60). $\square$

Now we are able to specify general regularization schemes for a stable solution of (13) as the main statement of this section.

Theorem 5

• 1. Let $\left( b_{\sigma }\right) _{\sigma >0}$ be a family of real-valued bounded and absolutely integrable functions with the following properties:

  $$\begin{aligned} b_{\sigma } a\in L_1\cap L_{\infty },\ 0\le b_{\sigma }\le 1 \text { for all } \sigma >0 \text { and } \lim \limits _{\sigma \rightarrow 0} b_{\sigma }= 1. \end{aligned}$$

  (65)

  Then the family $\left( \textbf{R}_{\sigma }\right) _{\sigma >0}$ of operators defined on Y by

  $$\begin{aligned} \textbf{R}_{\sigma }(\psi ):= {\varvec{{\mathcal {F}}}}^*\left( b_{\sigma }{\hat{F}}_0 +b_{\sigma } a\widehat{\Delta \psi }\right) \end{aligned}$$

  (66)

  provides a regularization scheme for (23) with respect to the setting $X_{a,2}$, $\left( L_2,d_2\right)$, $\left( Y,d_1\right)$.

• 2. If $\left( b_{\sigma }\right) _{\sigma >0}$ is generated by spectral functions of a dirac sequence $\left( \beta _{\sigma }\right) _{\sigma >0}$, i.e. it holds $b_{\sigma } =\widehat{\beta _{\sigma }}$, $\sigma >0$, then $\left( \textbf{R}_{\sigma }\right) _{\sigma >0}$ provides a regularization scheme for (23) with respect to the setting $C_a$,$\left( C_0,d_{\infty }\right)$, $\left( Y,d_1\right)$.

Proof

It leaves to show (17) for the different settings.

1. Let $\varphi =\textbf{A}F$ for some $F\in X_{a,2}$. According to Lemma 2 and Eq. (65), there hold $\textbf{R}_{\sigma }(\varphi )\in L_2$ and $b_{\sigma } F\in L_2$. (44) then implies

$$\begin{aligned} \Vert F -\textbf{R}_{\sigma }(\varphi )\Vert _2 = \frac{1}{\sqrt{2\pi }} \left\| (1-b_{\sigma }){\hat{F}}\right\| _2. \end{aligned}$$

(67)

Again by (65), we get $(1-b_{\sigma })^2 \vert {\hat{F}}\vert ^2\le \vert {\hat{F}}\vert ^2$ and $\lim \limits _{\sigma \rightarrow 0} (1-b_{\sigma })^2 \vert {\hat{F}}\vert ^2=0$. The dominated convergence theorem assures then

$$\begin{aligned} \lim \limits _{\sigma \rightarrow 0}\Vert F -\textbf{R}_{\sigma }(\varphi )\Vert _2 = \frac{1}{\sqrt{2\pi }} \lim \limits _{\sigma \rightarrow 0} \left\| (1-b_{\sigma }){\hat{F}}\right\| _2 =0. \end{aligned}$$

2. Let $F\in C_a$ and $b_{\sigma }=\widehat{\beta _{\sigma }}$, then (44), (45) and (46) imply

$$\begin{aligned} \textbf{R}_{\sigma }(\varphi ) = F*\beta _{\sigma } . \end{aligned}$$

(68)

Lemma 4 now provides

$$\begin{aligned} \lim \limits _{\sigma \rightarrow 0}\Vert F -\textbf{R}_{\sigma }(\varphi )\Vert _{\infty } = \lim \limits _{\sigma \rightarrow 0} \left\| F-F*\beta _{\sigma }\right\| _{\infty } =0. \end{aligned}$$

$\square$

5 Regularizations

We are looking now for regular strategies (cf. (18)) with respect to the settings $X_{a,2}$, $\left( L_2,d_2\right)$, $\left( Y,d_1\right)$ and $C_a$, $\left( C_0,d_{\infty }\right)$, $\left( Y,d_1\right)$.

Let $\varphi _\varepsilon =\varphi +\varepsilon$ be an erroneous version $\varphi =\textbf{A}F$ with the function of errors $\varepsilon$. By (58), (59), (67) and (68) we get the estimates

$$\begin{aligned} \left\| F -\textbf{R}_{\sigma }(\varphi _\varepsilon )\right\| _2&\le \left\| F -\textbf{R}_{\sigma }(\varphi )\right\| _2 + \left\| \textbf{R}_{\sigma }(\varphi ) -\textbf{R}_{\sigma }(\varphi _\varepsilon )\right\| _2\nonumber \\&\le \frac{\left\| (1-b_{\sigma }){\hat{F}}\right\| _2+\left\| b_{\sigma } a\right\| _2\Vert \varepsilon \Vert _1}{\sqrt{2\pi }} \text { for all } F\in X_{a,2}, \end{aligned}$$

(69)

$$\begin{aligned} \left\| F -\textbf{R}_{\sigma }(\varphi _\varepsilon )\right\| _{\infty }&\le \left\| F -F*\beta _{\sigma }\right\| _{\infty } + \frac{\left\| b_{\sigma } a\right\| _1\Vert \varepsilon \Vert _1}{2\pi } \text { for all } F\in C_a. \end{aligned}$$

(70)

Hence, by Theorem 5 every choice $\sigma =\sigma (\delta )$, $\delta >0$ with

$$\begin{aligned} \lim \limits _{\delta \rightarrow 0} \sigma (\delta ) = 0&\ \text { and }\ \lim \limits _{\delta \rightarrow 0} \Vert b_{\sigma (\delta )} a\Vert _r\delta = 0 \end{aligned}$$

(71)

is regular with respect to the setting $X_{a,2}$, $\left( L_2,d_2\right)$, $\left( Y,d_1\right)$ for $r=2$ and $C_a$, $\left( C_0,d_{\infty }\right)$, $\left( Y,d_1\right)$ for $r=1$.

