Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

A Sobolev Space Inroad to Riemann Integrability

Nassar H. S. Haidar

CRAMS: Center for Research in Applied Mathematics & Statistics, AUL, Beirut, Lebanon

Abstract: A conditioned equivalence is proved for a certain weighted Sobolev space to the space of Riemann integrable functions. An equivalence representing a new result that not only asserts the sufficiency (but non-necessity) nature of bounded variation of functions for their Riemann integrability, but also reveals a potential for some novel computational findings.

Keywords: bounded variation, Sobolev space, Stieltjes integrals, continuity of functions, Riemann integrability

Received June 04, 2020; Revised July 05, 2020; Accepted July 13, 2020

Citation: Nassar H. S. Haidar, A Sobolev Space Inroad to Riemann Integrability. Turkish Journal of Analysis and Number Theory, 8 (2). 34-38. January 2020. doi:10.12691/tjant-8-2-3

1. Introduction

The transparent classical definition of the Riemann sum type of integral, see e.g. [1], was shown by Graves in [2], to admit an easy extension to a following integration of Banach space-valued functions, [3]. Indeed, let f: I → X be given, where X is a Banach space with the norm ‖ .‖X and the compact interval I = [a,b]⊂ R = (-∞,∞) is endowed with a length measure λ. f(x) is said to be Riemann integrable and γ∈ X its Riemann integral if for every ε > 0, ∃ a constant gauge δ∈ R⁺ =[0,∞), such that for every δ-fine K-partition {xi , Ii : i∈ Γ}, see e.g. [3], the inequality

holds.

Moreover, if in the previous statement δ∈ R⁺ is replaced by δ:I →R⁺, f would be Henstock-Kurzweil, [4], integrable, whereas if both δ∈R⁺ and K- partition in the previous statement are respectively replaced by δ : I → R⁺, and M- partition, [3], f becomes MacShane, [5], integrable. The same approach was followed by Bochner, in [6], for Lebesgue integrability in Banach spaces, i.e. when the above λ is a Lebesgue measure, to introduce the Bochner integral.

Lebesgue integrability happens also, however, to be a defining operation for all Sobolev spaces, see e.g. [7], and Riemann integrability in Sobolev spaces may sound rather unusual. The goal of this note, is to demonstrate that a linearly weighted Sobolev space can serve, nonetheless, as a natural space for Riemann integrable functions. The obtained result not only constitutes an addition to the existing literature, e.g. [3], on vector-valued integration, but also reveals a potential for some novel computational findings.

2. Main Result

The space of ρ-weighted Riemann integrable functions over a closed interval I,

[picture] | (1)

where both integrals are conceived in the Riemann sense, is well known to be seminormed, see e.g. [1,7]. A seminorm on a functional vector space X, with f,g∈ X and c ∈ ℂ, is a function ξ: X→R with the properties:

i) ξ(cf) = |c| ξ(f), absolute homogeneity,

ii) ξ(f+g) ≤ ξ(f) + ξ(g), subadditivity.

It is clear that Ker(ξ) = {f ∈ X: ξ(f)=0} is a subspace of X. Moreover, a seminorm ξ becomes a norm υ if Ker(ξ)={0}. Hence a norm υ: X → R⁺ is a seminorm with the additional property:

iii) υ(f) ≥ 0, ∀f∈X, with the equality if f ≡ 0, point separability.

Both integrals in (1), namely

[picture] | (2)

or

[picture] | (3)

can be seminorms for ℜ(I;ρ), because they happen to satisfy the previous properties (i) and (ii) but may violate (iii). Indeed ξℜ; (f) = 0 if f = 0 only a.e., that is when f ≈ 0, while ζℜ; (f) = 0, when e.g.

This makes ℜ(I;ρ) a vector space of equivalence classes of ρ-weighted Riemann integrable functions on I, where f can be equivalent to g iff f = g a.e. Nonetheless, by application of the Riesz-Fischer theorem, [1,7], it can readily be proved that ℜ(I;ρ) is complete under the seminorm ℜ; , i.e. ℜ(I;ρ) is a Banach space.

