Negative Lipschitz (Hölder) exponent: Intuition

In summary, the conversation discusses the concept of negative Lipschitz (Hölder) exponent and its implications for functions. The usual definition states that a function is pointwise Lipschitz (local Hölder) if there exists a polynomial of certain degree that satisfies a specific condition. However, this definition only applies to functions with positive alpha values. When considering negative alpha values, the concept of function is no longer valid and the class of tempered distribution is used instead due to the presence of poles.
  • #1
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Negative Lipschitz (Hölder) exponent: Intuition!

Hi everybody!

Sorry for double posting... :(

I have some problem with Hölder (Lipschitz) exponent!
From what I know Lipschitz refer to integer values whereas Hölder to non integer ones.
The usual definition roughly states that:

A function [tex]f[/tex] is pointwise Lipschitz (local Hölder) [tex] \alpha \geq 0 [/tex] at [tex] \nu [/tex] if
there exist K>0 and a polynomial [tex]P_{\nu}[/tex] of degree [tex]m = \lfloor \alpha \rfloor [/tex], where [tex]\lfloor \alpha
\rfloor [/tex] is defined as the largest integer less than or equal to [tex]\alpha [/tex], such that
[tex] \forall x \in \mathbb{R} , |f(x)-P_{\nu}(x)|\leq
K|x-\nu|^{\alpha} . [/tex]

The above definition refers only to functions which have a positive alpha. I read somewhere , but I can't find no more references, that "if we want to consider negative values
too, as in this case the concept of function is not longer valid, because of the divergence at the singular
point, we have to shift to the class of tempered distribution"

What does it mean? Why if such exponent is negative the concept of function is no longer valid? Can you give me some intuition for that? :(

Many many thanks in advance!
:)
 
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  • #2
For ##\alpha < 0## we would have a condition ##|f(x)-P_\nu (x)|\leq K\cdot \dfrac{1}{|x-\nu|^{|\alpha|}}## which hasn't anything to do with continuity no longer. All of a sudden we are dealing with poles, the closer we get to ##x=\nu##. The boundary would become trivially true in case of defined functions, and poles otherwise.
 

1. What is a negative Lipschitz (Hölder) exponent?

A negative Lipschitz (Hölder) exponent refers to the exponent in the Hölder continuity condition, which describes the regularity of a function. It is a negative value that indicates the degree of smoothness or roughness of a function.

2. How is the negative Lipschitz (Hölder) exponent calculated?

The negative Lipschitz (Hölder) exponent is calculated by taking the negative of the Hölder exponent, which is the ratio of the change in the function's value over the change in its input variable. It is typically denoted as "α" and can range from 0 (constant function) to 1 (continuously differentiable function).

3. What does a negative Lipschitz (Hölder) exponent tell us about a function?

A negative Lipschitz (Hölder) exponent tells us about the regularity of a function. A higher negative exponent indicates a smoother function, while a lower negative exponent indicates a more irregular or rough function. It also gives us information about the rate of change of the function and its behavior near a point.

4. How is the negative Lipschitz (Hölder) exponent used in mathematics and science?

The negative Lipschitz (Hölder) exponent is used in various fields of mathematics and science, such as analysis, geometry, and signal processing. It is used to characterize the regularity of functions and to study their properties, such as continuity, differentiability, and integrability. It is also used in the study of fractals and self-similar sets.

5. What is the significance of a negative Lipschitz (Hölder) exponent in real-world applications?

The negative Lipschitz (Hölder) exponent has many real-world applications, such as in image and audio processing, where it is used to measure the smoothness of signals. It is also used in the analysis of data, such as time series and financial data, to understand the underlying patterns and trends. In addition, it plays a crucial role in the study of physical phenomena, such as chaotic systems and turbulence.

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