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 4 Minor addition edited Feb 22 at 17:14 Daniele Tampieri 77511 gold badge66 silver badges1717 bronze badges The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $$\delta$$-sequences, follows the path traced years ago by Stanis?aw ?ojasiewicz in the paper [1], so I describe his approach to the problem below. ?ojasiewicz analyzes the problem for functions of one variable, i.e. $$n=1$$: by using the definition of change of variables in a distribution (see for example [2], §1.9 pp. 21-22) and considering the change of variable $$y=x_0+\lambda x$$, for $$x,x_0,\lambda \in\Bbb R$$, i.e. $$\begin{split} T(x_0+\lambda x)&\triangleq \langle T(x_0+\lambda x),\varphi(x)\rangle\\ &=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x_0)\big)}{\lambda}\right\rangle \end{split} \quad \varphi\in\mathscr{D}(\Bbb R)$$ he defines the limit of a distribution at a point $$x_o$$ as ([1], §1 p. 2-3) $$\lim_{x\to x_0} T\triangleq \lim_{\lambda\to 0} T(x_0+\lambda x) \label{1}\tag{1}$$ and proves that $$\lim_{x\to x_0} T=\lim_{x\to x_0^+} T=\lim_{x\to x_0^-} T$$ by using an earlier result of Ziele?ny, if the limit \eqref{1} exists, it is necessarily a constant $$C\in \Bbb C$$, or more precisely a constant distribution $$C$$. a necessary and sufficient condition for the limit \eqref{1} to exist is (see [1], §2, theorem 2.2, pp. 5-7) that $$T=f^{(n)}$$, where $$f\in C^0(\Bbb R)$$ and $$\lim_{x\to x_0}\frac{f(x)}{(x-x_0)^n}=\frac{C}{n!}.$$ Then ?ojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point: note that this definition does not rely on any particular test function (or sequence of such) $$\varphi\in\mathscr{D}(\Bbb R)$$, as stated above. Now a few observations: ?ojasiewicz ([1], §1 p. 1) states that the case $$n>1$$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984) Multivariable Analysis, Springer-Verlag. ?ojasiewicz gives another necessary and sufficient condition for the limit \eqref{1} to exists, in terms of Denjoy differentials ([1], §2, corollary to theorem 2.2, p. 7). The term $$\lambda^{-1}$$, more or less intrinsically used in \eqref{1}, suggests the possible use of the Mellin transform: this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines a (generalized) spectral value of a function/distributions at a point and proves ([3], §12 p. 43) that it coincides with ?ojasiewicz point value \eqref{1} when this exists (by using the necessary and sufficient condition above): the construction of Ziemaian however does not apply to all distributions. [1] Stanis?aw ?ojasiewicz (1957-1958), "Sur la valeur et la limite d'une distribution en un point" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, MR0087905 Zbl 0086.09405. [2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029. [3] Bogdan Ziemian (1988), "Taylor formula for distributions", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, MR0931848, Zbl 0685.46025. The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $$\delta$$-sequences, follows the path traced years ago by Stanis?aw ?ojasiewicz in the paper [1], so I describe his approach to the problem below. ?ojasiewicz analyzes the problem for functions of one variable, i.e. $$n=1$$: by using the definition of change of variables in a distribution (see for example [2], §1.9 pp. 21-22) and considering the change of variable $$y=x_0+\lambda x$$, for $$x,x_0,\lambda \in\Bbb R$$, i.e. $$\begin{split} T(x_0+\lambda x)&\triangleq \langle T(x_0+\lambda x),\varphi(x)\rangle\\ &=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x_0)\big)}{\lambda}\right\rangle \end{split} \quad \varphi\in\mathscr{D}(\Bbb R)$$ he defines the limit of a distribution at a point $$x_o$$ as ([1], §1 p. 2-3) $$\lim_{x\to x_0} T\triangleq \lim_{\lambda\to 0} T(x_0+\lambda x) \label{1}\tag{1}$$ and proves that $$\lim_{x\to x_0} T=\lim_{x\to x_0^+} T=\lim_{x\to x_0^-} T$$ by using an earlier result of Ziele?ny, if the limit \eqref{1} exists, it is necessarily a constant $$C\in \Bbb C$$, or more precisely a constant distribution $$C$$. a necessary and sufficient condition for the limit \eqref{1} to exist is (see [1], §2, theorem 2.2, pp. 5-7) that $$T=f^{(n)}$$, where $$f\in C^0(\Bbb R)$$ and $$\lim_{x\to x_0}\frac{f(x)}{(x-x_0)^n}=\frac{C}{n!