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# Abel and Euler Summation Formulas for SBV (ℝ) Functions

Written By

Sergio Venturini

Reviewed: 09 September 2021 Published: 19 October 2021

DOI: 10.5772/intechopen.100373

From the Edited Volume

## Coding Theory - Recent Advances, New Perspectives and Applications

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## Abstract

The purpose of this paper is to show that the natural setting for various Abel and Euler-Maclaurin summation formulas is the class of special function of bounded variation. A function of one real variable is of bounded variation if its distributional derivative is a Radom measure. Such a function decomposes uniquely as sum of three components: the first one is a convergent series of piece-wise constant function, the second one is an absolutely continuous function and the last one is the so-called singular part, that is a continuous function whose derivative vanishes almost everywhere. A function of bounded variation is special if its singular part vanishes identically. We generalize such space of special function of bounded variation to include higher order derivatives and prove that the functions of such spaces admit a Euler-Maclaurin summation formula. Such a result is obtained by deriving in this setting various integration by part formulas which generalizes various classical Abel summation formulas.

### Keywords

• Euler summation
• Abel summation
• bounded variation functions
• special bounded variation functions

## 1. Introduction

Abel and the Euler-Maclaurin summation formulas are standard tool in number theory (see e.g. [1, 2]).

The space of special functions of bounded variation (SBV) is a particular subclass of the classical space of bounded variation functions which is the natural setting for a wide class of problems in the calculus of variations studied by Ennio De Giorgi and his school: see e.g. [3, 4].

The purpose of this paper is to show that this class of functions (and some subclasses introduced here of function of a single real variable) is the natural settings for (an extended version of) the Euler-Maclaurin formula.

Let us describe now what we prove in this paper.

In Section 2 we obtain some “integration by parts”-like formulas for functions of bounded variations which imply the various “Abel summation” techniques (Propositions (0.6), (0.7), and the relative examples) and in Section 3 we give some criterion for the absolute summability of some series obtained by sampling the values of a bounded variations function.

The last section contains the proofs of the main result of this paper (Theorem (0.1)) that we will now describe.

We denote by C1R (resp. Ckab), L1R and LR respectively the space of continuously differentiable functions (resp. k-times differentiables functions on the closed interval ab), the space of Lebesgue (absolutely) integrable functions and the space of essentially bounded Borel functions on R.

Given f:RC and xR we set

fx+=limh0+fx+h,E1
fx=limh0fx+h,E2
δfx=fx+fx.E3

We denote by BVR the space of bounded variation complex functions on R; we refer to [5, 6] for the main properties of functions in BVR.

Any real function of bounded variation can be written as a difference of two non decreasing functions. It follows that if fBVR then fx+, fx and δfx exist for each xR and the set xRδfx0 is an arbitrary at most countable subset of R. Moreover, the derivative fx exists for almost all xR and fxL1R.

Let fBVR. We denote by df the unique Radon measure on R such that for each open interval ]a,b[R

dfa,b=fafb+.E4

We recall that f is special if for any bounded Borel function u

Ruxdfx=Ruxfxdx+xRuxδfx.E5

We denote by SBVR the space of all special functions of bounded variation. We also say that fBVlocR (resp. fSBVlocR) if for each a,bR, with a<b the function

fx=0ifx<aorx>b,fxifaxb,E6

is in BVR (resp. SBVR).

We define SBVnR inductively setting

SBV1R=SBVR,E7

and for each integer n>1

SBVnR=fSBVRfSBVn1RE8

We denote by Bn and Bnx, n=1,2, respectively the Bernoulli numbers and the Bernoulli functions. Let us recall that

B1x=0ifxZ,xx12ifxR\Z,E9

where x stands for the greatest integer less than or equal to x and Bnx, n=2,3, are the unique continuous functions such that

Bnx+1=Bnx,E10
Bnx=nBn1x,E11
01Bnxdx=0.E12

Moreover B2n+1=0 for n>0 and Bn=Bn0 for n>1.

The main results of this paper is the following theorem.

Theorem 0.1 Let fSBVmR, m1 and suppose f,,fmL1R. Then

nZfn++fn2=Rfxdx+xRk=1m1k1k!Bkxδfk1x+1m1m!RBmxfmxdx.E13

Remark. The sum “xR” in the right hand side of the above “Euler-Maclaurin formula” (13) is actually a sum over the subset of the xR such that some of the terms Bkxδfk1x do not vanish. We point out that such a set can be an arbitrary at most countable subset of R.

