Differentiability of the Convolution

Diﬀerentiability of the convolution.

Pablo Jimenez´ Rodr´ıguez

Universidad de Valladolid

March, 8th, 2019

Pablo JiménezRodr´ıguez Differentiability of the convolution Definition

Let L1[−1, 1] be the set of the 2-periodic functions f so that R 1 −1 |f (t)| dt < ∞. Given f , g ∈ L1[−1, 1], the convolution of f and g is the function

Z 1 f ∗ g(x) = f (t)g(x − t) dt −1

Pablo JiménezRodr´ıguez Differentiability of the convolution Definition

Let L1[−1, 1] be the set of the 2-periodic functions f so that R 1 −1 |f (t)| dt < ∞. Given f , g ∈ L1[−1, 1], the convolution of f and g is the function

Z 1 f ∗ g(x) = f (t)g(x − t) dt −1

Pablo JiménezRodr´ıguez Differentiability of the convolution Definition

Let L1[−1, 1] be the set of the 2-periodic functions f so that R 1 −1 |f (t)| dt < ∞. Given f , g ∈ L1[−1, 1], the convolution of f and g is the function

Z 1 f ∗ g(x) = f (t)g(x − t) dt −1

Pablo JiménezRodr´ıguez Differentiability of the convolution Definition

Let L1[−1, 1] be the set of the 2-periodic functions f so that R 1 −1 |f (t)| dt < ∞. Given f , g ∈ L1[−1, 1], the convolution of f and g is the function

Z 1 f ∗ g(x) = f (t)g(x − t) dt −1

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution If f is Lipschitz, then f ∗ g is Lipschitz.

If f , g ∈ L2[−1, 1], then f ∗ g is continuous. There exist nowhere diﬀerentiable functions f , g so that f ∗ g is diﬀerentiable.

The previous examples, and some others in the same way, gives the idea that the convolution takes “the best” properties from its parent functions Results in this line have given the convolution operator the label of smoothening.

Lemma

Let f , g ∈ L1[−1, 1]. dk f If f is k times diﬀerentiable (k ≥ 0), with dxk continuous, then f ∗ g is k times diﬀerentiable, with dk dk f dxk (f ∗ g)(x) = ( dxk ∗ g)(x).

Pablo JiménezRodr´ıguez Differentiability of the convolution If f , g ∈ L2[−1, 1], then f ∗ g is continuous. There exist nowhere differentiable functions f , g so that f ∗ g is differentiable.

Lemma

Let f , g ∈ L1[−1, 1]. dk f If f is k times diﬀerentiable (k ≥ 0), with dxk continuous, then f ∗ g is k times diﬀerentiable, with dk dk f dxk (f ∗ g)(x) = ( dxk ∗ g)(x). If f is Lipschitz, then f ∗ g is Lipschitz.

Results in this line have given the convolution operator the label of smoothening.

Lemma

The previous examples, and some others in the same way, gives the idea that the convolution takes “the best” properties from its parent functions.

Pablo JiménezRodr´ıguez Differentiability of the convolution There exist nowhere differentiable functions f , g so that f ∗ g is differentiable.

Results in this line have given the convolution operator the label of smoothening.

Lemma

If f , g ∈ L2[−1, 1], then f ∗ g is continuous.

The previous examples, and some others in the same way, gives the idea that the convolution takes “the best” properties from its parent functions.

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Results in this line have given the convolution operator the label of smoothening.

Lemma

If f , g ∈ L2[−1, 1], then f ∗ g is continuous. There exist nowhere diﬀerentiable functions f , g so that f ∗ g is diﬀerentiable.

The previous examples, and some others in the same way, gives the idea that the convolution takes “the best” properties from its parent functions.

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Lemma

If f , g ∈ L2[−1, 1], then f ∗ g is continuous. There exist nowhere diﬀerentiable functions f , g so that f ∗ g is diﬀerentiable.

The previous examples, and some others in the same way, gives the idea that the convolution takes “the best” properties from its parent functions. Results in this line have given the convolution operator the label of smoothening.