}.$$ Then ?ojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point: note that this definition does not rely on any particular test function (or sequence of such) $$\varphi\in\mathscr{D}(\Bbb R)$$, as stated above. Now a few observations: ?ojasiewicz ([1], §1 p. 1) states that the case $$n>1$$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984) Multivariable Analysis, Springer-Verlag. ?ojasiewicz gives another necessary and sufficient condition for the limit \eqref{1} to exists, in terms of Denjoy differentials ([1], §2, corollary to theorem 2.2, p. 7). The term $$\lambda^{-1}$$, more or less intrinsically used in \eqref{1}, suggests the possible use of the Mellin transform: this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines a (generalized) spectral value of a function/distributions at a point and proves ([3], §12 p. 43) it coincides with ?ojasiewicz point value \eqref{1} when this exists: the construction of Ziemaian however does not apply to all distributions. [1] Stanis?aw ?ojasiewicz (1957-1958), "Sur la valeur et la limite d'une distribution en un point" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, MR0087905 Zbl 0086.09405. [2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029. [3] Bogdan Ziemian (1988), "Taylor formula for distributions", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, MR0931848, Zbl 0685.46025. The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $$\delta$$-sequences, follows the path traced years ago by Stanis?aw ?ojasiewicz in the paper [1], so I describe his approach to the problem below. ?ojasiewicz analyzes the problem for functions of one variable, i.e. $$n=1$$: by using the definition of change of variables in a distribution (see for example [2], §1.9 pp. 21-22) and considering the change of variable $$y=x_0+\lambda x$$, for $$x,x_0,\lambda \in\Bbb R$$, i.e. $$\begin{split} T(x_0+\lambda x)&\triangleq \langle T(x_0+\lambda x),\varphi(x)\rangle\\ &=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x_0)\big)}{\lambda}\right\rangle \end{split} \quad \varphi\in\mathscr{D}(\Bbb R)$$ he defines the limit of a distribution at a point $$x_o$$ as ([1], §1 p. 2-3) $$\lim_{x\to x_0} T\triangleq \lim_{\lambda\to 0} T(x_0+\lambda x) \label{1}\tag{1}$$ and proves that $$\lim_{x\to x_0} T=\lim_{x\to x_0^+} T=\lim_{x\to x_0^-} T$$ by using an earlier result of Ziele?ny, if the limit \eqref{1} exists, it is necessarily a constant $$C\in \Bbb C$$, or more precisely a constant distribution $$C$$. a necessary and sufficient condition for the limit \eqref{1} to exist is (see [1], §2, theorem 2.2, pp. 5-7) that $$T=f^{(n)}$$, where $$f\in C^0(\Bbb R)$$ and $$\lim_{x\to x_0}\frac{f(x)}{(x-x_0)^n}=\frac{C}{n!}.$$ Then ?ojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point: note that this definition does not rely on any particular test function (or sequence of such) $$\varphi\in\mathscr{D}(\Bbb R)$$, as stated above. Now a few observations: ?ojasiewicz ([1], §1 p. 1) states that the case $$n>1$$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984) Multivariable Analysis, Springer-Verlag. ?ojasiewicz gives another necessary and sufficient condition for the limit \eqref{1} to exists, in terms of Denjoy differentials ([1], §2, corollary to theorem 2.2, p. 7). The term $$\lambda^{-1}$$, more or less intrinsically used in \eqref{1}, suggests the possible use of the Mellin transform: this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines a (generalized) spectral value of a function/distributions at a point and proves ([3], §12 p. 43) that it coincides with ?ojasiewicz point value \eqref{1} when this exists (by using the necessary and sufficient condition above): the construction of Ziemaian however does not apply to all distributions. [1] Stanis?aw ?ojasiewicz (1957-1958), "Sur la valeur et la limite d'une distribution en un point" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, MR0087905 Zbl 0086.09405. [2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029. [3] Bogdan Ziemian (1988), "Taylor formula for distributions", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, MR0931848, Zbl 0685.46025. 