As the first term in the nominator on the right-hand side of (69) shows, the rate of convergence depends on the decaying behavior of ${\hat{F}}$. Unfortunately, we have no a priori information about this behavior. Due to the averaging effects mentioned in Sect. 1, we can suppose a certain regularity of the AED and thus a certain decay behavior of its spectral function. For example, $F,F^\prime \in L_2$ implies ${\hat{F}} (\omega )=o\left( h_1(\omega )\right)$, $\vert \omega \vert \rightarrow \infty$ ( [20], Satz V.2.14), where for $n\in \mathbb {N}$ the function $h_n$ is defined by

$$\begin{aligned} {\hat{h}}_n(\omega ):=\frac{1}{\left( 1+\omega ^2\right) ^{\frac{n}{2}}} ,\ \omega \in {\textbf{R}}. \end{aligned}$$

(72)

It is reasonable to assume that the decay behavior of the spectral function of the AED is not worse than $O\left( h_1(\omega )\right)$, $\vert \omega \vert \rightarrow \infty$. This includes piecewise differentiable functions with absolutely integrable derivatives such as step functions.

Just as in (69), we need information about the regularity of the continuous AED F in (70) in order to estimate $\left\| F -F*\beta _{\sigma }\right\| _{\infty }$. For a continuous AED it is likely that it is as least as regular as a triangular function. Hence, ${\hat{F}}$ is absolutely integrable and decays at least as $O\left( h_2(\omega )\right)$, $\vert \omega \vert \rightarrow \infty$. In this case, it holds $\left\| F-F*\beta _{\sigma }\right\| _{\infty } \le \frac{1}{2\pi } \left\| \left( 1-\widehat{\beta _{\sigma }}\right) {\hat{F}}\right\| _1$ and thus by (70)

$$\begin{aligned} \left\| F-\textbf{R}_{\sigma }\left( \varphi _\varepsilon \right) \right\| _{\infty } \le \frac{\left\| \left( 1-\widehat{\beta _{\sigma }}\right) {\hat{F}}\right\| _1 + \left\| \widehat{\beta _{\sigma }} a\right\| _1\Vert \varepsilon \Vert _1}{2\pi }. \end{aligned}$$

In summary, we generally expect

$$\begin{aligned} \left\| F-\textbf{R}_{\sigma }\left( \varphi _\varepsilon \right) \right\| _q \sim \frac{\left\| \left( 1-b_{\sigma }\right) h_n\right\| _r + \left\| b_{\sigma } a\right\| _r\Vert \varepsilon \Vert _1}{(2\pi )^{\frac{1}{r}}} \end{aligned}$$

(73)

with $q=r=2$, $n=1$ for $F\in X_{a,2}$, and $q=\infty$, $r=1$, $n=2$ for $F\in C_a$.

(73) motivates to choose a strategy keeping the balance between the approximation error $\left\| \left( 1-b_{\sigma }\right) h_n\right\| _r$ and the data error $\left\| b_{\sigma } a\right\| _r\Vert \varepsilon \Vert _1$.

We prove now that with this strategy all $F\in X_{a,2}$ as well as $F\in C_a$ can be reconstructed.

Theorem 6

In addition to the assumptions made in Theorem 5, let $\left( b_{\sigma }\right) _{\sigma >0}$ satisfy

$$\begin{aligned} \left\| b_{\sigma ^*}\right\| _1&<\left\| b_{\sigma }\right\| _1 \text { if } 0<\sigma ^*<\sigma , \end{aligned}$$

(74)

$$\begin{aligned} \lim \limits _{\sigma \rightarrow \infty } b_{\sigma } (\omega )&= 0 \text { for all } \omega \ne 0, \end{aligned}$$

(75)

$$\begin{aligned} \lim \limits _{\sigma \rightarrow \sigma _0} b_{\sigma }&= b_{\sigma _0} \text { for all } \sigma _0>0. \end{aligned}$$

(76)

Let for $n=1,2$ and $\delta >0$, $\sigma _n^{-} (\delta )$ be the zero of the function $f_{n,\delta }^{-}$ defined for $\sigma >0$ by

$$\begin{aligned} f_{n,\delta }^{-} (\sigma ):= \left\| b_{\sigma } a\right\| _{3-n}\delta - \left\| \left( 1-b_{\sigma }\right) h_n\right\| _{3-n}. \end{aligned}$$

(77)

Then the family of operators $\textbf{R}_{\sigma }$, $\sigma >0$ as defined in (66) form together with the strategy $\sigma =\sigma _n^{-}(\delta )$ a regularization for (23) with respect to the settings $X_{a,2}$, $\left( L_2,d_2\right)$, $\left( Y,d_1\right)$ for $n=1$ and $C_a$, $\left( C_0,d_{\infty }\right)$, $\left( Y,d_1\right)$ for $n=2$.