Since ℜ(I;ρ)⊂ L¹(I;ρ), one should not confuse ξℜ; (f) = ℜ; with the L¹ seminorm

[picture] | (4)

of the p =1 case of Lp spaces, in which the integral is conceived in the Lebesgue sense. Note moreover that L¹ happens to be unusual among the Lp spaces in that it is not reflexive. Incidentally, conventional ρ-weighted Sobolev spaces Wp,k(I;ρ) are related to these Lp spaces and have additional structure viz.

[picture] | (5)

as they are endowed with the norm

[picture] | (6)

Here, the D f derivatives, up to a given k-order, are understood in a weak sense, to make the Wp,k(I;ρ) space complete, thus a Banach space.

Definition 1. (Functions of bounded variation, [8,9]). Let us subdivide the I=[a,b] interval arbitrarily into n subintervals to form a finite partition

[picture] | (7)

with the sum

[picture] | (8)

A function f: I→R is of bounded variation (BV), or f∈ BV(I), is characterized by the norm

[picture] | (9)

and is defined when

[picture] | (10)

Definition 2. (Hölder continuous functions). f : I →R is said to be Hölder continuous (HC) if ∃ α > 0 such that

[picture] | (11)

, and L is some > 0 constant.

The special case of α = 1 corresponds to the case of Lipschitz continuous (LC) functions; and it is well known that Lipschitzianity is weaker than continuous differentiability (CD), and stronger than uniform continuity (UC) or continuity (C).

As for the ℜ(I;ρ) space of interest to this work, its related L¹(I;ρ) space naturally invokes W1,1(I;ρ), which is known, [5,7], to be isomorphic to the space of functions of BV, endowed with the normBV , defined by (9-10). Hence W1,1(I;ρ) is a space of absolutely continuous functions on I or equivalence classes of ρ-weighted Lebesgue integrable functions that are a.e. absolutely continuous. It is a Banach space w.r.t. the norm

[picture] | (12)

Replacement of the Lebesgue integrability condition in the definition of W1,1(I;ρ) by Riemann integrability transforms it to an unconventional Riemann-Sobolev space Wℜ,1 (I;ρ) endowed with the norm

[picture] | (13)

where the ℜ index refers to integration in the Riemann sense of functions essentially in BV(I), isomorphic with W1,1(I;ρ).

To demonstrate then that ℜ,1; is a norm, we examine it against the previous three related properties. Although the second term ℜ; in (13) is always a seminorm since it annihilates constants, i.e. violates the norm property (iii), the first term ℜ; does not. This guarantees, incidentally, satisfaction of (iii) by the entire ℜ,1;. Property (i) is clearly satisfied by both components of ℜ,1; , then by ℜ,1; itself. It is also straight forward to prove that both ℜ; and ℜ; satisfy property (ii). To illustrate this, let I=[0,1], ρ(x)=1 on I and let f, g ∈ Wℜ,1(I;ρ) be f(x) = x and g(x)= -x. Clearly,ℜ;1 =ℜ;1 = 1 , while ℜ;1 = 0, whence ℜ;1 +ℜ;1 >ℜ;1 is a satisfaction of the triangle inequality by ℜ;. The inequality ℜ,1 + ℜ,1 > ℜ,1 can similarly be established to confirm satisfaction of property (ii) by the sum, ℜ,1; , of these two terms.

Theorem 1. Let I = [a,b] ⊂ R, then

[picture] | (14)

Proof. Consider integration by parts of the Stieltjes integral , see e.g. [8,10], via

[picture] | (15)

On one hand, existence of either one of the Stielties integrals implies existence of the other, [11], with a satisfaction of

[picture] | (16)

On the other hand, classical analysis results, [4,12], show that the left Stieltjes integral of (15) always exists, with both (15) and (16) valid, if

i) f(x)∈ C(I) and ρ(x)∈ BV(I),

ii) f(x)∈ ℜ(I) and ρ(x)∈ LC(I),

iii) f(x)∈ α-HC(I) and ρ(x)∈ β-HC(I); α + β >1, and the three statements are reflexive by replacement of f with ρ. Results that cohere with the following classical inclusion over I, [13],

