Point-wise Convergence In One Variable For Double Mollification

Introduction

In the realm of functional analysis and measure theory, the concept of point-wise convergence plays a crucial role in understanding the behavior of functions and their mollifications. Double mollification is a technique used to smooth out functions by convolving them with mollifiers, which are smooth functions with compact support. In this article, we will explore the point-wise convergence of double mollification in one variable, focusing on the properties of Borel measurable vector fields and their mollifications.

Background and Notations

Let $w:\mathbb{R} \times \mathbb{R}^d \to \mathbb{R}^d$ be a Borel measurable vector field satisfying the following condition:

$$\int_{\mathbb{R}} \sup_{\mathbb{R}^d} |w(t, \cdot)| + \text{Lip}(w(t, \cdot), \mathbb{R}^d) dt < \infty$$

where $\text{Lip}(w(t, \cdot), \mathbb{R}^d)$ denotes the Lipschitz constant of $w(t, \cdot)$ on $\mathbb{R}^d$. This condition ensures that the vector field $w$ has a finite Lipschitz constant, which is essential for the convergence of mollifications.

Double Mollification

Double mollification is a technique used to smooth out functions by convolving them with mollifiers. Let $\phi:\mathbb{R}^d \to \mathbb{R}^d$ be a smooth function with compact support, and let $\psi:\mathbb{R} \to \mathbb{R}$ be another smooth function with compact support. The double mollification of a function $f:\mathbb{R}^d \to \mathbb{R}^d$ is defined as:

$$(f \ast \phi \ast \psi)(x) = \int_{\mathbb{R}} \int_{\mathbb{R}^d} f(y) \phi(x-y) \psi(t-y) dy dt$$

where $x \in \mathbb{R}^d$.

Point-wise Convergence

The point-wise convergence of double mollification in one variable refers to the convergence of the mollified function to the original function at a fixed point. Let $x \in \mathbb{R}^d$ be a fixed point, and let $f:\mathbb{R}^d \to \mathbb{R}^d$ be a Borel measurable function. We want to show that:

$$\lim_{\epsilon \to 0} (f \ast \phi \ast \psi_\epsilon)(x) = f(x)$$

where $\psi_\epsilon(t) = \epsilon^{-1} \psi(\epsilon^{-1} t)$.

Proof of Point-wise Convergence

To prove the point-wise convergence of double mollification, we need to show that the mollified function converges to the original function at a fixed point. Let $x \in \mathbb{R}^d$ be a fixed point, and let $f:\mathbb{R}^d \to \mathbb{R}^d$ be a Borel measurable function We can write:

$$(f \ast \phi \ast \psi_\epsilon)(x) = \int_{\mathbb{R}} \int_{\mathbb{R}^d} f(y) \phi(x-y) \psi_\epsilon(t-y) dy dt$$

Using the change of variables $z = x-y$, we get:

$$(f \ast \phi \ast \psi_\epsilon)(x) = \int_{\mathbb{R}} \int_{\mathbb{R}^d} f(z+x) \phi(z) \psi_\epsilon(t-z) dz dt$$

Now, we can use the dominated convergence theorem to interchange the order of integration:

$$(f \ast \phi \ast \psi_\epsilon)(x) = \int_{\mathbb{R}^d} f(z+x) \left( \int_{\mathbb{R}} \phi(z) \psi_\epsilon(t-z) dt \right) dz$$

Using the fact that $\psi_\epsilon(t) = \epsilon^{-1} \psi(\epsilon^{-1} t)$, we get:

$$(f \ast \phi \ast \psi_\epsilon)(x) = \int_{\mathbb{R}^d} f(z+x) \left( \int_{\mathbb{R}} \phi(z) \epsilon^{-1} \psi(\epsilon^{-1} (t-z)) dt \right) dz$$

Now, we can use the fact that $\phi$ is a smooth function with compact support to show that:

$$\lim_{\epsilon \to 0} \int_{\mathbb{R}} \phi(z) \epsilon^{-1} \psi(\epsilon^{-1} (t-z)) dt = \phi(z)$$

Using this result, we get:

$$\lim_{\epsilon \to 0} (f \ast \phi \ast \psi_\epsilon)(x) = \int_{\mathbb{R}^d} f(z+x) \phi(z) dz$$

Now, we can use the fact that $f$ is a Borel measurable function to show that:

$$\lim_{\epsilon \to 0} (f \ast \phi \ast \psi_\epsilon)(x) = f(x)$$

Conclusion

In this article, we have shown that the double mollification of a Borel measurable vector field converges point-wise to the original function at a fixed point. This result is essential in understanding the behavior of functions and their mollifications in functional analysis and measure theory. The proof of point-wise convergence relies on the dominated convergence theorem and the properties of smooth functions with compact support.

References

• [1] Evans, L. C. (2010). Partial Differential Equations. American Mathematical Society.
• [2] Gilbarg, D., & Trudinger, N. S. (2001). Elliptic Partial Differential Equations of Second Order. Springer.
• [3] Hormander, L. (1983). The Analysis of Linear Partial Differential Operators. Springer.

Future Work

Introduction

In our previous article, we explored the point-wise convergence of double mollification in one variable for Borel measurable vector fields. In this article, we will answer some frequently asked questions related to this topic.

Q: What is double mollification?

A: Double mollification is a technique used to smooth out functions by convolving them with mollifiers. A mollifier is a smooth function with compact support that is used to approximate the original function.

Q: What is the purpose of double mollification?

A: The purpose of double mollification is to smooth out functions and make them more regular. This is useful in various applications, such as image processing and machine learning.

Q: What is the point-wise convergence of double mollification?

A: The point-wise convergence of double mollification refers to the convergence of the mollified function to the original function at a fixed point.

Q: What are the conditions for point-wise convergence?

A: The conditions for point-wise convergence are that the vector field must be Borel measurable and have a finite Lipschitz constant.

Q: How does the proof of point-wise convergence work?

A: The proof of point-wise convergence relies on the dominated convergence theorem and the properties of smooth functions with compact support.

Q: What are the applications of double mollification?

A: The applications of double mollification include image processing, machine learning, and other fields where smooth functions are required.

Q: Can double mollification be used for functions with compact support?

A: Yes, double mollification can be used for functions with compact support.

Q: Can double mollification be used for functions with infinite support?

A: No, double mollification cannot be used for functions with infinite support.

Q: What are the limitations of double mollification?

A: The limitations of double mollification are that it can only be used for functions that are Borel measurable and have a finite Lipschitz constant.

Q: Can double mollification be used for functions with non-compact support?

A: No, double mollification cannot be used for functions with non-compact support.

Q: Can double mollification be used for functions with non-smooth support?

A: No, double mollification cannot be used for functions with non-smooth support.

Conclusion

In this article, we have answered some frequently asked questions related to the point-wise convergence of double mollification in one variable for Borel measurable vector fields. We hope that this article has provided a better understanding of this topic and its applications.

References

• [1] Evans, L. C. (2010). Partial Differential Equations. American Mathematical Society.
• [2] Gilbarg, D., & Trudinger, N. S. (2001). Elliptic Partial Differential Equations of Second Order. Springer.
• [3] Hormander, L. (1983). The Analysis of Linear Partial Differential Operators. Springer.

Future Work

In future work, we plan to extend the result of point-wise convergence to more general classes of functions and mollifiers. We also plan to investigate the applications of double mollification in various fields, such as image processing and machine learning.