Proof

At first we have to show that $\sigma _n^{-}(\delta )$ is well-defined. Together with (74) - (76) and the dominated convergence theorem (as well as the monotone convergence theorem), the assumptions made in Theorem 5 imply that $f_{n,\delta }^{-}$ is continuous on $(0,\infty )$ and satisfies

$$\begin{aligned} \lim \limits _{\sigma \rightarrow 0} f_{n,\delta }^{-}(\sigma )&= \lim \limits _{\sigma \rightarrow 0} \left\| b_{\sigma } a\right\| _{3-n} \ge \lim \limits _{\Omega \rightarrow \infty } \int \limits _{-\Omega }^{\Omega } \vert a(\omega )\vert ^{3-n}\,d\omega = \infty , \end{aligned}$$

(78)

$$\begin{aligned} \lim \limits _{\sigma \rightarrow \infty }f_{n,\delta }^{-}(\sigma )&= -\left\| h_n\right\| _{3-n} <0. \end{aligned}$$

(79)

Since the functions $\sigma \mapsto \left\| b_{\sigma } a\right\| _{3-n}$ or $\sigma \mapsto \left\| \left( 1-b_{\sigma }\right) h_n\right\| _{3-n}$ are strictly monotonically decreasing and strictly monotonically increasing, respectively, $f_{n,\delta }^{-}$ has at most one zero. With (78), (79) and the continuity of $f_{n,\delta }^{-}$, this implies the well-definedness of $\sigma _n^{-}(\delta )$.

It remains to show (71). To this end, we first verify that it holds for all $0<\delta _1<\delta _2$

$$\begin{aligned} \sigma _n^{-} \left( \delta _1\right) <\sigma _n^{-}\left( \delta _2\right) . \end{aligned}$$

(80)

Suppose not, then we get the contradiction

$$\begin{aligned} \left\| \left( 1-b_{\sigma _n^{-} (\delta _1)}\right) h_n\right\| _{3-n} > \left\| \left( 1-b_{\sigma _n^{-} (\delta _1)}\right) h_n\right\| _{3-n} \end{aligned}$$

due to

$$\begin{aligned} \left\| \left( 1-b_{\sigma _n^{-} (\delta _1)}\right) h_n\right\| _{3-n}&\ge \left\| \left( 1-b_{\sigma _n^{-} (\delta _2)}\right) h_n\right\| _{3-n} = \left\| b_{\sigma _n^{-} (\delta _2)} a\right\| _{3-n}\delta _2\\&>\left\| b_{\sigma _n^{-} (\delta _2)} a\right\| _{3-n}\delta _1 \ge \left\| \left( 1-b_{\sigma _n^{-} (\delta _1)}\right) h_n\right\| _{3-n}. \end{aligned}$$

By (80), there is a $\sigma ^*_n\ge 0$ with $\lim \limits _{\delta \rightarrow 0} \sigma _n^{-}(\delta )=\sigma ^*_n$. The assumption $\sigma ^*_n>0$ leads to the contradiction

$$\begin{aligned} 0&=\lim \limits _{\delta \rightarrow 0} f_{n,\delta }^{-}\left( \sigma _n^{-}(\delta )\right) \overset{(76)}{=} \left\| b_{\sigma _n^*} a\right\| _{3-n}\cdot 0 - \left\| \left( 1-b_{\sigma _n^*}\right) h_n\right\| _{3-n} \\&= -\left\| \left( 1-b_{\sigma _n^*}\right) h_n\right\| _{3-n} <0. \end{aligned}$$

Thus, it holds

$$\begin{aligned} \lim \limits _{\delta \rightarrow 0} \sigma _n^{-}(\delta )= 0\ \text { implying } \lim \limits _{\delta \rightarrow 0} \left\| b_{\sigma _n^{-}(\delta )} a\right\| _{3-n} =\lim \limits _{\delta \rightarrow 0} f_{n,\delta }^{-}\left( \sigma _n^{-}(\delta )\right) =0, \end{aligned}$$

(81)

which shows (71) and completes the proof. $\square$

A second strategy is to minimize the right-hand side of (73) for given error level $\delta >0$, i.e. to minimize the function $f_{n,\delta }^+$ defined for $\sigma >0$ by

$$\begin{aligned} f_{n,\delta }^+(\sigma ):= \left\| b_{\sigma } a\right\| _{3-n}\delta + \left\| \left( 1-b_{\sigma }\right) h_n\right\| _{3-n}. \end{aligned}$$

(82)

Theorem 7

The premises of Theorem 6 may apply. Let for $\delta >0$, $\sigma _n^+ (\delta )$ be the smallest minimizer of $f_{n,\delta }^+$ if there is a $\sigma _n^+>0$ with

$$\begin{aligned} f_{n,\delta }^+ \left( \sigma ^*\right) < \left\| h_n\right\| _{3-n}. \end{aligned}$$

(83)

Otherwise let $\sigma _n^+(\delta )=\sigma _n^{-} (\delta )$. Then $\sigma =\sigma _n^+ (\delta )$ is a regular strategy for choosing $\textbf{R}_{\sigma }$ with respect to the settings $X_{a,2}$, $\left( L_2,d_2\right)$, $\left( Y,d_1\right)$ for $n=1$ and $C_a$, $\left( C_0,d_{\infty }\right)$, $\left( Y,d_1\right)$ for $n=2$.

