Differentiability of the Convolution

Differentiability of the Convolution

Differentiability of the convolution. Pablo Jimenez´ Rodr´ıguez Universidad de Valladolid March, 8th, 2019 Pablo Jim´enezRodr´ıguez Differentiability of the convolution Definition Let L1[−1; 1] be the set of the 2-periodic functions f so that R 1 −1 jf (t)j dt < 1: Given f ; g 2 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 Definition Let L1[−1; 1] be the set of the 2-periodic functions f so that R 1 −1 jf (t)j dt < 1: Given f ; g 2 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 Definition Let L1[−1; 1] be the set of the 2-periodic functions f so that R 1 −1 jf (t)j dt < 1: Given f ; g 2 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 Definition Let L1[−1; 1] be the set of the 2-periodic functions f so that R 1 −1 jf (t)j dt < 1: Given f ; g 2 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 2 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 2 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 Jim´enezRodr´ıguez Differentiability of the convolution If f ; g 2 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 2 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. Pablo Jim´enezRodr´ıguez Differentiability of the convolution If f ; g 2 L2[−1; 1], then f ∗ g is continuous. 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 Let f ; g 2 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. 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 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 Let f ; g 2 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. If f ; g 2 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 Let f ; g 2 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. If f ; g 2 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 Let f ; g 2 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. If f ; g 2 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 Let f ; g 2 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.a If f ; g 2 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 Let f ; g 2 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.a If f ; g 2 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 Let f ; g 2 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.a If f ; g 2 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 (from Wikipedia) Theorem Let f ; g 2 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 2 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 2 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 2 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 2 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 2 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 Jim´enezRodr´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 2 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 2 L1[−1; 1]. If f is differentiable with integrable derivative, then f ∗ g is differentiable, with (f ∗ g)0(x) = (f 0 ∗ g)(x).

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