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

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.


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) χ = χ(u) B Steffen Arnrich sdrarnrich@gmail.com Grit Kalies kalies@htw-dresden.de or rather its density, i.e. the (differential) adsorption energy distribution (AED) F by means of the experimentally measured total adsorption isotherms t = t ( p, T ).
Here χ(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 χ(u b ) − χ(u a ), u a < u b is the percentage of adsorption sites releasing molar energies u a < u ≤ u b . t ( p, T ) represents the total coverage of the surface at pressure p and temperature T .
It is supposed that t is a superposition of the local isotherms l = 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 χ(0) = 0, the total isotherm is given by a Stieltjes integral: If χ has an integrable density F, that is then (1) reads as (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 · · · < u n , where the adsorption takes place. The corresponding CAED is a step function with a j ≥ 0 for j = 1, ..., n and n j=1 a j = 1, where a j is the percentage of adsorption sites releasing the molar energy u j and H denotes the Heaviside function (1) then reduces to a finite sum t ( p, T ) = n j=1 a j l ( p, T , u j ).
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 χ 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 ≤ u min < u max < ∞ ( [2], [3]) implying F(u) = 0 for all u / ∈ [u min , u max ]. With this condition, (3) reads as By definition of χ , F meets additionally the constraints Equation (3) as well as (8), where the temperature T is kept fixed, are Fredholm integral equations of the first kind with kernel l (·, T , ·), given data t (·, T , ·) 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 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 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 l describing type-I total isotherms t is the Langmuir model 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 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 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 in order to transform the Langmuir AIE into By means of an inversion formula for the transformed total (adsorption) isotherm ϕ, 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.