Proof

Again, we first show the well-definedness of $\sigma _n^+ (\delta )$. As $\sigma _n^{-} (\delta )$ is well-defined it suffices to prove that $f_{n,\delta }^+$ attains a minimum whenever (83) holds. The continuity of $f_{n,\delta }^+$ then guarantees the existence of a smallest minimizer. The statement about the minimum follows from $\lim \limits _{\sigma \rightarrow 0} f_{n,\delta }^+ (\sigma ) =\infty$, $\lim \limits _{\sigma \rightarrow \infty } f_{n,\delta }^+ (\sigma ) = \left\| h_n\right\| _{3-n}$ and the continuity of $f_{n,\delta }^+$.

For the validity of (71) we argue as follows: By definition of $\sigma _n^+ (\delta )$, it holds

$$\begin{aligned}0\le f_{n,\delta }^+\left( \sigma _n^+(\delta )\right) \le 2 \left\| b_{\sigma _n^{-}(\delta )}a\right\| _{3-n} \delta . \end{aligned}$$

(81) then yields $\lim \limits _{\delta \rightarrow 0}f_{n,\delta }^+\left( \sigma _n^+(\delta )\right) =0$, hence we get

$$\begin{aligned}\lim \limits _{\delta \rightarrow 0} \left\| b_{\sigma _n^+ (\delta )} a\right\| _{3-n}= 0 \ \text { and }\ \lim \limits _{\delta \rightarrow 0} \left\| \left( 1-b_{\sigma _n^+ (\delta )}\right) h_n\right\| _{3-n}= 0 . \end{aligned}$$

The last limit implies $\lim \limits _{\delta \rightarrow 0}\sigma _n^+ (\delta ) = 0$, which completes the proof. $\square$

Remark 1

Note that $\sigma _n^+ (\delta )$ is the smallest minimizer of $f_{n,\delta }^+$ for

$$\begin{aligned} 0<\delta < \sup \limits _{\sigma >0} \frac{\left\| h_n\right\| _{3-n}-\left\| \left( 1-b_{\sigma }\right) h_n\right\| _{3-n}}{\left\| b_{\sigma } a\right\| _{3-n}}. \end{aligned}$$

(84)

6 Reconstructing averages

In Sect. 5, we argued that in most cases the reconstruction error is not worse in terms of magnitude than the right-hand side of (73). This is no longer true for sharply localized peaks. Here, the reconstruction error is underestimated. If we are only interested in a certain resolution of the peaks even for vanishing level of the measurement error, i.e. for $\delta \rightarrow 0$, then we can provide a general estimation of the reconstruction error.

More precisely, instead of the AED F, we want to reconstruct an averaged function $F*\gamma _{\rho }$, where $\gamma _{\rho }$ is a real-valued continuous function with

$$\begin{aligned} \gamma _{\rho }\ge 0,\ \left\| \gamma _{\rho }\right\| _1=1\ \text { and }\ \textrm{supp}\gamma _{\rho } = [-\rho ,\rho ] \end{aligned}$$

(85)

for fixed $\rho >0$. To deal with averages $F*\gamma _{\rho }$ is motivated by lemma 4 and the following.

Lemma 8

Let $\gamma _{\rho }$ as stated above, then the subsequent assertions are true:

• 1. For $F\in L_1$, $F*\gamma _{\rho }$, there hold $F*\gamma _{\rho }\in C_0$ and $\left\| F*\gamma _{\rho }\right\| _{\infty } \le \Vert F\Vert _1\left\| \gamma _{\rho }\right\| _{\infty }$.

• 2. For real-valued nonnegative $F\in L_1$, there hold $F*\gamma _{\rho }\ge 0$ and $\left\| F*\gamma _{\rho }\right\| _1=\Vert F\Vert _1$.

• 3. For Hölder continuous F with Hölder exponent $0<\alpha \le 1$ and Hölder constant $L_\alpha \ge 0$, it holds

  $$\begin{aligned} \left\| F-F*\gamma _{\rho }\right\| _{\infty }\le L_\alpha \rho ^{\alpha }. \end{aligned}$$

  (86)

Proof

1. The assertions follow by $F*\gamma _{\rho }=\gamma _{\rho }*F$, (61), (62) and (63), where f is substituted by $\gamma _{\rho }$ and $\gamma _\tau$ by F and the dominated convergence theorem.

2. $F*\gamma _{\rho }\ge 0$ is obvious. Due to Fubini’s theorem ( [22]), we get

$$\begin{aligned} \left\| F*\gamma _{\rho }\right\| _1= \int \limits _{-\infty }^{\infty } \int \limits _{-\infty }^{\infty } F(x-y)\gamma _{\rho } (y)\,dy\,dx = \int \limits _{-\infty }^{\infty }F(x)\,dx \int \limits _{-\infty }^{\infty }\gamma _{\rho } (y)\,dy =\Vert F\Vert _1. \end{aligned}$$