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Lemma

If f , g ∈ L2[−1, 1], then f ∗ g is continuous. There exist nowhere differentiable functions f , g so that f ∗ g is differentiable. aLipschitz property is sometimes regarded as a ”middle way“ between continuity and differentiability.

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Lemma

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution (from Wikipedia)

Theorem

Let f , g ∈ L1[−1, 1]. If f is diﬀerentiable with bounded derivative, then f ∗ g is diﬀerentiable, with (f ∗ g)0(x) = (f 0 ∗ g)(x).

Theorem

Let f , g ∈ L1[−1, 1]. If f is diﬀerentiable with integrable derivative, then f ∗ g is diﬀerentiable, with (f ∗ g)0(x) = (f 0 ∗ g)(x).

What if we just ask diﬀerentiability?

With some more requirements over f 0 the property of diﬀerentiability can be extended to the convolution.

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution (from Wikipedia)

Theorem

Let f , g ∈ L1[−1, 1]. If f is diﬀerentiable with integrable derivative, then f ∗ g is diﬀerentiable, with (f ∗ g)0(x) = (f 0 ∗ g)(x).

What if we just ask diﬀerentiability?

With some more requirements over f 0 the property of diﬀerentiability can be extended to the convolution. Theorem

Let f , g ∈ L1[−1, 1]. If f is diﬀerentiable with bounded derivative, then f ∗ g is diﬀerentiable, with (f ∗ g)0(x) = (f 0 ∗ g)(x).

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution (from Wikipedia)

What if we just ask diﬀerentiability?

With some more requirements over f 0 the property of diﬀerentiability can be extended to the convolution. Theorem

Let f , g ∈ L1[−1, 1]. If f is diﬀerentiable with bounded derivative, then f ∗ g is diﬀerentiable, with (f ∗ g)0(x) = (f 0 ∗ g)(x).

Theorem

Let f , g ∈ L1[−1, 1]. If f is diﬀerentiable with integrable derivative, then f ∗ g is diﬀerentiable, with (f ∗ g)0(x) = (f 0 ∗ g)(x).

Pablo JiménezRodr´ıguez Differentiability of the convolution What if we just ask differentiability?

With some more requirements over f 0 the property of diﬀerentiability can be extended to the convolution. Theorem

Let f , g ∈ L1[−1, 1]. If f is diﬀerentiable with bounded derivative, then f ∗ g is diﬀerentiable, with (f ∗ g)0(x) = (f 0 ∗ g)(x).

Theorem (from Wikipedia)

Let f , g ∈ L1[−1, 1]. If f is diﬀerentiable with integrable derivative, then f ∗ g is diﬀerentiable, with (f ∗ g)0(x) = (f 0 ∗ g)(x).

Pablo JiménezRodr´ıguez Differentiability of the convolution With some more requirements over f 0 the property of differentiability can be extended to the convolution. Theorem

Let f , g ∈ L1[−1, 1]. If f is diﬀerentiable with bounded derivative, then f ∗ g is diﬀerentiable, with (f ∗ g)0(x) = (f 0 ∗ g)(x).

Theorem (from Wikipedia)

Let f , g ∈ L1[−1, 1]. If f is diﬀerentiable with integrable derivative, then f ∗ g is diﬀerentiable, with (f ∗ g)0(x) = (f 0 ∗ g)(x).

What if we just ask diﬀerentiability?

Pablo JiménezRodr´ıguez Differentiability of the convolution 1 f (x) = x2 sin x3 1 g(x) = sin x3 f ∗ g not differentiable?

First idea Take a function with unbounded derivative.

Pablo JiménezRodr´ıguez Differentiability of the convolution 1 g(x) = sin x3 f ∗ g not differentiable?

First idea Take a function with unbounded derivative. 1 f (x) = x2 sin x3

Pablo JiménezRodr´ıguez Differentiability of the convolution 1 g(x) = sin x3 f ∗ g not differentiable?

