# Some Particular Norm in the Sobolev Space \(H^{1}[a,b]\)

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## Abstract

This paper is a continuation of the recent paper of the author, where a certain reproducing kernel Hilbert space \(X_{\mathcal {S}}\) was constructed. The norm in \(X_{\mathcal {S}}\) is related to a certain generalized isoperimetric inequality in \({{\mathbb {R}}^2}\). In the present paper we give an alternative description of the space \(X_{\mathcal {S}}\), which appears to be a Sobolev space \(H^{1}[a,b]\) with some special norm.

## Keywords

Absolutely continuous functions Isoperimetric inequality Hilbert spaces Sobolev spaces Reproducing kernels## Mathematics Subject Classification

46E22## 1 Introduction

*K*[in our situation the function \(K(\varphi ,\psi )\) given by the formula (7)] so we may try to construct another, perhaps more convenient, model of the space \(\mathcal {H}(K)\) corresponding to the considered kernel. Such a situation occurs frequently in the theory of RKHS’s and has even his own name (

*the reconstruction problem,*cf [4]). The aim of this paper is to give such a concrete description of the space \(\mathcal {H}(K)\) for the kernel given by (7). It will appear that in our case the space \(\mathcal {H}(K)\) is a Sobolev space \(H^1[a,b]\) equipped with some special norm. The construction of ”our” \(\mathcal {H}(K)\) is contained in the three theorems: Theorem 4, Theorem 5 and the Theorem 6.

In Theorem 4 we define a certain bilinear form \(\langle ,\rangle _i\) (14) and we prove that this form is positively defined. In the proof we use the Wirtinger inequality (3) in a general version for the derivatives from \(L_2[a,b]\). This variant of the Wirtinger inequality is to be found in [3].

In Theorem 5 we check the reproducing property of the given kernel (7). The proof is easy to understand, but may be is heavy to read because of lengthy calculations. The kernel (7), as far as we know, was not studied before.

In the last Theorem 6 we prove, that the constructed space is an RKHS. For this we must check, that the evaluation functionals are commonly bounded, and that the constructed space is complete. This type of arguments is typical in examples of Sobolev spaces \(H^1[a,b]\) [2]. Summarizing, the main result of this paper says, that the space \(H^{1}[-\frac{\pi }{2},\frac{\pi }{2}]\) equipped with the norm \(||\cdot ||_i\) is a reproducing kernel Hilbert space corresponding to the same kernel (7) as the space \(X_{\mathcal {S}}\) constructed in [8].

We present also one more construction of the considered space \(\mathcal {H}(K)\), called here the *sequence model*. This sequence model, if one looks from strictly theoretical point of view, does not bring new information, but gives tools for proofs and for numerical calculations. This model represents also a kind of presentation of the density of polygons in the space of convex sets.

From a number of known examples of the kernels in the space \(H^1[a,b]\), we present here one see formulas (12) and (13) in order to compare it with the considered here kernel (7).

This paper is organized as follows. In the second part we give a summary of the paper [8] necessary to understand the main result of the present paper. We recall also some information on the Sobolev of the type \(H^{1}[a,b]\) in the aim to see the ”particularity” of the norm we are going to construct. In the third section we define some special (particular) norm \(||\cdot ||_i\) in the space \(H^{1}[-\frac{\pi }{2},\frac{\pi }{2}]\). In the last section we present the construction of the *sequence model*.

## 2 About the Space of Generalized Convex Sets

*Minkowski addition*and the scalar multiplication by non-negative real numbers. In the product \(\mathcal {S}\times \mathcal {S}\) we consider the equivalence relation

*isoperimetric norm*, which is constructed in a few steps. At first we prove, that the two dimensional Lebesgue measure \(m:\mathcal {S}\longrightarrow [0,\infty )\) can be extended in a unique way to the polynomial of the second degree \(m^{*}:\mathcal {S}\times \mathcal {S}/\diamondsuit \longrightarrow {\mathbb {R}}\) and the functional \(o:\mathcal {S}\longrightarrow \mathbb R\) where

*o*(

*U*) is a perimeter of U, can be extended to a linear functional on the whole \(\mathcal {S}\times \mathcal {S}/\diamondsuit \). Next we prove that for this extended measure \(m^{*}\) the

*generalized isoperimetric inequality*holds, which means that for each \([U,V]\in \mathcal {S}\times \mathcal {S}/\diamondsuit \) we have

*U*,

*V*] and (here and in the sequel) we write

*m*([

*U*,

*V*]) instead of \(m^{*}([U,V])\). Using this fact it was proved, that the formula

### 2.1 About the RKHS

*reproducing kernel Hilbert space*(RKHS for short). Although the reproducing property was discovered by Zaremba more than hundred years ago (1906), the first systematic lecture is due to Aronszajn in 1950 (see [1]). The necessary information concerning RKHS are to be found in the book of Berlinet and Thomas-Agnant [2] or in the recent book of Szafraniec [6]. The RKHS are function spaces, so at first one must recognize the elements of \(\mathcal {S}\times \mathcal {S}/\diamondsuit \) as real functions on the interval \(\Delta =[-\frac{\pi }{2},\frac{\pi }{2}]\). This may be done as follows. We consider the functions

*diangles*. Each diangle defines a line in \({\mathbb {R}}^2\) (\({\mathbb {R}} \cdot (\cos \psi ,\sin \psi ))\) which will be denoted also by \(I_\psi \) and for the set \(U\in \mathcal {S}\) the number \(\overline{U}(I_\psi )\) denotes the

*width*of the set

*U*with respect to the line \(I_\psi \). The correspondence

*U*,

*V*] as functions on the interval \(\Delta \) (the details in [8]). Now we are ready to formulate the main result from [8].