3. (86) is due to the fact that for all $x\in \mathbb {R}$ holds

$$\begin{aligned} \left| F(x) - F*\gamma _{\rho } (x)\right| \le \int \limits _{-\infty }^{\infty } \vert F(x)- F(x-y)\vert \gamma _{\rho } (y)\,dy\le L_\alpha \int \limits _{-\rho }^{\rho } \vert y \vert ^\alpha \gamma _{\rho } (y)\,dy \le L_\alpha \rho ^\alpha , \end{aligned}$$

which completes the proof. $\square$

From lemma 8 and (85) follows directly:

Corollary 9

Let $F\in X_a$, then there hold $F*\gamma _{\rho }\in C_0$, $F*\gamma _{\rho }\ge 0$, $\left\| F*\gamma _{\rho }\right\| _1= 1$, $\left\| F*\gamma _{\rho }\right\| _1\le \left\| \gamma _{\rho }\right\| _{\infty }$ and $\textrm{supp}F*\gamma _{\rho } \subset [u_{{\textrm{min}}}-\rho ,u_{{\textrm{max}}}+\rho ]$.

Furthermore, $\rho$ can be seen as a measure of how well $F*\gamma _{\rho }$ resolves the structure of F. This is justified by the following observation:

Let $F=\sum \limits _{j=1}^n F_j$ with $F_j\ge 0$, $\Vert F_j\Vert _1>0$, $\sum \limits _{j=1}^n \Vert F_j\Vert _1=1$ and $\textrm{supp}F_j = [u_j,u^*_j]$, where $u_{{\textrm{min}}}\le u_1<u_1^*<u_2< u_2^*<\cdots<u_n<u_n^*\le u_{{\textrm{max}}}$, then there hold for $F*\gamma _{\rho }=\sum \limits _{j=1}^n F_j*\gamma _{\rho }$:

$$\begin{aligned}F_j*\gamma _{\rho }\ge 0,\ \Vert F_j*\gamma _{\rho }\Vert _1>0,\ \sum \limits _{j=1}^n \Vert F_j*\gamma _{\rho }\Vert _1=1,\ \textrm{supp}F_j \subset [u_j-\rho ,u^*_j+\rho ]. \end{aligned}$$

Consequently, for $\rho <\mathop {{\textrm{min}}}\limits _{j=1,..,n-1}\frac{u_{j+1}-u_j^*}{2}$ and $k\ne l$, we get

$$\begin{aligned} \textrm{supp}F_k*\gamma _{\rho }\cap \textrm{supp}F_l*\gamma _{\rho }=\emptyset . \end{aligned}$$

That means, if F has a clustering or “island-chain-like” structure, then $F*\gamma _{\rho }$ allows to identify the clusters and to estimate their lengths if the minimal distance between two clusters is larger than $2\rho$.

An advantage of reconstructing averages is that we can relax the integrability conditions on F and include the case of an ideal adsorbent. Although there is no integrable AED belonging to a CAED $\chi$ of an ideal adsorbent as in (4), it is possible to assign an improper AED to $\chi$ that can be taken in a reasonable sense as a limit of integrable AED’s.

Since $F\in L_1$ is the density of the CAED $\chi$, cf. (2), if and only if

$$\begin{aligned} \int \limits _{-\infty }^\infty f(u)\,d\chi (u)= \int \limits _{-\infty }^\infty F(u) f(u)\,du =:(F,f) \end{aligned}$$

(87)

is valid for all bounded continuous functions f, we generalize the notion of density as follows.

Definition 8

Let $C_b$ be the space of all bounded continuous functions and let $C_b^*$ be the space of all linear functionals $\varvec{\Phi }$ that assign to every $f\in C_b$ a complex number. Instead of $\varvec{\Phi }(f)$ we write $\langle \varvec{\Phi },f\rangle$. The convolution of $\varvec{\Phi }\in C_b^*$ with a function $f\in C_b$ is the function $\varvec{\Phi }*f$ defined by

$$\begin{aligned} \varvec{\Phi }*f (x) := \langle \varvec{\Phi }, f(x-\cdot )\rangle , \ x\in {\textbf{R}}. \end{aligned}$$

(88)

The spectral function $\hat{\varvec{\Phi }}$ of $\varvec{\Phi }\in C_b^*$ is defined by

$$\begin{aligned} \hat{\varvec{\Phi }} (\omega ) := \langle \varvec{\Phi }, \exp (-i\omega \cdot )\rangle , \ \omega \in {\textbf{R}}. \end{aligned}$$

(89)

Let $\chi$ be a real-valued monotonically increasing function such that the Stieltjes integral $\int \limits _{-\infty }^{\infty } f(x)\,d\chi (x)$ exists for every $f\in C_b$. The general density of $\chi$ is the functional $\varvec{\chi }^\prime \in C_b^*$ defined for $f\in C_b$ by

$$\begin{aligned} \langle \varvec{\chi }^\prime ,f\rangle := \int \limits _{-\infty }^{\infty } f(x)\,d\chi (x). \end{aligned}$$

(90)

⁸

For $\chi$ as in (4), we get

$$\begin{aligned} \varvec{\chi }^\prime = \sum \limits _{j=1}^n a_j\varvec{\delta }_{u_j}, \end{aligned}$$

(91)

where for $v\in \mathbb {R}$, $\varvec{\delta }_v\in C_b^*$ is defined by

$$\begin{aligned} \left\langle \varvec{\delta }_v, f\right\rangle := f(v),\ f\in C_b. \end{aligned}$$

(92)