First idea Take a function with unbounded derivative. 1 f (x) = x2 sin x3

Pablo JiménezRodr´ıguez Differentiability of the convolution f ∗ g not differentiable?

First idea Take a function with unbounded derivative. 1 f (x) = x2 sin x3 1 g(x) = sin x3

Pablo JiménezRodr´ıguez Differentiability of the convolution First idea Take a function with unbounded derivative. 1 f (x) = x2 sin x3 1 g(x) = sin x3 f ∗ g not differentiable?

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Z 1 Z 1 2 1 1 2 2 1 f ∗ g(0) = t sin 3 sin 3 dt = t sin 3 dt −1 t (−t) −1 t

First idea Take a function with unbounded derivative. 1 f (x) = x2 sin x3 1 g(x) = sin x3 f ∗ g not diﬀerentiable?

Z 1 2 1 1 f ∗ g(x) = t sin 3 sin 3 dt −1 t (x − t)

Pablo JiménezRodr´ıguez Differentiability of the convolution First idea Take a function with unbounded derivative. 1 f (x) = x2 sin x3 1 g(x) = sin x3 f ∗ g not differentiable?

Z 1 2 1 1 f ∗ g(x) = t sin 3 sin 3 dt −1 t (x − t) Z 1 Z 1 2 1 1 2 2 1 f ∗ g(0) = t sin 3 sin 3 dt = t sin 3 dt −1 t (−t) −1 t

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Second idea Let’s see if there is anything we can do to simplify calculations...

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution 1 1 7i For each i ≥ 1, divide the interval 2i , 2i−1 into 2 sub-intervals 7i−1 of the same length For each k = 0, 1,..., 2 − 1, consider φi,k ∞ and ψi,k two C −hat functions so that

1 2k + 1 1 2k + 2 supp φ ⊆ + , + , i,k 2i 28i 2i 28i 1 2k 1 2k + 1 supp ψ ⊆ + , + , i,k 2i 28i 2i 28i 1 8k + 5 1 8k + 7 φ (x) = 1 for + ≤ x ≤ + and i,k 2i 28i+2 2i 28i+2 1 8k + 1 1 8k + 3 ψ (x) = 1 for + ≤ x ≤ + . i,k 2i 28i+2 2i 28i+2

Second idea Let’s see if there is anything we can do to simplify calculations...

Deﬁnition

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution 7i−1 For each k = 0, 1,..., 2 − 1, consider φi,k ∞ and ψi,k two C −hat functions so that

Second idea Let’s see if there is anything we can do to simplify calculations...

Deﬁnition 1 1 7i For each i ≥ 1, divide the interval 2i , 2i−1 into 2 sub-intervals of the same length.

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Second idea Let’s see if there is anything we can do to simplify calculations...

Deﬁnition 1 1 7i For each i ≥ 1, divide the interval 2i , 2i−1 into 2 sub-intervals 7i−1 of the same length. For each k = 0, 1,..., 2 − 1, consider φi,k ∞ and ψi,k two C −hat functions so that

Pablo JiménezRodr´ıguez Differentiability of the convolution Theorem The functions f and g defined above are differentiable functions for which f ∗ g is not differentiable at 0.

Second idea Let’s see if there is anything we can do to simplify calculations...

Deﬁnition Deﬁne ∞ 27i−1−1 2 X X f (x) = x φi,k (x), i=1 k=0 ∞ 27j−1−1 2 X X g(x) = x ψj,l (x) j=1 l=0

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Second idea Let’s see if there is anything we can do to simplify calculations...

Deﬁnition Deﬁne ∞ 27i−1−1 2 X X f (x) = x φi,k (x), i=1 k=0 ∞ 27j−1−1 2 X X g(x) = x ψj,l (x) j=1 l=0

Theorem The functions f and g defined above are differentiable functions for which f ∗ g is not differentiable at 0.

Pablo JiménezRodr´ıguez Differentiability of the convolution In fact, for the differentiable functions f and g in the previous theorem, one can prove that f 0 ∗ g(x) is well-defined for every −1 < x < 1.

