the classification of generalized riemann...

Definition and basic propertiesGeneralized vs. Ordinary Differentiation

The Classification of Real Generalized Riemann DerivativesThe Classification of Complex Generalized Riemann Derivatives

The Classification of Generalized RiemannDerivatives

Stefan Catoiu

DePaul University, Chicago

Groups, Rings and the Yang-Baxter EquationSpa, Belgium, June 18-24, 2017

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Outline

1 Definition and basic propertiesDefinition and basic properties

2 Generalized vs. Ordinary DifferentiationGeneralized vs. Ordinary Differentiation

3 The Classification of Real Generalized Riemann DerivativesThe Classification of Real Generalized RiemannDerivatives

4 The Classification of Complex Generalized RiemannDerivatives

The Classification of Complex Generalized RiemannDerivatives

Definition and basic properties

Outline

DefinitionAn nth generalized Riemann derivative or the A-derivative ofa real function f at x is defined by the limit

DAf (x) = limh→0

∑mi=0 Ai f (x + aih)

where the data vector A = {Ai ,ai : i = 1, . . . ,m} of 2m realnumbers satisfies the nth Vandermonde relations

m∑i=1

Aiaji =

{0 if j = 0,1, . . . ,n − 1,n! if j = n.

The numerator ∆Af (x ,h) is an nth generalized Riemanndifference.Linear algebra forces m > n.

The Vandermonde conditions assure that the ordinaryderivative implies the generalized derivative and these areequal whenever they both exist.Indeed, for n = 1, suppose that f is differentiable at x , andlet A be a data vector of a first GRD. We have

DAf (x) = limh→0

= limh→0

∑mi=0 Ai [f (x + aih)− f (x)] +

∑mi=0 Ai f (x)

= limh→0

m∑i=0

Aiaif (x + aih)− f (x)

)f (x)

)f ′(x) + 0 = f ′(x).

Thus f differentiable at x implies f is A-differentiable at x .Stefan Catoiu The Classification of Generalized Riemann Derivatives

The Vandermonde conditions assure that the ordinaryderivative implies the generalized derivative and these areequal whenever they both exist.Indeed, for n = 1, suppose that f is differentiable at x , andlet A be a data vector of a first GRD. We have

DAf (x) = limh→0

= limh→0

∑mi=0 Ai [f (x + aih)− f (x)] +

∑mi=0 Ai f (x)

= limh→0

m∑i=0

Aiaif (x + aih)− f (x)

)f (x)

)f ′(x) + 0 = f ′(x).

Thus f differentiable at x implies f is A-differentiable at x .Stefan Catoiu The Classification of Generalized Riemann Derivatives

Examples of Generalized Riemann Derivatives

First order:the ordinary derivative has A = {1,−1; 1,0}:

f ′(x) = limh→0

f (x + h)− f (x)

the symmetric derivative has A = {1,−1; 12 ,−

f ′s(x) = limh→0

f (x + h2 )− f (x − h

the “crazy” derivative has A = {2,−3,1; 1,0,−1}:

f ′∗(x) = limh→0

2f (x + h)− 3f (x) + f (x − h)

Higher order:the second symmetric (Schwarz) derivative hasA = {1,−2,1; 1,0,−1}:

f ′′s (x) = limh→0

f (x + h)− 2f (x) + f (x − h)

the nth forward Riemann derivative

Rnf (x) = limh→0

∑nk=0(−1)k(n

)f (x + kh)

the nth symmetric Riemann derivative

Rsnf (x) = lim

∑nk=0(−1)k(n

)f (x + (n

2 − k)h)

Quantum Riemann derivatives (after Ash, C, Rios, 2002):the nth forward quantum Riemann derivative

R[n]f (x) = limq→1

∑nk=0(−1)k[n

2)f (qn−kx)

(qx − x)n

the nth symmetric quantum Riemann derivative

Rs[n]f (x) = lim

∑nk=0(−1)k[n

2)f (qn2−kx)