3 Corrected the statement of an important theorem+ minor additions edited Feb 22 at 17:02 Daniele Tampieri 77511 gold badge66 silver badges1717 bronze badges The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $$\delta$$-sequences, follows the path traced years ago by Stanis?aw ?ojasiewicz in the paper [1], so I describe his approach to the problem below. ?ojasiewicz analyzes the problem for functions of one variable, i.e. $$n=1$$: by using the definition of change of variables in a distribution (see for example [2], §1.9 pp. 21-22) and considering the change of variable $$y=x_0+\lambda x$$, for $$x,x_0,\lambda \in\Bbb R$$, i.e. $$\begin{split} T(x_0+\lambda x)&\triangleq \langle T(x_0+\lambda x),\varphi(x)\rangle\\ &=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x_0)\big)}{\lambda}\right\rangle \end{split} \quad \varphi\in\mathscr{D}(\Bbb R)$$ he defines the limit of a distribution at a point $$x_o$$ as ([1], §1 p. 2-3) $$\lim_{x\to x_0} T\triangleq \lim_{\lambda\to 0} T(x_0+\lambda x) \label{1}\tag{1}$$ and proves that $$\lim_{x\to x_0} T=\lim_{x\to x_0^+} T=\lim_{x\to x_0^-} T$$ by using an earlier result of Ziele?ny, if the limit \eqref{1} exists, it is necessarily a constant $$C\in \Bbb C$$, or more precisely a constant distribution $$C$$. a necessary and sufficient condition for the limit \eqref{1} to exist is (see [1], §2, theorem 2.2, pp. 5-7) that $$T=f^{(n)}$$, where $$f\in C^0(\Bbb R)$$ and $$\lim_{x\to x_0}\frac{f(x)}{(x-x_0)^n}=\frac{C}{n!}.$$ Then ?ojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point: note that this definition does not rely on any particular test function (or sequence of such) $$\varphi\in\mathscr{D}(\Bbb R)$$, as stated above. Now a few observations: ?ojasiewicz ([1], §1 p. 1) states that the case $$n>1$$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984) Multivariable Analysis, Springer-Verlag. ?ojasiewicz gives another necessary and sufficient condition for the limit \eqref{1} to exists, in terms of Denjoy differentials ([1], §2, corollary to theorem 2.2, p. 7). The term $$\lambda^{-1}$$, more or less intrinsically used in \eqref{1}, suggests the possible use of the Mellin transform: this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines a (generalized) spectral value of a function/distributions at a point and proves ([3], §12 p. 43) it coincides with ?ojasiewicz point value \eqref{1} when this exists: the construction of Ziemaian however does not apply to all distributions. [1] Stanis?aw ?ojasiewicz (1957-1958), "Sur la valeur et la limite d'une distribution en un point" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, MR0087905 Zbl 0086.09405. [2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029. [3] Bogdan Ziemian (1988), "Taylor formula for distributions", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, MR0931848, Zbl 0685.46025. The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $$\delta$$-sequences, follows the path traced years ago by Stanis?aw ?ojasiewicz in the paper [1], so I describe his approach to the problem below. ?ojasiewicz analyzes the problem for functions of one variable, i.e. $$n=1$$: by using the definition of change of variables in a distribution (see for example [2], §1.9 pp. 21-22) and considering the change of variable $$y=x_0+\lambda x$$, for $$x,x_0,\lambda \in\Bbb R$$, i.e. $$\begin{split} T(x_0+\lambda x)&\triangleq \langle T(x_0+\lambda x),\varphi(x)\rangle\\ &=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x_0)\big)}{\lambda}\right\rangle \end{split} \quad \varphi\in\mathscr{D}(\Bbb R)$$ he defines the limit of a distribution at a point $$x_o$$ as ([1], §1 p. 2-3) $$\lim_{x\to x_0} T\triangleq \lim_{\lambda\to 0} T(x_0+\lambda x) \label{1}\tag{1}$$ and proves that $$\lim_{x\to x_0} T=\lim_{x\to x_0^+} T=\lim_{x\to x_0^-} T$$ by using an earlier result of Ziele?ny, if the limit \eqref{1} exists, it is necessarily a constant $$C\in \Bbb C$$, or more precisely a constant distribution $$C$$. a necessary and sufficient condition for the limit \eqref{1} to exist is (see [1], §2, theorem 2.2, pp. 5-7) that $$T=f^{(n)}$$, where $$f\in C^0(\Bbb R)$$ and $$\lim_{x\to x_0}\frac{f(x)}{(x-x_0)^n}=\frac{C}{n!