Remark 1

Theorem There exist diﬀerentiable functions f and g so that f ∗ g(x) is not diﬀerentiable at x = 0.

Remark 1

Pablo JiménezRodr´ıguez Differentiability of the convolution Theorem There exist differentiable functions f and g so that f ∗ g(x) is not differentiable at x = 0.

Remark 1 In fact, for the diﬀerentiable functions f and g in the previous theorem, one can prove that f 0 ∗ g(x) is well-deﬁned for every −1 < x < 1.

Pablo JiménezRodr´ıguez Differentiability of the convolution Theorem There exist differentiable functions f and g so that f ∗ g(x) is not differentiable at x = 0.

Remark 1 In fact, for the diﬀerentiable functions f and g in the previous theorem, one can prove that f 0 ∗ g(x) is well-deﬁned for every −1 < x < 1.

Pablo JiménezRodr´ıguez Differentiability of the convolution (f + g) ∗ (f + g) = f ∗ f + f ∗ g + g ∗ f + g ∗ g = f ∗ f + 2f ∗ g + g ∗ g 1 f ∗ g = [(f + g) ∗ (f + g) − (f ∗ f + g ∗ g)] 2 In particular, there exists a differentiable function h so that h ∗ h is not differentiable at 0.

Theorem There exist diﬀerentiable functions f and g so that f ∗ g(x) is not diﬀerentiable at x = 0.

Remark 2 Can we make f = g for the previous theorem?

Pablo JiménezRodr´ıguez Differentiability of the convolution 1 f ∗ g = [(f + g) ∗ (f + g) − (f ∗ f + g ∗ g)] 2 In particular, there exists a differentiable function h so that h ∗ h is not differentiable at 0.

Theorem There exist diﬀerentiable functions f and g so that f ∗ g(x) is not diﬀerentiable at x = 0.

Remark 2 Can we make f = g for the previous theorem?

(f + g) ∗ (f + g) = f ∗ f + f ∗ g + g ∗ f + g ∗ g = f ∗ f + 2f ∗ g + g ∗ g

Pablo JiménezRodr´ıguez Differentiability of the convolution In particular, there exists a differentiable function h so that h ∗ h is not differentiable at 0.

Theorem There exist diﬀerentiable functions f and g so that f ∗ g(x) is not diﬀerentiable at x = 0.

Remark 2 Can we make f = g for the previous theorem?

(f + g) ∗ (f + g) = f ∗ f + f ∗ g + g ∗ f + g ∗ g = f ∗ f + 2f ∗ g + g ∗ g 1 f ∗ g = [(f + g) ∗ (f + g) − (f ∗ f + g ∗ g)] 2

Pablo JiménezRodr´ıguez Differentiability of the convolution Theorem There exist differentiable functions f and g so that f ∗ g(x) is not differentiable at x = 0.

Remark 2 Can we make f = g for the previous theorem?

(f + g) ∗ (f + g) = f ∗ f + f ∗ g + g ∗ f + g ∗ g = f ∗ f + 2f ∗ g + g ∗ g 1 f ∗ g = [(f + g) ∗ (f + g) − (f ∗ f + g ∗ g)] 2 In particular, there exists a diﬀerentiable function h so that h ∗ h is not diﬀerentiable at 0.

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution What algebraic structure does this problem admit?

How many of such functions can we ﬁnd?

Pablo JiménezRodr´ıguez Differentiability of the convolution How many of such functions can we find? What algebraic structure does this problem admit?

Pablo JiménezRodr´ıguez Differentiability of the convolution Theorem There exist two closed cones V and W generated by two respective non-numerable sets of differentiable functions so that, if f ∈ V \{0} and g ∈ W \{0}, then f ∗ g is not differentiable at 0.

Theorem There exist two algebras A and B generated by two respective non-numerable sets of diﬀerentiable functions so that, if f ∈ A \{0} and g ∈ B \{0}, then f ∗ g is not diﬀerentiable at 0.

