Difference between revisions of "Diamond alpha derivative"

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(Created page with " =Properties= <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem:</strong> Let $0 \leq \alpha \leq 1$. If $f$ is both $\Delta$ and $\nab...")
 
 
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Let $\mathbb{T}$ be a [[time scale]] and let $0\leq \alpha \leq 1$. The $\Diamond_{\alpha}$-derivative of a function $f \colon \mathbb{T} \rightarrow \mathbb{C}$ is formally defined to be the number $f^{\Diamond_{\alpha}}(t)$ such that for all $\epsilon>0$ there is a neighborhood $U$ of $t$ such that for all $s \in U$,
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$$\left| \alpha[f^{\sigma}(t)-f(s)]\eta_{ts} + (1-\alpha)[f^{\rho}(t)-f(s)]\mu_{ts}-f^{\Diamond_{\alpha}}\mu_{ts}\eta_{ts} \right| < \epsilon |\mu_{ts}\eta_{ts}|,$$
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where $\mu_{ts}=\sigma(t)-s$ and $\eta_{ts}=\rho(t)-s$.
  
 
=Properties=
 
=Properties=
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<strong>Theorem:</strong> 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  
 
<strong>Theorem:</strong> 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  
 
$$f^{\Diamond_{\alpha}}(t)=\alpha f^{\Delta}(t) + (1-\alpha)f^{\nabla}(t).$$
 
$$f^{\Diamond_{\alpha}}(t)=\alpha f^{\Delta}(t) + (1-\alpha)f^{\nabla}(t).$$
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<div class="mw-collapsible-content">
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<strong>Proof:</strong> █
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</div>
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</div>
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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<strong>Theorem:</strong> Let $f \colon \mathbb{T} \rightarrow \mathbb{C}$. If $f$ is $\Diamond_{\alpha}$-differentiable at $t$, then $f$ is [[continuous]] at $t$.
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<div class="mw-collapsible-content">
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<strong>Proof:</strong> █
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</div>
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</div>
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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<strong>Theorem:</strong> Let $f \colon \mathbb{T} \rightarrow \mathbb{C}$. If $f$ is $\Diamond_{\alpha}$-differentiable at $t$, then $f$ is both [[Delta derivative|$\Delta$-differentiable]] and [[Nabla derivative|$\nabla$-differentiable]] at $t$.
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █  
 
<strong>Proof:</strong> █  
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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 />
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[http://arxiv.org/pdf/0902.1380%20%5Bmath.CA%5D]<br />

Latest revision as of 08:32, 12 April 2015

Let $\mathbb{T}$ be a time scale and let $0\leq \alpha \leq 1$. The $\Diamond_{\alpha}$-derivative of a function $f \colon \mathbb{T} \rightarrow \mathbb{C}$ is formally defined to be the number $f^{\Diamond_{\alpha}}(t)$ such that for all $\epsilon>0$ there is a neighborhood $U$ of $t$ such that for all $s \in U$, $$\left| \alpha[f^{\sigma}(t)-f(s)]\eta_{ts} + (1-\alpha)[f^{\rho}(t)-f(s)]\mu_{ts}-f^{\Diamond_{\alpha}}\mu_{ts}\eta_{ts} \right| < \epsilon |\mu_{ts}\eta_{ts}|,$$ where $\mu_{ts}=\sigma(t)-s$ and $\eta_{ts}=\rho(t)-s$.

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:

Theorem: Let $f \colon \mathbb{T} \rightarrow \mathbb{C}$. If $f$ is $\Diamond_{\alpha}$-differentiable at $t$, then $f$ is continuous at $t$.

Proof:

Theorem: Let $f \colon \mathbb{T} \rightarrow \mathbb{C}$. If $f$ is $\Diamond_{\alpha}$-differentiable at $t$, then $f$ is both $\Delta$-differentiable and $\nabla$-differentiable at $t$.

Proof:

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

References

[1]
[2]