The Classification of Generalized Riemann Derivatives Stefan Catoiu - - PowerPoint PPT Presentation

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

The Classification of Generalized Riemann Derivatives

Stefan Catoiu

DePaul University, Chicago

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

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Outline

1

Definition and basic properties Definition and basic properties

2

Generalized vs. Ordinary Differentiation Generalized vs. Ordinary Differentiation

3

The Classification of Real Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

4

The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Definition and basic properties

Outline

1

Definition and basic properties Definition and basic properties

2

Generalized vs. Ordinary Differentiation Generalized vs. Ordinary Differentiation

3

The Classification of Real Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

4

The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Definition and basic properties

Definition An nth generalized Riemann derivative or the A-derivative of a real function f at x is defined by the limit DAf(x) = lim

h→0

m

i=0 Aif(x + aih)

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

m

• i=1

Aiaj

i =

• if j = 0, 1, . . . , n − 1,

n! if j = n. The numerator ∆Af(x, h) is an nth generalized Riemann difference. Linear algebra forces m > n.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Definition and basic properties

The Vandermonde conditions assure that the ordinary derivative implies the generalized derivative and these are equal whenever they both exist. Indeed, for n = 1, suppose that f is differentiable at x, and let A be a data vector of a first GRD. We have DAf(x) = lim

h→0

m

i=0 Aif(x + aih)

h = lim

h→0

m

i=0 Ai[f(x + aih) − f(x)] + m i=0 Aif(x)

h = lim

h→0 m

• i=0

Aiai f(x + aih) − f(x) aih + m

• i=0

Ai

• f(x)

h = m

• i=0

Aiai

• 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

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Definition and basic properties

The Vandermonde conditions assure that the ordinary derivative implies the generalized derivative and these are equal whenever they both exist. Indeed, for n = 1, suppose that f is differentiable at x, and let A be a data vector of a first GRD. We have DAf(x) = lim

h→0

m

i=0 Aif(x + aih)

h = lim

h→0

m

i=0 Ai[f(x + aih) − f(x)] + m i=0 Aif(x)

h = lim

h→0 m

• i=0

Aiai f(x + aih) − f(x) aih + m

• i=0

Ai

• f(x)

h = m

• i=0

Aiai

• 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

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Definition and basic properties

Examples of Generalized Riemann Derivatives

First order: the ordinary derivative has A = {1, −1; 1, 0}: f ′(x) = lim

h→0

f(x + h) − f(x) h ; the symmetric derivative has A = {1, −1; 1

2, − 1 2}:

f ′

s(x) = lim h→0

f(x + h

2) − f(x − h 2)

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

∗(x) = lim h→0

2f(x + h) − 3f(x) + f(x − h) h ;

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Definition and basic properties

Examples of Generalized Riemann Derivatives

Higher order: the second symmetric (Schwarz) derivative has A = {1, −2, 1; 1, 0, −1}: f ′′

s (x) = lim h→0

f(x + h) − 2f(x) + f(x − h) h2 ; the nth forward Riemann derivative Rnf(x) = lim

h→0

n

k=0(−1)kn k

• f(x + kh)

hn the nth symmetric Riemann derivative Rs

nf(x) = lim h→0

n

k=0(−1)kn k

• f(x + ( n

2 − k)h)

hn

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Definition and basic properties

Examples of Generalized Riemann Derivatives

Quantum Riemann derivatives (after Ash, C, Rios, 2002): the nth forward quantum Riemann derivative R[n]f(x) = lim

q→1

n

k=0(−1)kn k

• q(k

2)f(qn−kx)

(qx − x)n the nth symmetric quantum Riemann derivative Rs

[n]f(x) = lim q→1

n

k=0(−1)kn k

• q(k

2)f(q n 2 −kx)

(qx − x)n

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

Outline

1

Definition and basic properties Definition and basic properties

2

Generalized vs. Ordinary Differentiation Generalized vs. Ordinary Differentiation

3

The Classification of Real Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

4

The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

Generalized vs. Ordinary Differentiation I

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

s(0) = lim h→0

f(0 + h

2) − f(0 − h 2)

h = lim

h→0

• h

2

• −
• − h

2

• h

= 0, while f ′(0) does not exist. Moreover, any even function is symmetric Riemann differentiable of any odd order but not symmetric Riemann defferentiable of any even order. In particular, higher order generalized differentiation does not imply lower order generalized differentiation. Since the pointwise relation between the ordinary and generalized derivative seemed pointless, the research focus moved to the a.e. relation.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