General solution strategy: regularization
Usually, integral equations are considered as operator equations Ax = y between normed (function) spaces (X , · X ) and (Y , · Y ). For integral equations of the first kind, the linear integral operator A is often one-to-one, while the inverse operator A −1 is not continuous (unbounded) on A(X ). In applications, the right-hand side y is usually erroneous with a known error level δ > 0. More precisely, instead of y we have to deal with a pertubed right-hand side y ε = y + ε, where ε Y ≤ δ applies to the pertubation ε. The challenge is now to find a reasonable approximation x ε to the exact solution x that depends continuously on the data y ε ("stability condition") [5].
Since y ε , in general, doesn't belong to the range of A, the inverse operator A −1 : A(X ) → X has to be approximated by a continuous (bounded) linear operator R : Y → X . Furthermore, x ε should converge to the exact solution x if the error level δ tends to 0. Hence, we need a family of continuous linear operators R σ : Y → X , σ > 0 that converges pointwise to A −1 on A(X ) as σ → 0. Now, each can be regarded as an approximation of x. The approximation error is estimated by It remains to find a strategy to choose σ dependent on the error level δ or more general dependent on δ and the given data y ε to assure x ε,σ → x, δ → 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 σ is required for accuracy, since R σ y → A −1 y = x, σ → 0. On the other hand, a larger σ is required for stability. This results from the following argument: By (14) it holds for two perturbations ε andε with ε Y ≤ δ and ε Y ≤ δ As a consequence of the unboundedness of A −1 we get R σ Y →X → ∞, σ → 0. Therefore, the approximation error (15) becomes large for small σ . 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 illposed (unstable) problem is called regularization. Mostly, generalized regularizations are specialized to Hilbert space settings, i.e. X and Y are Hilbert spaces and A : X → 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 A −1 needs only to be approximated on A(X a ). So we have more freedom to choose the R σ .
These considerations lead to the following definition of regularization motivated by [5]: Definition 1 Let A be an operator and X a be a subset of the domain of A such that A is one-to-one on X a . Furthermore, let (X , d X ) and (Y , d Y ) be metric spaces with X a ⊂ X and A(X a ) ⊂ Y .
A regularization scheme for the equation Ax = y with respect to the setting X a , The parameter σ is called the regularization parameter.
A strategy of choosing σ = σ (δ) in dependence of the error level δ > 0 is called regular if it satisfies for all y ∈ A (X a ) the condition: 1 Note that R σ y needs not to be an element of X a . Occasionally, we will replace lim A regularization scheme for Ax = y with respect to the setting X a , (X , including a regular strategy is called a regularization for Ax = y with respect to the setting X a , (X , To emphasize the importance of the setting, we paraphrase [5], p. 222: The instability of the equation Ax = y is a property of the setting, since the continuity of A −1 : A(X a ) → 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: We call a function f essentially bounded if there is c ≥ 0 such that there holds If for two functions f and g the integral dy does exist for every real x, then the convolution f * g of f and g is defined as the function By 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) = 0.
Let A be the operator that assigns to a function f the function whenever k * f exists. Then (13) can be formulated as with the constraint that F vanishes outside of [u min , u max ], i.e. it holds 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 : 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 Since k is bounded and continuous, A f is well-defined for every absolutely integrable function f . According to [4], A is one-to-one on X a and thus on C a , too. Obviously, it holds X a ⊂ X and C a ⊂ 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 ϕ ∈ A(X a ) holds that ϕ is bounded, continuous and that ϕ − k is absolutely integrable.
Proof Let F ∈ X a with ϕ = AF, then (9) and the fact that k is strictly monotonically decreasing and satisfies 0 < k < 1 yields for ξ ∈ R: which proves boundedness.
Hence |ϕ − k| = ϕ − k is bounded by an interable function and consequently, the continuous function ϕ − k is absolutely integrable.
The above result motivates: Definition 3 Let Y be the set of all essentially bounded functions ψ for which ψ − k is absolutely integrable.
By definition 3, we deal only with erroneous transformed total isotherms ϕ ε = ϕ + ε where the functions of errors ε 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 t between t ( p min , T ) and t ( p max , T ), 0 < p min < p max < ∞. So we can always find a ϕ ε as an approximation of the transformed total isotherm ϕ 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 t is then given be the function of the measurement errors. V tends to a limit volume V ,∞ for large pressures which is assumed to be larger than 0, otherwise the measurement has to be rejected. It is natural to assume that is controlled by V while the oscillations ∞ − around the limit error ∞ : 3 (27) yields together with V ∼ t and V ∞ − V ∼ 1 − t for := t − t (the error with respect to t ): This implies for ε = ϕ ε − ϕ (the error of the transformed total isotherm ϕ): The right-hand side of (28) is a positive and integrable function. This justifies to assume that ϕ ε 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 ≤ q < ∞ whenever the defining right-hand sides exists. For 1 ≤ r ≤ ∞ we denote by L r the set of all functions f for which f r is well-defined.
Since X = L 1 ∩ L ∞ and it holds ϕ 1 − ϕ 2 ∈ L 1 ∩ L ∞ for all ϕ 1 , ϕ 2 ∈ Y , d r defines a metric on X as well as on Y for every 1 ≤ r ≤ ∞. On C 0 , d ∞ defines a metric and it holds for all f , g ∈ C 0 This motivates to consider the settings for (23), where 1 ≤ q, r , s ≤ ∞ have to be further specified to assure the continuity of the used operators R σ .

Regularization schemes by means of Fourier transform
Starting point for the construction of the regularization operators R σ is the following heuristical argument: Applying formally the convolution theorem of fourier transform ( [19]) to (23) yieldsφ whereF denotes the spectral function of the function F, see definition 5 below. Again formally, we get by (34)F Inserting an erroneous ϕ ε = ϕ + ε into the right-hand side of (35) provides an approximation ofF:F Because of the instability of (13),k −1 and consequentlyF ε are unbounded. A stable solution can be found replacingk −1 by bounded functions κ σ with κ σ →k −1 , σ → 0. This suggests to choose the approximation operators R σ defined by R σ f := F −1 κ σf where 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 ρ ≥ 0 the functionsf ,f ,f ρ andf ρ defined bŷ whenever the right-hand sides exists for every real ω. Moreover, let the operators F , F * , F ρ and F * ρ be defined by f is called the spectral function of the function f and the operator F is called the fourier transform.