(qx − x)n

Generalized vs. Ordinary Differentiation

Outline

Generalized vs. Ordinary Differentiation I

Ordinary nth differentiability⇒ nth A-differentiabilityfor functions f at x comes by Taylor expansion.The converse is false: Function f (x) = |x | has

f ′s(0) = limh→0

f (0 + h2 )− f (0− h

h= lim

∣∣h2

∣∣− ∣∣−h2

∣∣h

while f ′(0) does not exist.Moreover, any even function is symmetric Riemanndifferentiable of any odd order but not symmetric Riemanndefferentiable of any even order. In particular, higher ordergeneralized differentiation does not imply lower ordergeneralized differentiation.Since the pointwise relation between the ordinary andgeneralized derivative seemed pointless, the researchfocus moved to the a.e. relation.

f ′s(0) = limh→0

f (0 + h2 )− f (0− h

h= lim

∣∣h2

∣∣− ∣∣−h2

∣∣h

f ′s(0) = limh→0

f (0 + h2 )− f (0− h

h= lim

∣∣h2

∣∣− ∣∣−h2

∣∣h

f ′s(0) = limh→0

f (0 + h2 )− f (0− h

h= lim

∣∣h2

∣∣− ∣∣−h2

∣∣h

A Short History

First symmetric derivative a.e. equivalent to first derivative:Kinchin, 1927.nth symmetric Riemann derivative equivalent a.e. to nthderivative: Marcinkiewicz and Zygmund, 1936.GRDs were introduced by A. Denjoy in 1935.nth A-differentiability a.e. equivalent to nth differentiabiltity:J. M. Ash, 1967.nth quantum Riemann a.e. equivalent to nth derivative:Ash, C. and Rios, 2002.nth quantum Riemann in Lp a.e. equivalent to nthderivative: Ash and C., 2008.

A Short History

Generalized vs. Ordinary Differentiation II

Revisiting the pointwise implication: (Ash, C. and Csörnyei,2017) Suppose that for a function f the “crazy” derivative at x

f ′∗(x) = limh→0

2f (x + h)− 3f (x) + f (x − h)

hexists. By changing h into −h we also have

f ′∗(x) = limh→0

f (x + h)− 3f (x) + 2f (x − h)

Multiplying the first equation by 2/3 and the second by 1/3 andadding yields

f ′∗(x) = limh→0

f (x + h)− f (x)

so f is ordinary differentiable at x and f ′(x) = f ′∗(x).Conclude that some generalized derivatives are equivalentto ordinary derivative and some are not.

f ′∗(x) = limh→0

2f (x + h)− 3f (x) + f (x − h)

f ′∗(x) = limh→0

f (x + h)− 3f (x) + 2f (x − h)

f ′∗(x) = limh→0

f (x + h)− f (x)

f ′∗(x) = limh→0

2f (x + h)− 3f (x) + f (x − h)

f ′∗(x) = limh→0

f (x + h)− 3f (x) + 2f (x − h)

f ′∗(x) = limh→0

f (x + h)− f (x)

f ′∗(x) = limh→0

2f (x + h)− 3f (x) + f (x − h)

f ′∗(x) = limh→0

f (x + h)− 3f (x) + 2f (x − h)

f ′∗(x) = limh→0

f (x + h)− f (x)

Question 1. Which generalized Riemann differentiationsare equivalent to ordinary differentiation?The answer is given in the following theorem:

A dilation by a non-zero parameter r of an order n difference∆Af (x ,h) is the difference 1

rn ∆Af (x , rh).

Question 1. Which generalized Riemann differentiationsare equivalent to ordinary differentiation?The answer is given in the following theorem:

A dilation by a non-zero parameter r of an order n difference∆Af (x ,h) is the difference 1

rn ∆Af (x , rh).