}.$$ Then ?ojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point: note that this definition does not rely on any particular test function (or sequence of such) $$\varphi\in\mathscr{D}(\Bbb R)$$, as stated above. Now a few observations: ?ojasiewicz ([1], §1 p. 1) states that the case $$n>1$$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984) Multivariable Analysis, Springer-Verlag. The term $$\lambda^{-1}$$, more or less intrinsically used in \eqref{1}, suggests the possible use of the Mellin transform: this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines a (generalized) spectral value of a function/distributions at a point and proves ([3], §12 p. 43) it coincides with ?ojasiewicz point value \eqref{1} when this exists: the construction of Ziemaian however does not apply to all distributions. [1] Stanis?aw ?ojasiewicz (1957-1958), "Sur la valeur et la limite d'une distribution en un point" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, MR0087905 Zbl 0086.09405. [2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029. [3] Bogdan Ziemian (1988), "Taylor formula for distributions", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, MR0931848, Zbl 0685.46025. The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $$\delta$$-sequences, follows the path traced years ago by Stanis?aw ?ojasiewicz in the paper [1], so I describe his approach to the problem below. ?ojasiewicz analyzes the problem for functions of one variable, i.e. $$n=1$$: by using the definition of change of variables in a distribution (see for example [2], §1.9 pp. 21-22) and considering the change of variable $$y=x_0+\lambda x$$, for $$x,x_0,\lambda \in\Bbb R$$, i.e. $$\begin{split} T(x_0+\lambda x)&\triangleq \langle T(x_0+\lambda x),\varphi(x)\rangle\\ &=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x_0)\big)}{\lambda}\right\rangle \end{split} \quad \varphi\in\mathscr{D}(\Bbb R)$$ he defines the limit of a distribution at a point $$x_o$$ as ([1], §1 p. 2-3) $$\lim_{x\to x_0} T\triangleq \lim_{\lambda\to 0} T(x_0+\lambda x) \label{1}\tag{1}$$ and proves that $$\lim_{x\to x_0} T=\lim_{x\to x_0^+} T=\lim_{x\to x_0^-} T$$ by using an earlier result of Ziele?ny, if the limit \eqref{1} exists, it is necessarily a constant $$C\in \Bbb C$$, or more precisely a constant distribution $$C$$. a necessary and sufficient condition for the limit \eqref{1} to exist is (see [1], §2, theorem 2.2, pp. 5-7) that $$T=f^{(n)}$$, where $$f\in C^0(\Bbb R)$$ and $$\lim_{x\to x_0}\frac{f(x)}{(x-x_0)^n}=\frac{C}{n!}.$$ Then ?ojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point: note that this definition does not rely on any particular test function (or sequence of such) $$\varphi\in\mathscr{D}(\Bbb R)$$, as stated above. Now a few observations: ?ojasiewicz ([1], §1 p. 1) states that the case $$n>1$$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984) Multivariable Analysis, Springer-Verlag. ?ojasiewicz gives another necessary and sufficient condition for the limit \eqref{1} to exists, in terms of Denjoy differentials ([1], §2, corollary to theorem 2.2, p. 7). The term $$\lambda^{-1}$$, more or less intrinsically used in \eqref{1}, suggests the possible use of the Mellin transform: this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines a (generalized) spectral value of a function/distributions at a point and proves ([3], §12 p. 43) it coincides with ?ojasiewicz point value \eqref{1} when this exists: the construction of Ziemaian however does not apply to all distributions. [1] Stanis?aw ?ojasiewicz (1957-1958), "Sur la valeur et la limite d'une distribution en un point" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, MR0087905 Zbl 0086.09405. [2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029. [3] Bogdan Ziemian (1988), "Taylor formula for distributions", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, MR0931848, Zbl 0685.46025. 