Generalized vs. Ordinary Differentiation I

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

s(0) = lim h→0

f(0 + h

2) − f(0 − h 2)

h = lim

h→0

• h

2

• −
• − h

2

• h

= 0, while f ′(0) does not exist. Moreover, any even function is symmetric Riemann differentiable of any odd order but not symmetric Riemann defferentiable of any even order. In particular, higher order generalized differentiation does not imply lower order generalized differentiation. Since the pointwise relation between the ordinary and generalized derivative seemed pointless, the research focus moved to the a.e. relation.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

Generalized vs. Ordinary Differentiation I

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

s(0) = lim h→0

f(0 + h

2) − f(0 − h 2)

h = lim

h→0

• h

2

• −
• − h

2

• h

= 0, while f ′(0) does not exist. Moreover, any even function is symmetric Riemann differentiable of any odd order but not symmetric Riemann defferentiable of any even order. In particular, higher order generalized differentiation does not imply lower order generalized differentiation. Since the pointwise relation between the ordinary and generalized derivative seemed pointless, the research focus moved to the a.e. relation.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

Generalized vs. Ordinary Differentiation I

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

s(0) = lim h→0

f(0 + h

2) − f(0 − h 2)

h = lim

h→0

• h

2

• −
• − h

2

• h

= 0, while f ′(0) does not exist. Moreover, any even function is symmetric Riemann differentiable of any odd order but not symmetric Riemann defferentiable of any even order. In particular, higher order generalized differentiation does not imply lower order generalized differentiation. Since the pointwise relation between the ordinary and generalized derivative seemed pointless, the research focus moved to the a.e. relation.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

A Short History

First symmetric derivative a.e. equivalent to first derivative: Kinchin, 1927. nth symmetric Riemann derivative equivalent a.e. to nth derivative: 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 nth derivative: Ash and C., 2008.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

A Short History

First symmetric derivative a.e. equivalent to first derivative: Kinchin, 1927. nth symmetric Riemann derivative equivalent a.e. to nth derivative: 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 nth derivative: Ash and C., 2008.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

A Short History

First symmetric derivative a.e. equivalent to first derivative: Kinchin, 1927. nth symmetric Riemann derivative equivalent a.e. to nth derivative: 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 nth derivative: Ash and C., 2008.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

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) = lim h→0

2f(x + h) − 3f(x) + f(x − h) h

• exists. By changing h into −h we also have

f ′

∗(x) = lim h→0

f(x + h) − 3f(x) + 2f(x − h) −h . Multiplying the first equation by 2/3 and the second by 1/3 and adding yields f ′

∗(x) = lim h→0

f(x + h) − f(x) h , so f is ordinary differentiable at x and f ′(x) = f ′

∗(x).

Conclude that some generalized derivatives are equivalent to ordinary derivative and some are not.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

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) = lim h→0

2f(x + h) − 3f(x) + f(x − h) h

• exists. By changing h into −h we also have

f ′

∗(x) = lim h→0

f(x + h) − 3f(x) + 2f(x − h) −h . Multiplying the first equation by 2/3 and the second by 1/3 and adding yields f ′

∗(x) = lim h→0

f(x + h) − f(x) h , so f is ordinary differentiable at x and f ′(x) = f ′

∗(x).

Conclude that some generalized derivatives are equivalent to ordinary derivative and some are not.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

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) = lim h→0

2f(x + h) − 3f(x) + f(x − h) h

• exists. By changing h into −h we also have

f ′

∗(x) = lim h→0

f(x + h) − 3f(x) + 2f(x − h) −h . Multiplying the first equation by 2/3 and the second by 1/3 and adding yields f ′

∗(x) = lim h→0

f(x + h) − f(x) h , so f is ordinary differentiable at x and f ′(x) = f ′

∗(x).

Conclude that some generalized derivatives are equivalent to ordinary derivative and some are not.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

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) = lim h→0

2f(x + h) − 3f(x) + f(x − h) h

• exists. By changing h into −h we also have

f ′

∗(x) = lim h→0

f(x + h) − 3f(x) + 2f(x − h) −h . Multiplying the first equation by 2/3 and the second by 1/3 and adding yields f ′

∗(x) = lim h→0

f(x + h) − f(x) h , so f is ordinary differentiable at x and f ′(x) = f ′

∗(x).

Conclude that some generalized derivatives are equivalent to ordinary derivative and some are not.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

Question 1. Which generalized Riemann differentiations are 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

r n ∆Af(x, rh).

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

Question 1. Which generalized Riemann differentiations are 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

r n ∆Af(x, rh).

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

Generalized vs. Ordinary Differentiation

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

h→0

[f(x + h) − f(x − h)] + A [f(x + rh) − 2f(x) + f(x − rh)] 2h , where Ar = 0 are equivalent to ordinary differentiation. (ii) Given any other A-derivative of any order n = 1, 2, . . . , there is a measurable function f (x) such that DAf (0) exists, but the

• rdinary derivative f (n) (0) does not.