### Theorem 1

The elements of \(X_\mathcal {S}\) are, roughly speaking, the ”differences” of convex sets (differences with respect to the Minkowski addition). This makes it possible to understand their geometrical character, but on the other hand, given a function \(f:\Delta \longrightarrow {\mathbb {R}}\) it is difficult to prove (or disprove), that \(f\in X_\mathcal {S}\). In this paper we will solve this problem. More exactly we will prove, that \(X_\mathcal {S}\) is a certain Sobolev space.

### 2.2 About the Sobolev spaces

We shall start by recalling some commonly known definitions and theorems concerning the Sobolev spaces.

### Definition 2

*absolutely continuous*on \(\Delta \) if for every \(\epsilon > 0\) there exists \(\delta >0\) such that

*K*(

*x*,

*y*) where

In the present paper we construct another inner product \(\langle ,\rangle _i\) defined in a subspace \(H^{1}_{0}(\Delta )\subset H^{1}(\Delta )\), related to another reproducing kernel \(K_i(x,y),\) which will be defined in the next section.

## 3 An Inner Product in the Sobolev Space \(H_0^{1}(\Delta )\)

It is clear, that the formula (14) is well defined (because of \(f',g' \in L_2(\Delta )\)) and defines a bilinear form in \(H_0^{1}(\Delta )\). The coefficient \(\frac{1}{\pi ^2}\) is clearly without importance and is chosen to have \(\langle \mathbf{1},\mathbf{1}\rangle _i = 1\) (where \(\mathbf{1}\) denotes the constant (and equal 1) function). Hence, to have a norm related to the form (14), it remains to show that \(\langle ,\rangle _i\) is positively defined. The proof of this positivity is similar in fact to the known proof of the isoperimetric inequality based on the Wirtinger inequality.

Let us start by recalling a variant of the Wirtinger inequality (for the proof and some other formulations see [3, 7]).

### Theorem 3

*f*be a continuous and periodic function with the period \(\pi \). Let \(f'\in L_2(\Delta )\) and let

Now we have the following

### Theorem 4

### Proof

We will prove now the reproducing property.

### Theorem 5

*the isoperimetric kernel*.

### Proof

*y*and

*f*as above. We must verify that

*B*.

*C*is more complicated. We have

It remains to prove now the main result of this section, which says that \(H^{1}_{0}(\Delta )\) with the norm (22) is an RKHS.

### Theorem 6

### Proof

The argumentation is similar for example to that in [4]. First we check that the evaluation functionals \(e_x(f)=f(x)\) are bounded with respect to the isoperimetric norm. Since \(H_0(\Delta )=H_{00}+\mathbb {R}\cdot {\mathbf{1}}\) then it is sufficient to prove, that each \(e_x\) is bounded on the subspace \(H_{00}\subset H^1_{0}\), composed of those functions which vanish at 0, (i.e. we may additionally assume, that \(f(0)=0)\).

*f*such that \(f(0)=0\) we obtain

*h*from \(L^2[-\frac{\pi }{2}, \frac{\pi }{2}]\) with respect to the \(L^2\) norm. But \(f_n\) tends uniformly to a continuous function

*f*. Since

*f*is absolutely continuous and since \(f_n(-\frac{\pi }{2})=f_n(\frac{\pi }{2})\) then also \(f(-\frac{\pi }{2})=f(\frac{\pi }{2})\). Thus \(f\in H^1_{0}(\Delta )\) and \(||f_n-f||_i\) tends to 0. Hence \( H^1_{0}(\Delta )\) is complete with respect to the isoperimetric norm. \(\square \)

We will end this chapter by a number of remarks.

### Remark 7

*U*with respect to the straight line generated by the diangle \(I_\varphi \), (see [8]).

*U*,

*V*] (in particular of

*U*) and the term

*U*,

*V*]. This fact is known for a long time as the so-called Cauchy formula (true not only for centrally symmetric, but for all convex and compact sets). Let us notice that (6) may be considered as an extension of the Cauchy formulas for generalized measure.

### Remark 8

### Remark 9

## 4 A Sequence Model

The construction of the RKHS space \(H^1_0(\Delta )\) presented above is frequently called *from space to kernel* (see e.g. [6]). In this section we will construct the same function space—i.e. \(X_\mathcal {S}\)—but in a way, which is called *from kernel to space*. The aim for which we repeat this commonly known construction is that it gives a possibility to see a kind of finite dimensional version of the generalized isoperimetric inequality. As usual, in this model the space \(X_\mathcal {S}\) will appear to be the completion of a certain function space \(\mathcal {F}\) with respect to a suitable inner product (suitable norm), corresponding to a given isoperimetric kernel. The idea we will describe below in details, was used in [8] in many places, but in an implicit form.

### 4.1 Definition of a Certain Sequence Space

### Definition 10

The coefficients in the formula (42) are chosen in such a way, that one may easily check directly the reproducing property of the form \(\langle ,\rangle _i\). In other words it is easy to verify, that \(\mathcal {F}\) equipped with the scalar product given by the (42) is identical (isometric) with the subspace of the space \(X_{\mathcal {S}}\) spanned by the unit disc and all diangles. In consequence \(\langle ,\rangle _i\) given by (42) is really an inner product (is positively defined) and the completion of \(\mathcal {F}\) equals \(X_{\mathcal {S}}\).

*the isoperimetric inequality for sequences*. If we put \(x_0=0\) we obtain the form

*W*the last inequality gives

*W*as above)

## Notes

## References

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