2 Corrected the statement of an important theorem. edited Feb 22 at 16:57 Daniele Tampieri 77511 gold badge66 silver badges1717 bronze badges The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $$\delta$$-sequences, follows the path traced years ago by Stanis?aw ?ojasiewicz in the paper [1], so I describe his approach to the problem below. ?ojasiewicz analyzes the problem for functions of one variable, i.e. $$n=1$$: by using the definition of change of variables in a distribution (see for example [2], §1.9 pp. 21-22) and considering the change of variable $$y=x_0+\lambda x$$, for $$x,x_0,\lambda \in\Bbb R$$, i.e. $$\begin{split} T(x_0+\lambda x)&\triangleq \langle T(x_0+\lambda x),\varphi(x)\rangle\\ &=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x_0)\big)}{\lambda}\right\rangle \end{split} \quad \varphi\in\mathscr{D}(\Bbb R)$$ he defines the limit of a distribution at a point $$x_o$$ as ([1], §1 p. 2-3) $$\lim_{x\to x_0} T\triangleq \lim_{\lambda\to 0} T(x_0+\lambda x) \label{1}\tag{1}$$ and proves that $$\lim_{x\to x_0} T=\lim_{x\to x_0^+} T=\lim_{x\to x_0^-} T$$ by using an earlier result of Ziele?ny, if the limit \eqref{1} exists, it is necessarily a constant $$C\in \Bbb C$$, or more precisely a constant distribution $$C$$. a necessary and sufficient condition for the limit \eqref{1} to exists,exist is (see [1], §2, theorem 2.2, pp. 5-7) that $$T=f^{(n)}$$, where $$f\in C^0(\Bbb R)$$, i.e. $$T$$ as a value at a point $$x_0\in \Bbb R$$ in the sense of \eqref{1} iff $$T$$ is the derivative of order $$n$$ for a fixed $$n\in\Bbb N$$ of a continuous function $$f$$. and $$\lim_{x\to x_0}\frac{f(x)}{(x-x_0)^n}=\frac{C}{n!}.$$ Then ?ojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point: note that this definition does not rely on any particular test function (or sequence of such) $$\varphi\in\mathscr{D}(\Bbb R)$$, as stated above. Now a few observations: ?ojasiewicz ([1], §1 p. 1) states that the case $$n>1$$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984) Multivariable Analysis, Springer-Verlag. The term $$\lambda^{-1}$$, more or less intrinsically used in \eqref{1}, suggests the possible use of the Mellin transform: this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines a (generalized) spectral value of a function/distributions at a point and proves ([3], §12 p. 43) it coincides with ?ojasiewicz point value \eqref{1} when this exists: the construction of Ziemaian however does not apply to all distributions. [1] Stanis?aw ?ojasiewicz (1957-1958), "Sur la valeur et la limite d'une distribution en un point" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, MR0087905 Zbl 0086.09405. [2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029. [3] Bogdan Ziemian (1988), "Taylor formula for distributions", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, MR0931848, Zbl 0685.46025. The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $$\delta$$-sequences, follows the path traced years ago by Stanis?aw ?ojasiewicz in the paper [1], so I describe his approach to the problem below. ?ojasiewicz analyzes the problem for functions of one variable, i.e. $$n=1$$: by using the definition of change of variables in a distribution (see for example [2], §1.9 pp. 21-22) and considering the change of variable $$y=x_0+\lambda x$$, for $$x,x_0,\lambda \in\Bbb R$$, i.e. $$\begin{split} T(x_0+\lambda x)&\triangleq \langle T(x_0+\lambda x),\varphi(x)\rangle\\ &=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x_0)\big)}{\lambda}\right\rangle \end{split} \quad \varphi\in\mathscr{D}(\Bbb R)$$ he defines the limit of a distribution at a point $$x_o$$ as ([1], §1 p. 2-3) $$\lim_{x\to x_0} T\triangleq \lim_{\lambda\to 0} T(x_0+\lambda x) \label{1}\tag{1}$$ and proves that $$\lim_{x\to x_0} T=\lim_{x\to x_0^+} T=\lim_{x\to x_0^-} T$$ by using an earlier result of Ziele?ny, if the limit \eqref{1} exists, it is necessarily a constant $$C\in \Bbb C$$, or more precisely a constant distribution $$C$$. a necessary and sufficient condition for the limit \eqref{1} to exists, is