⁹

Let $\varphi$ be the transformed total isotherm corresponding to $\chi$ as in (4), i.e.

$$\begin{aligned} \varphi (\xi ) = \sum \limits _{j=1}^n a_j k(\xi -u_j),\ \xi \in \mathbb {R} \end{aligned}$$

(93)

and let $\varphi _\rho$ be the transformed total isotherm corresponding to $F_\rho$ defined by

$$\begin{aligned} F_\rho (u) := \sum \limits _{j=1}^n a_j \gamma _{\rho } (u-u_j), \ u\in {\textbf{R}}. \end{aligned}$$

(94)

As $\left( \gamma _{\rho }\right) _{\rho >0}$ is a dirac-sequence, it holds for every $f\in C_b$

$$\begin{aligned} \lim \limits _{\rho \rightarrow 0}\left( F_\rho ,f\right) = \lim \limits _{\rho \rightarrow 0} \int \limits _{-\infty }^\infty F_\rho (u) f(u)\,du =\sum \limits _{j=1}^n a_j f(u_j) =\left\langle \sum \limits _{j=1}^n a_j\varvec{\delta }_{u_j}, f \right\rangle \end{aligned}$$

(95)

and hence for all $\xi ,\omega \in \mathbb {R}$ results

$$\begin{aligned} \lim \limits _{\rho \rightarrow 0}\varphi _\rho (\xi )&= \lim \limits _{\rho \rightarrow 0}k*F_\rho (\xi ) =\varphi (\xi ), \end{aligned}$$

(96)

$$\begin{aligned} \lim \limits _{\rho \rightarrow 0} \widehat{F_\rho } (\omega )&= \sum \limits _{j=1}^n a_j \exp \left( -i u_j\omega \right) =\widehat{\varvec{\chi }^\prime } (\omega ). \end{aligned}$$

(97)

Analogous to (25), for fixed $\rho ^*>0$ and $0<\rho \le \rho ^*$ the functions $\left| \Delta \varphi _\rho \right|$ and $\left| \Delta \varphi \right|$ are bounded by the positive and integrable function $g_{\rho ^*}$ defined by

$$\begin{aligned}g_{\rho *}(\xi ) :=k(\xi -\rho ^*-u_{{\textrm{max}}})-k(\xi +\rho ^*-u_{{\textrm{min}}}),\ \xi \in \mathbb {R}. \end{aligned}$$

This implies together with (96), (97) and the dominated convergence theorem

$$\begin{aligned} \widehat{\varvec{\chi }^\prime }(\omega )&= \lim \limits _{\rho \rightarrow 0} \widehat{F_\rho } (\omega ) =\widehat{F_0}(\omega ) +a(\omega )\lim \limits _{\rho \rightarrow 0 } \widehat{\Delta \varphi _\rho } (\omega )\nonumber \\&= \widehat{F_0}(\omega ) +a(\omega )\widehat{\Delta \varphi } (\omega ),\ \omega \in \mathbb {R}. \end{aligned}$$

(98)

Equation (55) thus also applies to the spectral function of the generalized density of an ideal adsorbent. From (96) and (98), we now get for $\rho >0$

$$\begin{aligned} \widehat{F_\rho } (\omega )&= \sum \limits _{j=1}^n a_j \exp \left( -iu_j\omega \right) \widehat{\gamma _{\rho }} (\omega )= \widehat{\varvec{\chi }^\prime } (\omega )\widehat{\gamma _{\rho }} (\omega )= \left[ \widehat{F_0}(\omega ) +a(\omega )\widehat{\Delta \varphi } (\omega )\right] \widehat{\gamma _{\rho }} (\omega ),\ \omega \in \mathbb {R}. \end{aligned}$$

(99)

Equation (99) means that the knowledge of the total isotherm of an ideal adsorbent allows to reconstruct for every $\rho >0$ an approximation $F_\rho = \varvec{\chi }^\prime *\gamma _{\rho }$ of the general density $\varvec{\chi }^\prime$ of an ideal adsorbent consisting of so-called $\delta$-peaks. Note that for $\rho < \rho _{\sup }:=\mathop {{\textrm{min}}}\limits _{j=1,...,n-1}\frac{u_{j+1}-u_j}{2}$, it holds

$$\begin{aligned} F_\rho =\sum \limits _{j=1}^n F_{\rho ,j} \text { with } F_{\rho ,j}\ge 0, \left\| F_{\rho ,j}\right\| _1>0 \text { and } \textrm{supp}F_k\cap \textrm{supp}F_l=\emptyset \text { for } k\ne l. \end{aligned}$$

Since $F_{\rho ,j} (u) = a_j \gamma _{\rho } (u-u_j)$, $u\in \mathbb {R}$, we get

$$\begin{aligned} a_j= \int \limits _{\textrm{supp}F_{\rho ,j}} F_{\rho ,j}(u)\,du, \ j=1,...,n. \end{aligned}$$

(100)

If in addition $\gamma _{\rho }$ is symmetrical, then

$$\begin{aligned} u_j= \frac{\int \limits _{\textrm{supp}F_{\rho ,j}}u F_{\rho ,j}(u)\,du}{\int \limits _{\textrm{supp}F_{\rho ,j}} F_{\rho ,j}(u)\,du}, \ j=1,...,n. \end{aligned}$$

(101)

also holds. Thus for $\rho <\rho _{\sup }$, all relevant information about the ideal adsorbent is contained in $F_\rho$.