[picture] | (17)

Next assume in (15) that ρ(x) = x to reduce it to

[picture] | (18)

Relation (18) happens remarkably to operate as a "functional quadrupole" with two inputs , ρ and f, and two outputs, ℜ(I) and Wℜ,¹(I;x). In this quadrupole, the possible {CD, LC, HC, UC, BV, C, ℜ}(I) spaces for ρ and f combine, according to the results (i)-(iii) and their reflections, in order to maintain the outputs fixed. In particular, (18) guarantees existence of f∈ Wℜ,¹(I;x) simultaneously in any space of the set { LC, HC, UC, BV, C}(I), and not only in the isomorphic BV(I), when ρ(x) = x∈ CD(I) = {CD ∩ LC ∩ HC ∩ UC ∩ BV ∩ C∩ℜ}(I).

Clearly therefore if f∈ Wℜ,¹(I;x), then f∈ ℜ(I), which is the required result.

We should underline here that BV of f(x) over I is not necessary, though sufficient for its Riemann integrability. The same is true if the BV condition is replaced by the stronger LC condition, [3] or [9]. Moreover, LC(I) implies CD(I) a.e., and many functions can ∈ LC(I) while ∉ CD(I). So LC is weaker than CD, while being stronger than BV or C. Now it is time to state the important definition that follows, on Riemann integrability in a linearly-weighted Riemann-Sobolev space.

Definition 3. f : I→Wℜ,¹(I;x) is Riemann integrable in Wℜ,¹(I;x), whenever exists.

Remark 1. It should be noted that satisfaction of x f(x) < ∞ alone in (18) does not necessarily mean existence of the , i.e. that f(x) is of BV. But with added integrability of Df(x) over I, w.r.t. the x weight, f(x) becomes, by Theorem 1, Riemann integrable over I, even if it is not of BV.

Remark 2. Lipschitzianity happens to guarantee that f(x) is of BV, or even is absolutely continuous over I.

3. Demonstrative Examples

The first example is meant to illustrate the sufficiency nature of BV of f(x) over I for its Riemann integrability over the same I.

Example 1. Consider the unique infinitely discontinuous and wildly oscillatory function :

which is neither differentiable nor Lipschitzian over I = [0,1]. Indeed, this function, being continuous up to a discrete set of points, i.e. is continuous a.e., should obviously be Riemann integrable. Our purpose here, however, is to arrive at the same conclusion via theorem 1. Hence it can be graphically demonstrated that

[picture] | (19)

i.e. it is of BV over [0,1]. Obviously,

and

Then by (18) we may write

Now since x f(x)= 0, the previous relation becomes

Substitution of the identity

in the previous relation reduces it to

The above Dirichlet's Eta function η(s), of argument 2, is related to the associated Riemann Zeta function ξ(2), via η(2) = (1 -) ξ(2). So as ξ(2) = ²/6, [14], we may rewrite the last relation as

i.e. f(x) is Riemann integrable.

3.1. Novel Rational Expansion

One of the rather unexpected applications of theorem1 when integrating certain strongly oscillatory functions in Wℜ,1(I;x) is the possibility of generating expansions of some fractions in certain unusual partial fractions. Such a possibility is highlighted in the example that follows.

Example 2. According to Theorem 1, localization of nondifferentiability is not a required restriction. For an illustration, consider a function f(x) defined viz

[picture] | (20)

with linear chords, of distinct slopes, between consecutive vertices. This f(x) is a saw-like oscillation which, though continuous, is nondifferentiable at an infinite number of points on I = [0,1]. It can be graphically verified that

i.e. f(x) is not of BV on I. Moreover, the behavior of its chord's slopes as k→∞ is a reason for its non-Lipschitzianity over this I. Nevertheless, it is not difficult to verify the Riemann integrability of this function over I, either directly graphically, or via the application of Theorem 1.