The proof uses linear algebra of infinite systems of linear equations.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

Generalized vs. Ordinary Differentiation

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

h→0

[f(x + h) − f(x − h)] + A [f(x + rh) − 2f(x) + f(x − rh)] 2h , where Ar = 0 are equivalent to ordinary differentiation. (ii) Given any other A-derivative of any order n = 1, 2, . . . , there is a measurable function f (x) such that DAf (0) exists, but the

• rdinary derivative f (n) (0) does not.

The proof uses linear algebra of infinite systems of linear equations.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

Generalized vs. Ordinary Differentiation

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

h→0

[f(x + h) − f(x − h)] + A [f(x + rh) − 2f(x) + f(x − rh)] 2h , where Ar = 0 are equivalent to ordinary differentiation. (ii) Given any other A-derivative of any order n = 1, 2, . . . , there is a measurable function f (x) such that DAf (0) exists, but the

• rdinary derivative f (n) (0) does not.

The proof uses linear algebra of infinite systems of linear equations.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation

In particular, the following computations

f(x+h)−f(x) h

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

2h

,

f(x+ h

2 )−f(x− h 2 )

h

is the dilate (h → 1

2h) 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” differentiation are equivalent to ordinary differentiation, while the symmetric differentiation is not.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

Outline

1

Definition and basic properties Definition and basic properties

2

Generalized vs. Ordinary Differentiation Generalized vs. Ordinary Differentiation

3

The Classification of Real Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

4

The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

Two more questions: Question 2. For which pairs of data vectors (A, B) of

• rders (m, n), A-differentiation is equivalent to

B-differentiation? Question 3. For which pairs of data vectors (A, B) of

• rders (m, n), A-differentiation implies B-differentiation?

The answer to Q2 amounts to describing the partition of all generalized Riemann derivatives into equivalence classes. Theorem 1 describes the equivalence class of the ordinary first derivative.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

Two more questions: Question 2. For which pairs of data vectors (A, B) of

• rders (m, n), A-differentiation is equivalent to

B-differentiation? Question 3. For which pairs of data vectors (A, B) of

• rders (m, n), A-differentiation implies B-differentiation?

The answer to Q2 amounts to describing the partition of all generalized Riemann derivatives into equivalence classes. Theorem 1 describes the equivalence class of the ordinary first derivative.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

Two more questions: Question 2. For which pairs of data vectors (A, B) of

• rders (m, n), A-differentiation is equivalent to

B-differentiation? Question 3. For which pairs of data vectors (A, B) of

• rders (m, n), A-differentiation implies B-differentiation?

The answer to Q2 amounts to describing the partition of all generalized Riemann derivatives into equivalence classes. Theorem 1 describes the equivalence class of the ordinary first derivative.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

Even and Odd Functions

Every function f is expressed uniquely as a sum f = f1 + f2

• f an odd function f1 and an even function f2. We have

f1(x) = f(x) − f(−x) 2 and f2(x) = f(x) + f(−x) 2 .

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

Even and Odd Differences

Every difference ∆Af(x, h) = m

i=1 Aif(x + aih) is

expressed uniquely as the sum ∆Af(x, h) = ∆Af(x, h) + ∆Af(x, −h) 2 +∆Af(x, h) − ∆Af(x, −h) 2

• f an h-even difference ∆ev

A f(x, h) and an h-odd difference

∆odd

A f(x, h).

Let ∆Af(x, h) be a generalized Riemann difference of

• rder n and let ∆ǫ

Af(x, h) and ∆ǫ′ Af(x, h) be the two

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

Af(x, h) is a generalized Riemann difference of

• rder n and ∆ǫ′

Af(x, h) has order > n.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

Even and Odd Differences

Every difference ∆Af(x, h) = m

i=1 Aif(x + aih) is

expressed uniquely as the sum ∆Af(x, h) = ∆Af(x, h) + ∆Af(x, −h) 2 +∆Af(x, h) − ∆Af(x, −h) 2

• f an h-even difference ∆ev

A f(x, h) and an h-odd difference

∆odd

A f(x, h).

Let ∆Af(x, h) be a generalized Riemann difference of

• rder n and let ∆ǫ

Af(x, h) and ∆ǫ′ Af(x, h) be the two

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

Af(x, h) is a generalized Riemann difference of

• rder n and ∆ǫ′

Af(x, h) has order > n.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

Even and Odd Differences

Every difference ∆Af(x, h) = m

i=1 Aif(x + aih) is

expressed uniquely as the sum ∆Af(x, h) = ∆Af(x, h) + ∆Af(x, −h) 2 +∆Af(x, h) − ∆Af(x, −h) 2

• f an h-even difference ∆ev

A f(x, h) and an h-odd difference

∆odd

A f(x, h).