Pablo JiménezRodr´ıguez Differentiability of the convolution Theorem There exist two algebras A and B generated by two respective non-numerable sets of differentiable functions so that, if f ∈ A \{0} and g ∈ B \{0}, then f ∗ g is not differentiable at 0.

Theorem There exist two closed cones V and W generated by two respective non-numerable sets of diﬀerentiable functions so that, if f ∈ V \{0} and g ∈ W \{0}, then f ∗ g is not diﬀerentiable at 0.

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution For the algebras, consider

2 ∞ 2i −i−1−1 λ X X λ fλ(x) = |x| φi,k (|x|) and i=1 k=0 2 ∞ 2i −i−1−1 λ X X λ gλ(x) = |x| ψi,k (|x|), i=1 k=0

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution For the algebras, consider

2 ∞ 2i −i−1−1 λ X X λ fλ(x) = |x| φi,k (|x|) and i=1 k=0 2 ∞ 2i −i−1−1 λ X X λ gλ(x) = |x| ψi,k (|x|), i=1 k=0 For the closed cones, consider

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution For the algebras, consider

2 ∞ 2i −i−1−1 λ X X λ fλ(x) = |x| φi,k (|x|) and i=1 k=0 2 ∞ 2i −i−1−1 λ X X λ gλ(x) = |x| ψi,k (|x|), i=1 k=0 For the closed cones, consider

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution For the algebras, consider

2 ∞ 2i −i−1−1 λ X X λ fλ(x) = |x| φi,k (|x|) and i=1 k=0 2 ∞ 2i −i−1−1 λ X X λ gλ(x) = |x| ψi,k (|x|), i=1 k=0 For the closed cones, consider

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution For the algebras, consider

2 ∞ 2i −i−1−1 λ X X λ fλ(x) = |x| φi,k (|x|) and i=1 k=0 2 ∞ 2i −i−1−1 λ X X λ gλ(x) = |x| ψi,k (|x|), i=1 k=0

Theorem (V.I. Gurariy) If V ⊆ D[−1, 1] is a closed vector space, then V is of ﬁnite dimension.

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution What about more points?

Pablo JiménezRodr´ıguez Differentiability of the convolution Pablo JiménezRodr´ıguez Differentiability of the convolution Pablo JiménezRodr´ıguez Differentiability of the convolution (of zero measure)

There exist diﬀerentiable functions f and g so that f ∗ g is not diﬀerentiable on a Perfect set

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution (of zero measure)

There exist diﬀerentiable functions f and g so that f ∗ g is not diﬀerentiable on a Perfect set

Pablo JiménezRodr´ıguez Differentiability of the convolution There exist differentiable functions f and g so that f ∗ g is not differentiable on a Perfect set (of zero measure)

Pablo JiménezRodr´ıguez Differentiability of the convolution If f and g are differentiable functions and x ∈/ ∆f + ∆g , then f ∗ g is differentiable at x and (f ∗ g)0(x) = (f 0 ∗ g)(x).

Theorem Let f be a diﬀerentiable function, 0 ∆f = {x ∈ [−1, 1] : f is locally unbounded}.

Any positive result?

Theorem Let f be a diﬀerentiable function, 0 ∆f = {x ∈ [−1, 1] : f is locally unbounded}.

Pablo JiménezRodr´ıguez Differentiability of the convolution then f ∗ g is differentiable at x and (f ∗ g)0(x) = (f 0 ∗ g)(x).

Any positive result?

Theorem Let f be a diﬀerentiable function, 0 ∆f = {x ∈ [−1, 1] : f is locally unbounded}. If f and g are diﬀerentiable functions and x ∈/ ∆f + ∆g ,

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Any positive result?

Theorem Let f be a differentiable function, 0 ∆f = {x ∈ [−1, 1] : f is locally unbounded}. If f and g are differentiable functions and x ∈/ ∆f + ∆g , then f ∗ g is differentiable at x and (f ∗ g)0(x) = (f 0 ∗ g)(x).

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Any positive result?