Remark. Let p and q, p<q be two integers and let f be a function of class Cm on the interval pq. Set fx=0 when x is outside of the interval pq. Then the classical Euler-Maclaurin formula (see, e.g. Section 9.5 of [7])

k=pq1fk=pqfxdx+k=1mBkk!fk1qfk1p+1m1Bmm!pqBmxfmxdx,E14

follows easily from Theorem 0.1.

Remark. Any fBVR decomposes uniquely as f=f1+f2+f3, where f1x can be written in the form

f1x=n=1+φnxE15

where each φnx is a piece-wise constant function, f2x is an absolutely continuous function and f3x is a singular function, that is f3x is continuous and f3x=0 for almost all xR. Then f=f1+f2+f3 is special if, and only if, f3=0 and in this case, for each bounded Borel function ux,

Ruxdf1x=xRuxδfx,E16
Ruxdf2x=Ruxfxdx.E17

In this paper we do not need of the existence of such a decomposition.

## 2. Integration by parts formulas

Our starting point is the following theorem:

Theorem 0.2 Let f,g:RC two complex function. Assume that fBVRL1R and gBVlocRLR. Then

Rfx+dgx+Rgxdfx=0,E18
Rfxdgx+Rgx+dfx=0,E19
Rfx++fx2dgx+Rgx++gx2dfx=0.E20

Proof: Let a,bR with a<b. Theorem 7.5.9 of [5] yields

]a,b[fx+dgx+]a,b[gxdfx=fbgbfa+ga+,E21
]a,b[fxdgx+]a,b[gx+dfx=fbgbfa+ga+.E22

Since fL1R then necessarily

limb+fb=limafa+=0.E23

Since gLR then gx+ and gx are bounded and we also have

limb+fbgb=limafa+ga+=0.E24

and hence one obtains the formulas (18) and (19) taking the limits as a and b+ respectively in (21) and (22).

Formula (20) is obtained summing memberwise (18) and (19) and dividing by two. □

Next we prove:

Theorem 0.3 Let f,g:RC two complex function. Assume that fBVRL1R and gSBVlocRLR and suppose that gLR. Then

Rfxgxdx+xR'fx+δgx+Rgxdfx=0,E25
Rfxgxdx+xR'fxδgx+Rgx+dfx=0,E26
Rfxgxdx+xR'fx++fx2δgx+Rgx++gx2dfx=0.E27

where

xR'limab+a<x<b.E28

Moreover, if the function f also is continuous then

Rfxgxdx+Rgxdfx=0,E29

Proof: Given a,bR, a<b set

gabx=0xa,gxa<x<b,0xb.E30

The function hx=gabx is in SBVRLR. Hence, formula (18) yields

Rfx+dhx+Rhxdfx=0.E31

Since hSBVR we have

Rfx+dhx=abfx+gxdx+xRfx+δgabx.E32

But fx+=fx for almost all xR and hence

Rfx+dhx=abfxgxdx+xRfx+δgabx,E33

which combined with (31) yields

abfxgxdx+xRfx+δgabx+Rgabxdfx=0.E34

Using the definition of gabx we have

xRfx+δgabx=fa+ga++a<x<bfx+δgx,E35

and hence

a<x<bfx+δgx=fa+ga+abfxgxdxRgabxdfx.E36

As in the proof of the previous theorem we have

limafa+ga+=0.E37

Since fL1R and gLR then fgL1R and hence

limab+abfxgxdx=Rfxgxdx.E38

The Radon measure dfx is bounded and the functions xgabx are equibounded with respect to a and b; by the Lebesgue dominated convergence we have

limab+Rgabxdfx=Rgxdfx.E39

From (36) it follows that

limab+a<x<bfx+δgx=xR'fx+δgx=RfxgxdxRgxdfxE40

which is equivalent to (25).

The proof of (26) is obtained in a similar manner using (19) instead of (18), and (27) is obtained summing memberwise (25) and (26) and dividing by two.

If the function g is continuous then gx+=gx=gx for each xR,

xR'fx+δgx=0,E41

and (29) follows from, e.g., (25).

Example. This example shows that in the hypoteses of Theorem (0.3) the series

xR'fx+δgxE42

is not, in general, absolutely convergent. Indeed, set

fx=0ifx1/2,1/x2ifx>1/2,E43

and

gx=1if2n1<x2n,nZ,0if2n<x2n+1,nZ.E44

Then the integral

Rfxgxdx=n=1+2n12ndfx=n=1+12n2n1E45

is absolutely convergent, but the series

xR'fx+δgx=n=1+1nnE46

is convergent but not absolutely convergent.