Let ∆Af(x, h) be a generalized Riemann difference of

• rder n and let ∆ǫ

Af(x, h) and ∆ǫ′ Af(x, h) be the two

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

Af(x, h) is a generalized Riemann difference of

• rder n and ∆ǫ′

Af(x, h) has order > n.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

The Classification of Real Generalized Riemann Derivatives: The Equivalence Relation

Theorem 2 (Ash, C. and Chin, 2017) Let A and B be the data vectors of Generalized Riemann differences 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,

• ∆ǫ

Af(x, h) = ∆ǫ Bf(x, h) and

∆ǫ′

Af(x, h) = A∆ǫ′ Bf(x, h), A = 0.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

The Classification of Real Generalized Riemann Derivatives: 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 ∆ǫ

Af(x, h) and ∆ǫ′ Af(x, h) are finite linear combinations

• ∆ǫ

Af(x, h) = i Ui∆ǫ Bf(x, uih) and

∆ǫ′

Af(x, h) = i Vi∆ǫ′ Bf(x, vih)

• f non-zero ui-dilates of ∆ǫ

Bf(x, h) and vi-dilates of

∆ǫ′

Bf(x, h).

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Outline

1

Definition and basic properties Definition and basic properties

2

Generalized vs. Ordinary Differentiation Generalized vs. Ordinary Differentiation

3

The Classification of Real Generalized Riemann Derivatives The Classification of Real Generalized Riemann Derivatives

4

The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Generalized even and odd functions

Fix an integer ℓ > 1, and let ω be a primitive ℓth root of unity. Every complex function f is expressed uniquely as a sum f(x) =

ℓ−1

• i=0

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

ℓ−1

• j=0

ω−ijf(ωjx).

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Generalized even and odd functions

Fix an integer ℓ > 1, and let ω be a primitive ℓth root of unity. Every complex function f is expressed uniquely as a sum f(x) =

ℓ−1

• i=0

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

ℓ−1

• j=0

ω−ijf(ωjx).

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Generalized even and odd differences

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

ℓ−1

• i=0

∆i

Af(x, h),

where ∆Af(x, ωh) = ωi∆i

Af(x, h), for i = 0, 1, . . . , ℓ − 1.

Each component ∆i

Af(x, h) can be written explicitly.

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

Af(x, h)

• is an nth GR Difference,

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

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Generalized even and odd differences

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

ℓ−1

• i=0

∆i

Af(x, h),

where ∆Af(x, ωh) = ωi∆i

Af(x, h), for i = 0, 1, . . . , ℓ − 1.

Each component ∆i

Af(x, h) can be written explicitly.

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

Af(x, h)

• is an nth GR Difference,

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

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

Generalized even and odd differences

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

ℓ−1

• i=0

∆i

Af(x, h),

where ∆Af(x, ωh) = ωi∆i

Af(x, h), for i = 0, 1, . . . , ℓ − 1.

Each component ∆i

Af(x, h) can be written explicitly.

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

Af(x, h)

• is an nth GR Difference,

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

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

The Classification of Complex Generalized Riemann Derivatives: The Equivalence Relation

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

1

f is A-differentiable iff f is B-differentiable;

2

m = n and up to ℓ independent non-zero dilations, ∆i

Af(x, h) =

• ∆i

Bf(x, h)

if i = ¯ n αi∆i

Bf(x, h), αi = 0

if i = ¯ n. where ¯ n = n mod ℓ and i = 0, 1, . . . , ℓ − 1.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

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

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

References

1

• 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.
• rdinary differentiation, Proc. Amer. Math. Soc. 145

(2017), no. 4, 1553–1565.

4

• J. M. Ash, S. Catoiu and W. Chin, The classification of

generalized Riemann derivatives, preprint.

5

• J. M. Ash, S. Catoiu and W. Chin, The classification of

complex generalized Riemann derivatives, in progress.

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives 1

• J. M. Ash, S. Catoiu, R. Rios, On the nth quantum

derivative, J. London Math. Soc., 66 (2002), no. 1, 114–130.

2

• A. Denjoy, Sur l’intégration des coefficients différentiels

d’ordre supérieur, Fund. Math., 25 (1935), 273–326.

3

• A. Khintchine, Recherches sur la structure des fonctions

mesurables, Fund. Math., 9 (1927), 212–279.

4

• J. Marcinkiewicz and A. Zygmund, On the differentiability of

functions and summability of trigonometric series, Fund.

• Math. 26 (1936), 1–43

Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives

THANK YOU!

Stefan Catoiu The Classification of Generalized Riemann Derivatives