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution A dense set? Of positive measure? The whole [−1, 1]?

1 Let f and g be diﬀerentiable functions. How big can the set where f ∗ g is not diﬀerentiable be?

0 2 What are the weakest conditions over f to ensure that f ∗ g is differentiable? 3 If f and g are such that f ∗ g is differentiable at x and f 0 ∗ g(x) is well-defined, is it (f ∗ g)0(x) = f 0 ∗ g(x)?

Some open questions.

Pablo JiménezRodr´ıguez Differentiability of the convolution A dense set? Of positive measure? The whole [−1, 1]? 0 2 What are the weakest conditions over f to ensure that f ∗ g is differentiable? 3 If f and g are such that f ∗ g is differentiable at x and f 0 ∗ g(x) is well-defined, is it (f ∗ g)0(x) = f 0 ∗ g(x)?

Some open questions.

1 Let f and g be diﬀerentiable functions. How big can the set where f ∗ g is not diﬀerentiable be?

Pablo JiménezRodr´ıguez Differentiability of the convolution Of positive measure? The whole [−1, 1]? 0 2 What are the weakest conditions over f to ensure that f ∗ g is differentiable? 3 If f and g are such that f ∗ g is differentiable at x and f 0 ∗ g(x) is well-defined, is it (f ∗ g)0(x) = f 0 ∗ g(x)?

Some open questions.

1 Let f and g be diﬀerentiable functions. How big can the set where f ∗ g is not diﬀerentiable be? A dense set?

Pablo JiménezRodr´ıguez Differentiability of the convolution The whole [−1, 1]? 0 2 What are the weakest conditions over f to ensure that f ∗ g is differentiable? 3 If f and g are such that f ∗ g is differentiable at x and f 0 ∗ g(x) is well-defined, is it (f ∗ g)0(x) = f 0 ∗ g(x)?

Some open questions.

1 Let f and g be diﬀerentiable functions. How big can the set where f ∗ g is not diﬀerentiable be? A dense set? Of positive measure?

Pablo JiménezRodr´ıguez Differentiability of the convolution 0 2 What are the weakest conditions over f to ensure that f ∗ g is differentiable? 3 If f and g are such that f ∗ g is differentiable at x and f 0 ∗ g(x) is well-defined, is it (f ∗ g)0(x) = f 0 ∗ g(x)?

Some open questions.

1 Let f and g be diﬀerentiable functions. How big can the set where f ∗ g is not diﬀerentiable be? A dense set? Of positive measure? The whole [−1, 1]?

Pablo JiménezRodr´ıguez Differentiability of the convolution 3 If f and g are such that f ∗ g is differentiable at x and f 0 ∗ g(x) is well-defined, is it (f ∗ g)0(x) = f 0 ∗ g(x)?

Some open questions.

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Some open questions.

1 Let f and g be differentiable functions. How big can the set where f ∗ g is not differentiable be? A dense set? Of positive measure? The whole [−1, 1]? 0 2 What are the weakest conditions over f to ensure that f ∗ g is differentiable? 3 If f and g are such that f ∗ g is differentiable at x and f 0 ∗ g(x) is well-defined, is it (f ∗ g)0(x) = f 0 ∗ g(x)?

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution PUBLICATIONS

.J.R., G.A. Muñoz-Fernández,E. SáezMaestro, J.B. Seoane-Sepúlveda, The convolution of two differentiable functions on the circle need not be differentiable, Rev. Mat. Compl., ??, (2018) ??–??. .J.R., G.A. Muñoz-Fernández,E. SáezMaestro, J.B. Seoane-Sepúlveda, Algebraic structures and the differentiability of the convolution., J. Approx. Theory, ??, (2019) ??–??. .C. Ciesielski, P.J.R., G.A. Muñoz-Fernándezand J.B. Seoane-Sepúlveda, Nondifferentiability of the convolution of differentiable real functions., preprint..

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution Thank you for your attention!!

Pablo Jim´enezRodr´ıguez Diﬀerentiability of the convolution