Lemma 2
The following statements about the fourier transform or spectral functions, respectively, are true: 2. For f , g ∈ L 1 , it holds f * g =fĝ. 3.
For continuous differentiable f ∈ L 1 with f ∈ L 1 , it holds 4. For f ∈ L 2 , there are (uniquely determined)f ,f * ∈ L 2 with For the operators F 2 and F * 2 defined for all g ∈ L 2 by F 2 (g) :=ḡ and F * 2 (g) = g * and for all f ∈ L 2 , there hold Since k and thus ϕ 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 ϕ = k * F. Since k is continuous and absolutely integrable, we get for absolutely integrable F with continuous and absolutely integrable ϕ . 6 (40) can therefore be applied to (47). This yields For F ∈ X a 7 , (48) can be inverted by means of an inversion formula proved in [4]. It holds 5 The fourier transform can also be defined for tempered distributions [19]. In the sense of distribu- (34) is therefore not practical for our purposes. 6 This can be easily shown directly or by using [21], Lemma 16.2. and [19], p. 397.
for all globally analytical functions ζ defined on C, whose derivatives grow polynomially at most on the real axis, where Z satisfies Z = ζ . Inserting the functions ζ ω = exp(−iω·), ω ∈ R into (49) yields together with where sinhc is the function defined by In applications, we have to deal with erroneous ϕ ε . 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 F 0 ∈ C a and ϕ 0 := k * F 0 (53) and set for arbitrary G ∈ X and ψ ∈ Y G := G − F 0 and ψ := ψ − ϕ 0 .

Lemma 3 Let b be a function such that b and ba are bounded and absolutely integrable. Let R be the operator defined on Y by
then there hold for all ψ, Proof (57) results from (40), (46) and the fact that bF 0 +ba ψ is a bounded, absolutely integrable and thus also square-integrable function. Let now r = 1 for q = ∞ and r = 2 for q = 2 and let χ := ψ 1 − ψ 2 , then we get by (40), (44) and (46) Lemma 3 allows constructing regularization schemes for the settings X a , (X , (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.
We need the following statements: Lemma 4 Let (γ τ ) τ >0 be a dirac-sequence and f ∈ C 0 , then there hold f * γ τ ∈ C 0 and Proof As f ∈ C 0 , there hold for τ > 0: f * γ τ ∈ C 0 is then a consequence of the dominated convergence theorem [22].
Let now > 0. Due to the uniform continuity of f ∈ C 0 , there is a ρ > 0 such that Choose now τ such that |y|≥ρ γ τ (y) dy ≤ for all 0 < τ ≤ τ . It follows by (64) Now we are able to specify general regularization schemes for a stable solution of (13) as the main statement of this section.

Regularizations
We are looking now for regular strategies (cf. (18)) with respect to the settings X a,2 , Let ϕ ε = ϕ + ε be an erroneous version ϕ = AF with the function of errors ε. By (58), (59), (67) and (68) we get the estimates Hence, by Theorem 5 every choice σ = σ (δ), δ > 0 with lim δ→0 σ (δ) = 0 and lim is regular with respect to the setting X a,2 , (L 2 , d 2 ), (Y , d 1 ) for r = 2 and C a , As the first term in the nominator on the right-hand side of (69) shows, the rate of convergence depends on the decaying behavior ofF. 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 ∈ L 2 impliesF(ω) = o (h 1 (ω)), |ω| → ∞ ( [20], Satz V.2.14), where for n ∈ N the function h n is defined bŷ It is reasonable to assume that the decay behavior of the spectral function of the AED is not worse than O (h 1 (ω)), |ω| → ∞. 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 F − F * β σ ∞ . For a continuous AED it is likely that it is as least as regular as a triangular function. Hence,F is absolutely integrable and decays at least as O (h 2 (ω)), |ω| → ∞. In this case, it holds F − F * β σ ∞ ≤ 1 2π 1 − β σ F 1 and thus by (70) .
In summary, we generally expect with q = r = 2, n = 1 for F ∈ X a,2 , and q = ∞, r = 1, n = 2 for F ∈ C a . (73) motivates to choose a strategy keeping the balance between the approximation error (1 − b σ ) h n r and the data error b σ a r ε 1 .
We prove now that with this strategy all F ∈ X a,2 as well as F ∈ C a can be reconstructed.

