Difference between revisions of "Diamond alpha derivative"

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=References=
 
=References=
[http://www.sciencedirect.com/science/article/pii/S0022247X06002344]
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[http://www.sciencedirect.com/science/article/pii/S0022247X06002344]<br />
 +
[http://arxiv.org/pdf/0902.1380%20%5Bmath.CA%5D]<br />

Revision as of 08:23, 12 April 2015


Properties

Theorem: Let $0 \leq \alpha \leq 1$. If $f$ is both $\Delta$ and $\nabla$ differentiable at $t \in \mathbb{T}$ then $f$ is $\Diamond_{\alpha}$-differentiable at t and $$f^{\Diamond_{\alpha}}(t)=\alpha f^{\Delta}(t) + (1-\alpha)f^{\nabla}(t).$$

Proof:

Problem with the $\Diamond_{\alpha}$-integral

References

[1]
[2]