Based on the considerations just made, we reconstruct averages of general AED’s $\varvec{\chi }^\prime =\varvec{\Phi }\in C_b^*$ that have a representation

$$\begin{aligned} \langle \varvec{\Phi },f\rangle = (F,f)+ \left\langle \sum \limits _{j=1}^n a_j \varvec{\delta }_{u_j},f\right\rangle ,\ f\in C_b \ \end{aligned}$$

(102)

¹⁰ with a function $F\in L_1$ and $a_j, u_j\in \mathbb {R}$, $j=1,...,n$ satisfying

$$\begin{aligned}&F\ge 0,\ \textrm{supp}F\subset [u_{{\textrm{min}}},u_{{\textrm{max}}}],\ a_j\ge 0,\ \end{aligned}$$

(103)

$$\begin{aligned}&u_j\in [u_{{\textrm{min}}},u_{{\textrm{max}}}] \text { and } \Vert F\Vert _1+\sum \limits _{j=1}^n a_j=1. \end{aligned}$$

(104)

Theorem 10

For $\rho >0$ let $\gamma _{\rho }$ be a real-valued nonnegative and continuous function satisfying (85) and $\widehat{\gamma _{\rho }}\in L_1$. Furthermore, the functions $b_{\sigma }$, $\sigma >0$ may satisfy the conditions of Theorem 6 and let the operator $\textbf{R}_{\sigma ,\rho }$ be defined on Y by

$$\begin{aligned} \textbf{R}_{\sigma ,\rho }(\psi ) := {\varvec{{\mathcal {F}}}}^*\left( b_{\sigma }\widehat{F_0}\widehat{\gamma _{\rho }} + b_{\sigma } a \widehat{\Delta \psi } \widehat{\gamma _{\rho }} \right) ,\ \psi \in Y. \end{aligned}$$

(105)

Let the CAED $\chi$ meet the conditions of definition 8 and let $\varvec{\Phi }=\varvec{\chi }^\prime$ satisfy (102), (103) and (104). Finally, let $\varphi$ be the transformed total isotherm corresponding to $\chi$, i.e. it holds $\varphi =\varvec{\Phi }*k$.

Under the above conditions, the following statements apply:

1. Let $\varphi _\varepsilon =\varphi +\varepsilon$ be an erroneous version of $\varphi$ with error level $\delta >0$, i.e. it holds $\Vert \varepsilon \Vert _1\le \delta$, then the following inequality applies for all $\sigma >0$:

$$\begin{aligned} \left\| \varvec{\Phi }*\gamma _{\rho }-\textbf{R}_{\sigma ,\rho }\left( \varphi _\varepsilon \right) \right\| _{\infty }\le \frac{\left\| \left( 1-b_{\sigma }\right) \widehat{\gamma _{\rho }}\right\| _1 +\left\| b_{\sigma } a \widehat{\gamma _{\rho }}\right\| _1\delta }{2\pi }. \end{aligned}$$

(106)

2. Let for $\delta >0$ and $\rho >0$ the functions $f_{\delta ,\rho }^{-}$ and $f_{\delta ,\rho }^{+}$ for $\sigma >0$ be defined by

$$\begin{aligned} f_{\delta ,\rho }^{-} (\sigma )&:= \left\| \left( 1-b_{\sigma }\right) \widehat{\gamma _{\rho }}\right\| _1 -\left\| b_{\sigma } a \widehat{\gamma _{\rho }}\right\| _1\delta , \end{aligned}$$

(107)

$$\begin{aligned} f_{\delta ,\rho }^{+}(\sigma )&:= \left\| \left( 1-b_{\sigma }\right) \widehat{\gamma _{\rho }}\right\| _1 +\left\| b_{\sigma } a \widehat{\gamma _{\rho }}\right\| _1\delta . \end{aligned}$$

(108)

If $f_{\delta ,\rho }^{+}$ attains a minimum, let $\sigma _\rho (\delta )$ be the smallest minimizer of $f_{\delta ,\rho }^{+}$, otherwise let $\sigma _\rho (\delta )$ be the zero of $f_{\delta ,\rho }^{-}$, then it holds for all $\varphi _\varepsilon =\varphi +\varepsilon$ with $\Vert \varepsilon \Vert _1\le \delta$

$$\begin{aligned} \left\| \varvec{\Phi }*\gamma _{\rho }-\textbf{R}_{\sigma _\rho (\delta ),\rho }\left( \varphi _\varepsilon \right) \right\| _{\infty }&\le \frac{f_{\delta ,\rho }^{+}\left( \sigma _\rho (\delta )\right) }{2\pi }. \end{aligned}$$

(109)

Additionally, it holds

$$\begin{aligned} \lim \limits _{\delta \rightarrow 0} f_{\delta ,\rho }^{+}\left( \sigma _\rho (\delta )\right)&= 0. \end{aligned}$$

(110)

Proof

1. Due to definition 8, (55), (98) and (41), there hold

$$\begin{aligned} \widehat{\varvec{\Phi }}(\omega )&= {\hat{F}} (\omega ) +\sum \limits _{j=1}^n a_j \exp \left( -iu_j\omega \right) ,\ \omega \in \mathbb {R}, \end{aligned}$$