that $$T=f^{(n)}$$ where $$f\in C^0(\Bbb R)$$, i.e. $$T$$ as a value at a point $$x_0\in \Bbb R$$ in the sense of \eqref{1} iff $$T$$ is the derivative of order $$n$$ for a fixed $$n\in\Bbb N$$ of a continuous function $$f$$. Then ?ojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point: note that this definition does not rely on any particular test function (or sequence of such) $$\varphi\in\mathscr{D}(\Bbb R)$$, as stated above. Now a few observations: ?ojasiewicz ([1], §1 p. 1) states that the case $$n>1$$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984) Multivariable Analysis, Springer-Verlag. The term $$\lambda^{-1}$$, more or less intrinsically used in \eqref{1}, suggests the possible use of the Mellin transform: this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines a (generalized) spectral value of a function/distributions at a point and proves ([3], §12 p. 43) it coincides with ?ojasiewicz point value \eqref{1} when this exists: the construction of Ziemaian however does not apply to all distributions. [1] Stanis?aw ?ojasiewicz (1957-1958), "Sur la valeur et la limite d'une distribution en un point" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, MR0087905 Zbl 0086.09405. [2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029. [3] Bogdan Ziemian (1988), "Taylor formula for distributions", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, MR0931848, Zbl 0685.46025. The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $$\delta$$-sequences, follows the path traced years ago by Stanis?aw ?ojasiewicz in the paper [1], so I describe his approach to the problem below. ?ojasiewicz analyzes the problem for functions of one variable, i.e. $$n=1$$: by using the definition of change of variables in a distribution (see for example [2], §1.9 pp. 21-22) and considering the change of variable $$y=x_0+\lambda x$$, for $$x,x_0,\lambda \in\Bbb R$$, i.e. $$\begin{split} T(x_0+\lambda x)&\triangleq \langle T(x_0+\lambda x),\varphi(x)\rangle\\ &=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x_0)\big)}{\lambda}\right\rangle \end{split} \quad \varphi\in\mathscr{D}(\Bbb R)$$ he defines the limit of a distribution at a point $$x_o$$ as ([1], §1 p. 2-3) $$\lim_{x\to x_0} T\triangleq \lim_{\lambda\to 0} T(x_0+\lambda x) \label{1}\tag{1}$$ and proves that $$\lim_{x\to x_0} T=\lim_{x\to x_0^+} T=\lim_{x\to x_0^-} T$$ by using an earlier result of Ziele?ny, if the limit \eqref{1} exists, it is necessarily a constant $$C\in \Bbb C$$, or more precisely a constant distribution $$C$$. a necessary and sufficient condition for the limit \eqref{1} to exist is (see [1], §2, theorem 2.2, pp. 5-7) that $$T=f^{(n)}$$, where $$f\in C^0(\Bbb R)$$ and $$\lim_{x\to x_0}\frac{f(x)}{(x-x_0)^n}=\frac{C}{n!}.$$ Then ?ojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point: note that this definition does not rely on any particular test function (or sequence of such) $$\varphi\in\mathscr{D}(\Bbb R)$$, as stated above. Now a few observations: ?ojasiewicz ([1], §1 p. 1) states that the case $$n>1$$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984) Multivariable Analysis, Springer-Verlag. The term $$\lambda^{-1}$$, more or less intrinsically used in \eqref{1}, suggests the possible use of the Mellin transform: this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines a (generalized) spectral value of a function/distributions at a point and proves ([3], §12 p. 43) it coincides with ?ojasiewicz point value \eqref{1} when this exists: the construction of Ziemaian however does not apply to all distributions. [1] Stanis?aw ?ojasiewicz (1957-1958), "Sur la valeur et la limite d'une distribution en un point" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, MR0087905 Zbl 0086.09405. [2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029. [3] Bogdan Ziemian (1988), "Taylor formula for distributions", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, MR0931848, Zbl 0685.46025. 1 answered Feb 22 at 6:47 Daniele Tampieri 77511 gold badge66 silver badges1717 bronze badges
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