Graphical integration. When integrating the posing saw-like oscillation f(x) on [0,1], we encounter different directions of flow in the oscillation numbers, k, and in values of the x-axis. It turns out to be more convenient to use the map z = 1-x and integrate f(z) over z∈ [0,1]. In fact x = 1/k becomes z =. Then for odd k only, i.e. for k = 2m-1; m = 1,2,3,......, z =. Accordingly, any m-th oscillation on the z-axis becomes

[picture] | (21)

where the middle vertex is below the z-axis, = (z) and = (z) are straight lines with opposite slopes. Direct evaluation of the sum, Am, of the areas of the hanging on the z-axis three triangles, which is the integral of the m-th oscillation, can, by some lengthy calculations, be shown to be

[picture] | (22)

Hence

[picture] | (23)

with = 0.113605, = 0.113689, and ≈ 0.115.

Integration via Theorem 1. Here and need to be represented analytically viz

to allow for expressing the associated function f(z) as

[picture] | (24)

Since z f(z) = 0 , then application of theorem 1 results with

which can be shown, after some lengthy details, to lead to

[picture] | (25)

with

[picture] | (26)

It is straight forward to verify that Bm ≡ Am and that (25) and (27) are identical. Interestingly, however, Bm is not a rational expansion of the rational Am in simple partial fractions viz

[picture] | (27)

in which a = 1, b = -2, c = -3/2 , d = -2, and e = 3/2, as a must.

Clearly the first term in (26) is not a simple partial fraction for our Am. Hence comparison of (26) and (27) illustrates that Bm, in (26), is remarkably a novel rational expansion of Am, of (23), in unusual partial fractions. This fact can perhaps motivate other "recreational" or unforeseen applications for Theorem 1, along the arguments used in [15], for example.

4. Conclusion

The result of this note is meant to complement (and not to replace) the theory of Riemann integration of a.e. continuous functions. It implements for the first time, however, Sobolev space arguments to assert the well known result that BV is not necessary (though sufficient) for Riemann integrability. The distinguishing feature of the function of example 2, from that of example 1, is its non BV on the respective I. It is demonstrated that BV is not necessary for the Riemann integrability of example 2 over [0.1], indicating the surplus in its sufficiency (non-necessity) implied by Theorem 1. The computational potential of Theorem 1, as a technique of integration, raises, according to example 2, an issue that calls for further exploration.

References

[1]   | Royden, H. R, Real analysis, Macmillan, London, 1988.
[2]   | Graves, L. M, “Riemann Integration and Taylor's theorem in general analysis,” Transactions of the AMS, 27. 163-177. 1927.
[3]   | Gordon, R, “Riemann integrability in Banach spaces,” Rocky Mountain Journal of Mathematics, 21(3). 923-949. 1991.
[4]   | Rejouani, R, “On the question of the Riemann integrability of functions with values in a Banach space,” Vestnik Moskov. Univ., Ser. I, Mat. Mech. 26(4). 75-79.1971.
[5]   | McShane, E, A Riemann-type integral that includes Lebesgue-Stieltjes, Bochner and stochastic integrals, Vol.88, AMS, Providence, RI, 1969.
[6]   | Diestel, J. and Uhl, J. J, Vector measures, AMS, Providence, RI, 1977.
[7]   | Kufner, A., John, O. and Fučík, S, Function spaces, Noordhoff, Leyden, 1977.
[8]   | Rudin,W, Principles of mathematical analysis, McGraw-Hill, Auckland, 1964.
[9]   | Titchmarsh,E. C, The theory of functions, Oxford University Press, London,1939.
[10]   | Haidar, N. H. S, “A geometric note on integration,” Computers & Mathematics With Applications, 59(9). 3130-3136. 2010.
[11]   | Hille, E and Philips, R. S, Functional analysis on semigroups, AMS, Providence, RI, 1974.
[12]   | Young, L. C, “An equality of the Hölder type, connected with Stieltjes integration,” Acta Mathematica, 67(1). 251-282. 1936.
[13]   | Kolmogorov, A. N and Fomin, S. V, Introductory real analysis, Dover Publications, 1975.
[14]   | Janke, E., Emde, F. and Lösch, F, Taflen hoherer funktionen, Tubner, Stuttgart, 1960.
[15]   | Haidar, N. H. S, “A rational triplic form for the exponential,” Journal of Computational Analysis and Applications, 4(4). 389-404. 2002.