Looks like nobody mentioned the work A contribution to the theory of generalized functions by Egorov, which proposes an interesting approach to generalize the concept of distribution in a possibly simpler (yet more general) way than the one by Colombeau.

A nice introductory short-essay on the same argument is in the Appendix (again by Egorov) of the book: Demidov, A. S. (2001). Generalized functions in mathematical physics: main ideas and concepts (Vol. 237, Nova Publishers).

The author first introduces the concept of generalized real (or complex) number as equivalence classes of eventually coinciding sequences: $\{c_k\}\equiv \{z_k\}$ iff they concede for all $k$ large enough. Then he defines a generalized function as an equivalence class of sequences of $C^\infty$ functions defined on a domain $\Omega\subset \mathbb{R}^n$, where $\{f_k\}\equiv \{g_k\}$ iff they eventually coincide on every $K\Subset \Omega$. Clearly the values of generalized functions at a point make sense only as generalized real numbers.

Then the usual Dirac $\delta$ can be recovered as the equivalence class containing $\{k\phi(kx)\}$, where $\phi\in C^\infty_0(\Omega)$ such that $\int\phi(x)dx=1$. The product of two generalized functions is naturally defined through the equivalence class containing the product of (any) two representatives of the factors. Notice that $x\delta(x)\ne 0$, contrary to what happens in the "usual" distribution theory, and that is essential by a well-known no-go theorem by Schwartz:

Let $A$ be an associative algebra, in which a derivation operator (i.e., a linear operator $D : A \to A$ such that $D(f · g) = f · D(g)+D(f)·g)$ is defined. Suppose that the space $C(\mathbb{R})$ of continuous functions on the real line is a subalgebra in $A$, and $D$ coincides with the usual derivation operator on the set of continuously differentiable functions, and the function, which is identically equal to 1, is the unit of the algebra $A$. Then $A$ cannot contain an element $\delta$ such that $x ·  \delta(x) = 0$.

Looks like nobody mentioned the work A contribution to the theory of generalized functions by Egorov, which proposes an interesting approach to generalize the concept of distribution in a possibly simpler (yet more general) way than the one by Colombeau.

A nice introductory short-essay on the same argument is in the Appendix (again by Egorov) of the book: Demidov, A. S. (2001). Generalized functions in mathematical physics: main ideas and concepts (Vol. 237, Nova Publishers).

The author first introduces the concept of generalized real (or complex) numbers as equivalence classes of eventually coinciding sequences: $\{c_k\}\equiv \{z_k\}$ iff they coincide for all $k$ large enough. Then he defines a generalized function as an equivalence class of sequences of $C^\infty$ functions defined on a domain $\Omega\subset \mathbb{R}^n$, where $\{f_k\}\equiv \{g_k\}$ iff they eventually coincide on every $K\Subset \Omega$. Clearly the values of generalized functions at a point make sense only as generalized real numbers.

Then the usual Dirac $\delta$ can be recovered as the equivalence class containing $\{k\phi(kx)\}$, where $\phi\in C^\infty_0(\Omega)$ such that $\int\phi(x)dx=1$. The product of two generalized functions is naturally defined through the equivalence class containing the product of (any) two representatives of the factors. Notice that $x\delta(x)\ne 0$, contrary to what happens in the "usual" distribution theory, and that is essential by a well-known no-go theorem by Schwartz:

Let $A$ be an associative algebra, in which a derivation operator (i.e., a linear operator $D : A \to A$ such that $D(f\cdot g) = f\cdot D(g)+D(f)\cdot g$) is defined. Suppose that the space $C(\mathbb{R})$ of continuous functions on the real line is a subalgebra in $A$, and $D$ coincides with the usual derivation operator on the set of continuously differentiable functions, and the function, which is identically equal to 1, is the unit of the algebra $A$. Then $A$ cannot contain an element $\delta$ such that $x\cdot\delta(x) = 0$.

Looks like nobody mentioned the work A contribution to the theory of generalized functions by Egorov, which proposes an interesting approach to generalize the concept of distribution in a possibly simpler (yet more general) way than the one by Colombeau.

A nice introductory short-essay on the same argument is in the Appendix (again by Egorov) of the book: Demidov, A. S. (2001). Generalized functions in mathematical physics: main ideas and concepts (Vol. 237, Nova Publishers).

The author first introduces the concept of generalized real (or complex) number as equivalence classes of eventually coinciding sequences: $\{c_k\}\equiv \{z_k\}$ iff they concede for all $k$ large enough. Then he defines a generalized function as an equivalence class of sequences of $C^\infty$ functions defined on a domain $\Omega\subset \mathbb{R}^n$, where $\{f_k\}\equiv \{g_k\}$ iff they eventually coincide on every $K\Subset \Omega$. Clearly the values of generalized functions at a point make sense only as generalized real numbers.

Then the usual Dirac $\delta$ can be recovered as the equivalence class containing $\{k\phi(kx)\}$, where $\phi\in C^\infty_0(\Omega)$ such that $\int\phi(x)dx=1$. And in general, the product of generalized functions is defined through the class containing the product of any two representatives of each factor. Notice that $x\delta(x)\ne 0$, contrary to what happens in the "usual" distribution theory, and that is essential by a well-known no-go theorem by Schwartz:

Let $A$ be an associative algebra, in which a derivation operator (i.e., a linear operator $D : A \to A$ such that $D(f · g) = f · D(g)+D(f)·g)$ is defined. Suppose that the space $C(\mathbb{R})$ of continuous functions on the real line is a subalgebra in $A$, and $D$ coincides with the usual derivation operator on the set of continuously differentiable functions, and the function, which is identically equal to 1, is the unit of the algebra $A$. Then $A$ cannot contain an element such that $x ·  (x) = 0$.

Looks like nobody mentioned the work A contribution to the theory of generalized functions by Egorov, which proposes an interesting approach to generalize the concept of distribution in a possibly simpler (yet more general) way than the one by Colombeau.

A nice introductory short-essay on the same argument is in the Appendix (again by Egorov) of the book: Demidov, A. S. (2001). Generalized functions in mathematical physics: main ideas and concepts (Vol. 237, Nova Publishers).

The author first introduces the concept of generalized real (or complex) number as equivalence classes of eventually coinciding sequences: $\{c_k\}\equiv \{z_k\}$ iff they concede for all $k$ large enough. Then he defines a generalized function as an equivalence class of sequences of $C^\infty$ functions defined on a domain $\Omega\subset \mathbb{R}^n$, where $\{f_k\}\equiv \{g_k\}$ iff they eventually coincide on every $K\Subset \Omega$. Clearly the values of generalized functions at a point make sense only as generalized real numbers.

Then the usual Dirac $\delta$ can be recovered as the equivalence class containing $\{k\phi(kx)\}$, where $\phi\in C^\infty_0(\Omega)$ such that $\int\phi(x)dx=1$. The product of two generalized functions is naturally defined through the equivalence class containing the product of (any) two representatives of the factors. Notice that $x\delta(x)\ne 0$, contrary to what happens in the "usual" distribution theory, and that is essential by a well-known no-go theorem by Schwartz:

Let $A$ be an associative algebra, in which a derivation operator (i.e., a linear operator $D : A \to A$ such that $D(f · g) = f · D(g)+D(f)·g)$ is defined. Suppose that the space $C(\mathbb{R})$ of continuous functions on the real line is a subalgebra in $A$, and $D$ coincides with the usual derivation operator on the set of continuously differentiable functions, and the function, which is identically equal to 1, is the unit of the algebra $A$. Then $A$ cannot contain an element $\delta$ such that $x ·  \delta(x) = 0$.