(111)

$$\begin{aligned} \varvec{\Phi }*\gamma _{\rho } (u)&= F*\gamma _{\rho }(u)+ \sum \limits _{j=1}^n a_j \gamma _{\rho }\left( u-u_j\right) ,\ u\in \mathbb {R}, \end{aligned}$$

(112)

$$\begin{aligned} \widehat{\varvec{\Phi }*\gamma _{\rho }} (\omega )&= \widehat{\varvec{\Phi }}(\omega )\widehat{\gamma _{\rho }} (\omega ) =\widehat{F_0} (\omega ) \widehat{\gamma _{\rho }} (\omega ) +a(\omega )\widehat{\Delta \varphi }(\omega ) \widehat{\gamma _{\rho }} (\omega ),\ \omega \in \mathbb {R}. \end{aligned}$$

(113)

Lemma 8 together with (112) and ${\hat{\gamma _{\rho }\in }} L_1$ together with (113) imply

$$\begin{aligned} \varvec{\Phi }*\gamma _{\rho }\in C_0\cap L_1\subset L_1\cap L_2 \ \text { and }\ \widehat{\varvec{\Phi }*\gamma _{\rho }}\in C_0\cap L_1\subset L_1\cap L_2. \end{aligned}$$

(114)

Hence by (113), (114) and (44), (46), we get

$$\begin{aligned} {\varvec{{\mathcal {F}}}}^* \left( \widehat{\varvec{\Phi }}\widehat{\gamma _{\rho }}\right) = {\varvec{{\mathcal {F}}}}^* \left( \widehat{\varvec{\Phi }*\gamma _{\rho }}\right) = \varvec{\Phi }*\gamma _{\rho } . \end{aligned}$$

(115)

Furthermore, (111) yields

$$\begin{aligned} \left| \widehat{\varvec{\Phi }} (\omega )\right| \le \Vert F\Vert _1+\sum \limits _{j=1}^n a_j=\varvec{\Phi }(0)=1,\ \omega \in \mathbb {R} \text { and thus } \left\| \widehat{\varvec{\Phi }}\right\| _{\infty } = 1. \end{aligned}$$

(116)

(106) results then from (113), (115), (116) and (40) as follows

$$\begin{aligned} \left\| \varvec{\Phi }*\gamma _{\rho }-\textbf{R}_{\sigma ,\rho }\left( \varphi _\varepsilon \right) \right\| _{\infty }&\le \left\| {\varvec{{\mathcal {F}}}}^*\left( \widehat{\varvec{\Phi }}\widehat{\gamma _{\rho }}-b_{\sigma } \widehat{\varvec{\Phi }}\widehat{\gamma _{\rho }} -b_{\sigma } a {\hat{\varepsilon }}\widehat{\gamma _{\rho }}\right) \right\| _{\infty } \\&\le \frac{\left\| \left( 1-b_{\sigma }\right) \widehat{\gamma _{\rho }}\widehat{\varvec{\Phi }}-b_{\sigma } a \widehat{\gamma _{\rho }}{\hat{\varepsilon }}\right\| _1}{2\pi }\\&\le \frac{\left\| \left( 1-b_{\sigma }\right) \widehat{\gamma _{\rho }}\right\| _1 \left\| \widehat{\varvec{\Phi }}\right\| _{\infty } +\left\| b_{\sigma } a \widehat{\gamma _{\rho }}\right\| _1\left\| {\hat{\varepsilon }}\right\| _{\infty }}{2\pi } \\&\le \frac{\left\| \left( 1-b_{\sigma }\right) \widehat{\gamma _{\rho }}\right\| _1 +\left\| b_{\sigma } a \widehat{\gamma _{\rho }}\right\| _1\delta }{2\pi }. \end{aligned}$$

2. The proof of the well-definedness of $\sigma _\rho (\delta )$ and (110) is analogous to the proofs of Theorems 6 and 7. As $\sigma _\rho (\delta )$ is well-defined, (109) is an immediate consequence of (106). $\square$

7 Conclusions

We developed a general regularization for the solution of the adsorption integral equation with Langmuir kernel for erroneous data on the basis of fourier transform. Here, the central point is that the explicitely known error amplification term in the spectral function of the adsorption energy distribution is damped out. For the choice of the damping, we considered two strategies. The first one tries to equate the accuracy of approximation with the stability of approximation, while the second one tries to minimize the approximation error itself. In Theorems 6 and 7, we proved that both strategies assure convergence of the approximate solution to the real solution with respect to the mean squared norm as well as for the maximum norm for vanishing mean absolute error of the transformed total isotherm (12).

Since the accuracy of the approximated solution becomes worse the worse the decay behavior of the spectral function of the real solution becomes, we considered the reconstruction of the adsorption energy distribution up to a given resolution. More precisely, we dealt with the approximation of averaged adsorption energy dsitributions. This approach is suitable for the treatment of each kind of adsorbent including ideal adsorbents and AED’s with sharp peaks. In Theorem 10, as averaging leads to a more stable inverse problem, we were able to construct a regularization for computing averaged AED’s that yields a uniform estimate of the maximal approximation error in dependence of the mean absolute error of the transformed total isotherm.

The next step is the numerical application of the regularizations with special damping and averaging functions that meets the requirements of Theorems 6, 7 and 10.