Question

Let \(F\) be a distribution on \(\mathbb{R}^{n}\) such that \(\operatorname{supp}(F)=\\{0\\}\). a. There exist \(N \in \mathbb{N}, C>0\) such that for all \(\phi \in C_{e}^{\infty}\). $$ |\langle F, \phi\rangle| \leq C \sum_{|a| \leq N} \sup _{|x| \leq 1}\left|\partial^{\alpha} \phi(x)\right| $$ b. Fix \(\psi \in C_{e}^{\infty}\) with \(\psi(x)=1\) for \(|x| \leq 1\) and \(\psi(x)=0\) for \(|x| \geq 2\). If \(\phi \in C_{e}^{\infty}\), let \(\phi_{k}(x)=\phi(x)[1-\psi(k x)]\). If \(\partial^{\alpha} \phi(0)=0\) for \(|\alpha| \leq N\), then \(\partial^{\alpha} \phi_{k} \rightarrow \partial^{\alpha} \phi\) uniformly as \(k \rightarrow \infty\) for \(|\alpha| \leq N\). (Hint: By 'Taylor's theorem. \(\left|\partial^{\alpha} \phi(x)\right| \leq C|x|^{N+1-|\alpha|}\) for \(\left.|\alpha| \leq N_{.}\right)\) c. If \(\phi \in C_{c}^{\infty}\) and \(\partial^{\alpha} \phi(0)=0\) for \(|\alpha| \leq N\), then \(\langle F, \phi\rangle=0\). d. There exist constants \(c_{\alpha}(|\alpha| \leq N)\) such that \(F=\sum_{|a| \leq N} c_{\alpha} \partial^{\alpha} \delta\).

Short answer

The distribution \( F \) can be expressed as a sum of derivatives of the Dirac delta function at the origin.

Step-by-step solution

Understanding the Support Property

The problem states that the support of the distribution \( F \) is \( \{0\} \). This implies that \( F \) can be non-zero only at the origin. Therefore, \( F \) must be some combination of derivatives of the Dirac delta function centered at \( 0 \).

Establishing the Inequality

We need to show that there exist constants \( N \) and \( C \) such that the inequality holds for all \( \phi \in C_{e}^{\infty} \). By the definition of distributions with support in \( \{0\} \), it is known these distributions can behave like finite combinations of delta derivatives. The inequality essentially bounds the action of \( F \) on \( \phi \) by a finite number of its derivatives.

Using Taylor's Theorem

Given the hint, apply Taylor's theorem to show, for part b, that \( \partial^{\alpha} \phi(x) \) can be bounded as provided. Since \( \partial^{\alpha} \phi(0) = 0 \), the remainder of Taylor's series provides the required estimate for the derivatives of \( \phi \_k \) as \( k \rightarrow \infty \). This ensures the uniform convergence mentioned in the problem statement.

Prove \( \langle F, \phi \rangle = 0 \) Under Conditions

For \( \phi \in C_{c}^{\infty} \) with the condition \( \partial^{\alpha} \phi(0) = 0 \), \( F \) evaluated on \( \phi \) must be zero since \( F \) depends solely on the derivatives present at zero. If these derivatives evaluate to zero, the impact of \( F \) on \( \phi \) also nullifies, hence \( \langle F, \phi \rangle = 0 \).

Characterize \( F \) as a Sum of Delta Derivatives

Because \( F \)'s support is localized at \( \{0\} \), \( F \) can be expressed as \( \sum_{|\alpha| \leq N} c_{\alpha} \partial^{\alpha} \delta \). This reflects that the distribution is made of higher-order point derivatives at zero, according to the earlier parts of the problem, where \( c_{\alpha} \) represents the suitable constants.

Key concepts

Support of a Distribution

In distribution theory, 'support' refers to the subset of the domain where a distribution is non-zero. For a distribution \( F \) with support \( \{0\} \), it means \( F \) is only non-zero at the origin. Understanding 'support' is crucial because it dictates where and how a distribution can affect test functions.
The support of a distribution characterizes its behavior over its domain. When dealing with distributions, such as those in \( \mathbb{R}^n \), knowing the support helps us determine their influence and locality.

• If the support is at a single point, as in this case, it often involves derivatives of the Dirac delta function, which are centered at that point.
• This property allows us to use various mathematical tools to describe the distribution's action on smooth test functions \( \phi \).
• Since these functions are infinitely differentiable, the interaction with the distribution is often represented through higher-order derivatives confined to the support's location.

This idea is particularly relevant in the given problem, where the distribution \( F \) is at the origin, leading to an outcome expressed through these derivatives.

Dirac Delta Function

The Dirac delta function, often denoted as \( \delta(x) \), is a fundamental concept in distribution theory. It acts as a point distribution, with significant applications in various fields such as physics and engineering.
It is not a function in the traditional sense but instead a distribution that assigns a value of infinity at a single point, typically at zero, while being zero elsewhere. This property makes it useful for idealizing point sources and effects.

• The Dirac delta function is defined such that its integral over the entire real line is equal to one:
\[\int_{-\infty}^{\infty} \delta(x) \ dx = 1\]
• Despite its unusual characteristics, the Dirac delta's role is crucial when interacting with test functions \( \phi(x) \). It "samples" the value of these functions at the specific point where the delta function is centered.
• Moreover, when dealing with distributions like \( F \) in the given exercise, representing \( F \) as combinations of derivatives of \( \delta(x) \) is common when its support is localized at a point.
• This representation ties back to the concept of support because it signifies that all actions on test functions are concentrated at that point.

Taylor's Theorem

Taylor's theorem is a powerful tool that allows us to estimate and approximate functions in the vicinity of a point using polynomials. For smooth functions, it provides a series expansion that expresses the function using its derivatives at a point.
This theorem is especially helpful when dealing with distributions like in the exercise, where test functions \( \phi \) have derivatives that vanish at zero.

• According to Taylor's theorem, for a smooth function \( f \), it can be expanded around the point \( a \) as:\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^n(a)}{n!}(x-a)^n + R_n(x)\]
• In this expression, \( R_n(x) \) is the remainder term, which becomes negligible for sufficiently small \( x \).
• In the exercise, Taylor's theorem is used to bound derivatives of \( \phi \_k(x) \), showing that if certain derivatives of \( \phi \) vanish, so do those of \( \phi \_k \) as \( k \to \infty \).
• This provides uniform convergence necessary to prove claims like \( \langle F, \phi \rangle = 0 \) when certain conditions are met.

Understanding Taylor’s theorem facilitates grasping how the influence of a distribution concentrated at one point affects surrounding values through a polynomial expansion of test functions.