Theorem 6 In addition to the assumptions made in Theorem 5, let
Let for n = 1, 2 and δ > 0, σ − n (δ) be the zero of the function f − n,δ defined for σ > 0 by Then the family of operators R σ , σ > 0 as defined in (66) form together with the strategy σ = σ − n (δ) a regularization for (23) with respect to the settings X a,2 , (L 2 , d 2 ), (Y , d 1 ) for n = 1 and C a , Proof At first we have to show that σ − n (δ) 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,δ is continuous on (0, ∞) and satisfies Since the functions σ → b σ a 3−n or σ → (1 − b σ ) h n 3−n are strictly monotonically decreasing and strictly monotonically increasing, respectively, f − n,δ has at most one zero. With (78), (79) and the continuity of f − n,δ , this implies the well-definedness of σ − n (δ). It remains to show (71). To this end, we first verify that it holds for all 0 < δ 1 < δ 2 Suppose not, then we get the contradiction By (80), there is a σ * n ≥ 0 with lim δ→0 σ − n (δ) = σ * n . The assumption σ * n > 0 leads to the contradiction which shows (71) and completes the proof.
A second strategy is to minimize the right-hand side of (73) for given error level δ > 0, i.e. to minimize the function f + n,δ defined for σ > 0 by

Remark 1
Note that σ + n (δ) is the smallest minimizer of f + n,δ for

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 δ → 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 * γ ρ , where γ ρ is a real-valued continuous function with γ ρ ≥ 0, γ ρ 1 = 1 and suppγ ρ = [−ρ, ρ] for fixed ρ > 0. To deal with averages F * γ ρ is motivated by lemma 4 and the following.
3. (86) is due to the fact that for all x ∈ R holds which completes the proof.
Furthermore, ρ can be seen as a measure of how well F * γ ρ resolves the structure of F. This is justified by the following observation: where u min ≤ u 1 < u * 1 < u 2 < u * 2 < · · · < u n < u * n ≤ u max , then there hold for F * γ ρ = n j=1 F j * γ ρ : Consequently, for ρ < min where for v ∈ R, δ v ∈ C * b is defined by 9 Let ϕ be the transformed total isotherm corresponding to χ as in (4), i.e.
and let ϕ ρ be the transformed total isotherm corresponding to F ρ defined by As γ ρ ρ>0 is a dirac-sequence, it holds for every and hence for all ξ, ω ∈ R results Analogous to (25), for fixed ρ * > 0 and 0 < ρ ≤ ρ * the functions ϕ ρ and | ϕ| are bounded by the positive and integrable function g ρ * defined by This implies together with (96), (97) and the dominated convergence theorem 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 ρ > 0 Equation (99) means that the knowledge of the total isotherm of an ideal adsorbent allows to reconstruct for every ρ > 0 an approximation F ρ = χ * γ ρ of the general density χ of an ideal adsorbent consisting of so-called δ-peaks. Note that for ρ < ρ sup := min j=1,...,n−1 u j+1 −u j 2 , it holds F ρ = n j=1 F ρ, j with F ρ, j ≥ 0, F ρ, j 1 > 0 and suppF k ∩ suppF l = ∅ for k = l.
If in addition γ ρ is symmetrical, then also holds. Thus for ρ < ρ sup , all relevant information about the ideal adsorbent is contained in F ρ . Based on the considerations just made, we reconstruct averages of general AED's χ = ∈ C * b that have a representation 10 with a function F ∈ L 1 and a j , u j ∈ R, j = 1, ..., n satisfying 2. The proof of the well-definedness of σ ρ (δ) and (110) is analogous to the proofs of Theorems 6 and 7. As σ ρ (δ) is well-defined, (109) is an immediate consequence of (106).

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.