We also have the following theorem.

Theorem 0.4 Let f,g:RC two complex function. Assume that fSBVRL1R and gSBVlocRLR and suppose that gLR. Then

Rfxgxdx+Rfxgxdx+xRδfxgx++xR'fxδgx=0,E47
Rfxgxdx+Rfxgxdx+xRδfxgx+xR'fx+δgx=0,E48
Rfxgxdx+Rfxgxdx+xRgx++gx2δfx+xR'fx++fx2δgx=0,E49

where

xR'limab+a<x<bE50

If the function g also is continuous then

Rfxgxdx+Rfxgxdx+xRgxδfx=0,E51

Proof: Let f and g be as in the theorem. By formula (26) we have

Rfxgxdx+xR'fxδgx+Rgx+dfx=0.E52

Since fSBVR, using the fact that gx+=gx for almost all xR, we obtain

Rgx+dfx=Rgxfxdx+xRgx+δfx.E53

Then (52) and (53) yield (47). Formulas (48) and (49) are obtained in a similar manner using respectively Formulas (25) and (27) instead of (26).

If the function g is continuous then gx+=gx=gx for each xR,

xR'fx+δgx=0,E54

and (51) follows from, e.g., (47).□

Theorem 0.4 generalizes to high order derivatives.

Theorem 0.5 Let f,g:RC two complex function. Let m>0 be a positive integer. Assume that fSBVmR with f,,fmL1R and gSBVlocmR with g,,gmLR. Then

1m1Rfmxgxdx+Rfxgmxdx+xRk=1m1k1δfk1xgmkx++xR'k=1m1k1fk1xδgmkx=0,E55
1m1Rfmxgxdx+Rfxgmxdx+xRk=1m1k1δfk1xgmkx+xR'k=1m1k1fk1x+δgmkx=0,E56
1m1Rfmxgxdx+Rfxgmxdx+xRk=1m1k1δfk1xgmkx+gmkx+2+xR'k=1m1k1fk1x+fk1x+2δgmkx=0,E57

Proof: We prove first the formula (55). The proof is by induction on m. When m=1(55) reduces to (47). Assume that (55) holds for m1, that is

1m2Rfm1xgxdx+Rfxgm1xdx+xRk=1m11k1δfk1xgmk1x++xR'k=1m11k1fk1xδgmk1x=0.E58

Replacing f with f, k with k+1 and changing the sign we obtain

1m1RfmxgxdxRfxgm1xdx+xRk=2m1k1δfk1xgmkx++xR'k=2m1k1fk1xδgmkx=0.E59

Replacing g with gm1 in (47) we obtain

Rfxgm1)xdx+Rfxgmxdx+xRδfxgm1x++xR'fx+δgm1x=0.E60

Summing (59) and (60) we obtain (55).

The proofs of (56) and (57) are similar.

We say that a function fSBVlocR is a step function if fx=0 for almost every xR.

The following propositions are easy consequences of Theorem (0.4).

Proposition 0.6 Let uvR be a bounded closed interval and let f be an absolutely continuous function on the closed interval uv. Let gSBVlocR be a step function. Then

uvfxgxdx=fvgvfugu+u<x<vfxδgx.E61

Proof: First we extend the functions f as zero outside of the interval uv. We may also assume that the function g is zero outside of a bounded open interval containing the closed interval uv. Observe that then fu+=fu, fv=fv and fu=fv+=0 and therefore δfu=fu, δfv=fv and δfx=0 for xu,v. By (47), we have

Rfxgxdx+Rfxgxdx+xR'fx+δgx+xRgxδfx=0.E62

Since g is a step function then gx=0 for almost all xR and hence it follows that

Rfxgxdx=xR'fx+δgxxRgxδfx.E63

The function f by construction has compact support, and hence, as fv+=0, we have

xR'fx+δgx=fu+gu+gu+u<x<vfx+δgx=fugu+fugu+u<x<vfx+δgx,E64

and

xRgxδfx=guδfu+gvδfv=fugufvgv.E65

Summing memberwise the last two formulas we obtain

xR'fx+δgx+xRgxδfx=fvgv+fugu++u<x<vfx+δgx,E66

as desired.□

Proposition 0.7 Let f,gSBVlocR be two step function. Let uvR be a bounded closed interval. Then

u<x<vgx+δfx=fvgvfu+gu+u<x<vfxδgx.E67

Proof: Set both the functions f and g to zero outside the closed interval uv. Then formula (47) yields

xRfx+δgx+xRgxδfx=0.E68

But then

xRfx+δgx=fu+gu++u<x<vfx+δgx,E69

and

xRgxδfx=fvgv+u<x<vgxδfx;E70

hence

fu+gu++u<x<vfx+δgxfvgv+u<x<vgxδfx=0,E71

which is equivalent to (67).□

Example 1. (Abel summation I) Let an, nZ be a sequence of complex numbers such that an=0 for n<<0. Then the function