Theorem 1 (Ash, C. and Csörnyei, 2017)(i) The first order A-derivatives which are dilates (h→ sh, forsome s 6= 0) of

limh→0

[f (x + h)− f (x − h)] + A [f (x + rh)− 2f (x) + f (x − rh)]

where Ar 6= 0 are equivalent to ordinary differentiation.(ii) Given any other A-derivative of any order n = 1,2, . . . , thereis a measurable function f (x) such that DAf (0) exists, but theordinary derivative f (n) (0) does not.

The proof uses linear algebra of infinite systems of linearequations.

limh→0

In particular, the following computationsf (x+h)−f (x)

h = [f (x+h)−f (x−h)]+[f (x+h)−2f (x)+f (x−h)]2h ,

f (x+ h2 )−f (x− h

h is the dilate (h 7→ 12h) of [f (x+h)−f (x−h)]+0[··· ]

2h ,2f (x+h)−3f (x)+f (x−h)

h = [f (x+h)−f (x−h)]+3[f (x+h)−2f (x)+f (x−h)]2h

and Theorem 1 confirm that ordinary and “crazy” differentiationare equivalent to ordinary differentiation, while the symmetricdifferentiation is not.

The Classification of Real Generalized Riemann Derivatives

Outline

Two more questions:Question 2. For which pairs of data vectors (A,B) oforders (m,n), A-differentiation is equivalent toB-differentiation?Question 3. For which pairs of data vectors (A,B) oforders (m,n), A-differentiation implies B-differentiation?

The answer to Q2 amounts to describing the partition of allgeneralized Riemann derivatives into equivalence classes.Theorem 1 describes the equivalence class of the ordinary firstderivative.

Even and Odd Functions

Every function f is expressed uniquely as a sum f = f1 + f2of an odd function f1 and an even function f2. We have

f1(x) =f (x)− f (−x)

2and f2(x) =

f (x) + f (−x)

Even and Odd Differences

Every difference ∆Af (x ,h) =∑m

i=1 Ai f (x + aih) isexpressed uniquely as the sum

∆Af (x ,h) =∆Af (x ,h) + ∆Af (x ,−h)

∆Af (x ,h)−∆Af (x ,−h)

of an h-even difference ∆evA f (x ,h) and an h-odd difference

∆oddA f (x ,h).

Let ∆Af (x ,h) be a generalized Riemann difference oforder n and let ∆ε

Af (x ,h) and ∆ε′Af (x ,h) be the two

previous terms that have the same or opposite parity as n.Then ∆ε

Af (x ,h) is a generalized Riemann difference oforder n and ∆ε′

Af (x ,h) has order > n.

∆Af (x ,h) =∆Af (x ,h) + ∆Af (x ,−h)

∆Af (x ,h)−∆Af (x ,−h)

∆oddA f (x ,h).

∆Af (x ,h) =∆Af (x ,h) + ∆Af (x ,−h)

∆Af (x ,h)−∆Af (x ,−h)

∆oddA f (x ,h).

The Classification of Real Generalized RiemannDerivatives: The Equivalence Relation

Theorem 2 (Ash, C. and Chin, 2017)Let A and B be the data vectors of Generalized Riemanndifferences of orders m and n. For measurable functions f ,TFAE:

1 f is A-differentiable iff f is B-differentiable;2 m = n and up to independent non-zero dilations,{

∆εAf (x ,h) = ∆ε

Bf (x ,h) and∆ε′Af (x ,h) = A∆ε′

B f (x ,h), A 6= 0.

The Classification of Real Generalized RiemannDerivatives: The Implication Relation

Theorem 3 (Ash, C. and Chin, 2017)Let A and B be GRD data vectors of orders m and n. TFAE:

1 B-differentiability⇒ A-differentiability;2 m = n and for each measurable function f , the components

∆εAf (x ,h) and ∆ε′

Af (x ,h) are finite linear combinations{∆εAf (x ,h) =

∑i Ui∆

εBf (x ,uih) and

∆ε′Af (x ,h) =

∑i Vi∆

ε′B f (x , vih)

of non-zero ui -dilates of ∆εBf (x ,h) and vi -dilates of

∆ε′B f (x ,h).

The Classification of Complex Generalized Riemann Derivatives

Outline

Generalized even and odd functions

Fix an integer ` > 1, and let ω be a primitive `th root of unity.Every complex function f is expressed uniquely as a sum

f (x) =`−1∑i=0

where fi(ωx) = ωi fi(x), for i = 0,1, . . . , `− 1.The function fi has the expression

fi(x) =1`

`−1∑j=0

ω−ij f (ωjx).

Generalized even and odd functions

Fix an integer ` > 1, and let ω be a primitive `th root of unity.Every complex function f is expressed uniquely as a sum

f (x) =`−1∑i=0

where fi(ωx) = ωi fi(x), for i = 0,1, . . . , `− 1.The function fi has the expression

fi(x) =1`

`−1∑j=0

ω−ij f (ωjx).

Generalized even and odd differences

Every difference ∆Af (x ,h) is expressed uniquely as

∆Af (x ,h) =`−1∑i=0

∆iAf (x ,h),

where ∆Af (x , ωh) = ωi∆iAf (x ,h), for i = 0,1, . . . , `− 1.

Each component ∆iAf (x ,h) can be written explicitly.

If ∆Af (x ,h) is an nth GR Difference, then

∆iAf (x ,h)

{is an nth GR Difference, if i = n mod `

is a GR Difference of order > n, if i 6= n mod `.

∆Af (x ,h) =`−1∑i=0

∆iAf (x ,h),

∆iAf (x ,h)

∆Af (x ,h) =`−1∑i=0

∆iAf (x ,h),

∆iAf (x ,h)

The Classification of Complex Generalized RiemannDerivatives: The Equivalence Relation

Theorem 4 (Ash, C. and Chin, 2017)Let A and B be the data vectors of complex generalizedRiemann differences of orders m and n. For measurablecomplex functions f , TFAE:

1 f is A-differentiable iff f is B-differentiable;2 m = n and up to ` independent non-zero dilations,

∆iAf (x ,h) =

{∆iBf (x ,h) if i = n̄

αi∆iBf (x ,h), αi 6= 0 if i 6= n̄.

where n̄ = n mod ` and i = 0,1, . . . , `− 1.

The proof of Theorems 2-4 involves the group algebra kG of thegroup G = k× over the field k = R or k = C.

References1 J. M. Ash, Generalizations of the Riemann derivative,

Trans. Amer. Math. Soc. 126 (1967), 181–199.2 J. M. Ash and S. Catoiu, Quantum symmetric Lp

derivatives, Trans. Amer. Math. Soc., 360 (2008), no. 2,959–987.

3 J. M. Ash, S. Catoiu and M. Csörnyei, Generalized vs.ordinary differentiation, Proc. Amer. Math. Soc. 145(2017), no. 4, 1553–1565.

4 J. M. Ash, S. Catoiu and W. Chin, The classification ofgeneralized Riemann derivatives, preprint.

5 J. M. Ash, S. Catoiu and W. Chin, The classification ofcomplex generalized Riemann derivatives, in progress.

1 J. M. Ash, S. Catoiu, R. Rios, On the nth quantumderivative, J. London Math. Soc., 66 (2002), no. 1,114–130.

2 A. Denjoy, Sur l’intégration des coefficients différentielsd’ordre supérieur, Fund. Math., 25 (1935), 273–326.

3 A. Khintchine, Recherches sur la structure des fonctionsmesurables, Fund. Math., 9 (1927), 212–279.

4 J. Marcinkiewicz and A. Zygmund, On the differentiability offunctions and summability of trigonometric series, Fund.Math. 26 (1936), 1–43

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