Ax=n<xanE72

is a step function in SBVlocR. If fC1uv then Proposition (0.6) yields

uvfxAxdx=fvAvfuAu+u<n<vfnan.E73

Example 2. (Abel summation II) Let an,bn, nZ be two sequence of complex numbers. Let f,gC be defined respectively setting fx=an and gx=bn when nx<n+1, nZ. Clearly f,gSBVlocR and they are two step functions. Let be given two integers p and q, p<q. Set u=p and v=q+1. Then it is easy to show that

u<x<vgx+δfx=n=p+1qbnanan1E74

and

u<x<vfxδgx=n=p+1qan1bnbn1;E75

hence, Proposition (0.7) yields

n=p+1qbnanan1=aqbqapbpn=p+1qan1bnbn1.E76

## 3. Sampling estimates

In this section we give some conditions which ensures the absolute convergence of series of the form xEfx+fx+/2 where f is a function absolutely integrable of bounded variation and E is a countable subset of R.

The basic estimate is given in the following lemma.

Lemma 0.8 Let AR be an open subset and let FA be a finite subset of A. Assume that there exist a>0 such that

x1,x2F,x1x2x1x2a,xF,yR\Axya/2.E77

Then, for any complex functionfBVRL1R we have

Proof: Let define

gx=0,ifx<1/2orx=0orx1/2,x+1/2,if1/2x<0,x1/2,if0x<1/2,E79

and set

Gx=yFgxya.E80

For each xR we have

Gx+Gx+2=GxE81

By Eq. (27)

xR'fx++fx2δGx=RfxGxdx+RGxdfx.E82

We also have

δGx=1ifxF,0ifxR\F,E83

which implies

xR'fx++fx2δGx=xFfx+fx+2.E84

Set

E=xF]xa,x+a[.E85

Then FEA and

Gx=1/aifxE,0ifxR\E,E86

and hence

RfxGxdx=1aEfxdx.E87

Moreover we have Gx=0 if xR\E and hence

xFfx+fx+2=RfxGxdx+RGxdfx=1aEfxdx+EGxdfx.E88

Taking modules, and observing that Gx1/2 for each xE, we obtain

as required.

Corollary 0.9 Let fBVRL1R and let ER be a countable subset. If there exists a real constant a>0 such that for each pair of distinct x1,x2E we have x1x2a then

xEfx+fx+2<+.E90

Proof: It suffices to choose A=R; lemma (0.8) yields easily the assertion.

## 4. Proof of Theorem 0.1

Inserting Bmx instead of gmx in formula (57) of Theorem 0.5 we easily obtain

nZ'fn++fn2=Rfxdx+xRk=1m1k1k!Bkxδfk1x+1m1m!RBmxfmxdx.E91

By Corollary 0.9 it follows that

nZ'fn++fn2=nZfn++fn2E92

is an absolutely convergent series, and hence Theorem 0.1 follows.

## References

1. 1. Graham Everest and Thomas Ward. An Introduction to Number Theory. Springer-Verlag, 2005.
2. 2. Henri Cohen. Number Theory Volume II: Analytic and Modern Tools, volume 240 of Graduate Texts in Mathematics. Springer-Verlag, 2007.
3. 3. Ennio De Giorgi and Luigi Ambrosio. New functionals in the calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 82(2):199–210 (1989), 1988.
4. 4. L. Ambrosio, M. Miranda, Jr., and D. Pallara. Special functions of bounded variation in doubling metric measure spaces. In Calculus of variations: topics from the mathematical heritage of E. De Giorgi, volume 14 of Quad. Mat., pages 1–45. Dept. Math., Seconda Univ. Napoli, Caserta, 2004.
5. 5. Stanislaw Lojasiewicz. An Introduction to the Theory of Real Functions. Wiley, 1988.
6. 6. Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Function of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, 2000.
7. 7. Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics. Addison-Wesley Publishing Company, 1989.

Written By

Sergio Venturini

Reviewed: 09 September 2021 Published: 19 October 2021