We assign a complex number Am to each n-tuple of non-negative integers m = (m₁, ..., mn) arbitrarily. Show that there exists an f(x1,...,xn) E C (R") satisfying D"f(0) = Am for any m, where 0 = (0,..., 0). The one variable case was shown by Borel (1895). Later Rosenthal (1953) gave a simpler proof by considering 8(x)=ane-lanin!x2 xh n=0 is determined according to the given value of g)(0). Mirkil (1956) gave a proof for the n-dimensional case.
We assign a complex number Am to each n-tuple of non-negative integers m = (m₁, ..., mn) arbitrarily. Show that there exists an f(x1,...,xn) E C (R") satisfying D"f(0) = Am for any m, where 0 = (0,..., 0). The one variable case was shown by Borel (1895). Later Rosenthal (1953) gave a simpler proof by considering 8(x)=ane-lanin!x2 xh n=0 is determined according to the given value of g)(0). Mirkil (1956) gave a proof for the n-dimensional case.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 35E
Related questions
Question
![We assign a complex number Am to each n-tuple of non-negative integers
m = (m₁, ..., mn) arbitrarily. Show that there exists an f(x1,...,xn) = C(R")
satisfying
Dm f(0) = Am
for any m, where 0 = (0,..., 0).
The one variable case was shown by Borel (1895). Later Rosenthal (1953)
gave a simpler proof by considering
8(x)= ane-lanin!x²
n=0
where an, is determined according to the given value of g)(0). Mirkil (1956)
gave a proof for the n-dimensional case.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F823623bd-2dc2-416d-aaca-55607fdf9c03%2F27839b44-d223-4e9a-8e8a-c263715c8f26%2Fkrsa2ei_processed.jpeg&w=3840&q=75)
Transcribed Image Text:We assign a complex number Am to each n-tuple of non-negative integers
m = (m₁, ..., mn) arbitrarily. Show that there exists an f(x1,...,xn) = C(R")
satisfying
Dm f(0) = Am
for any m, where 0 = (0,..., 0).
The one variable case was shown by Borel (1895). Later Rosenthal (1953)
gave a simpler proof by considering
8(x)= ane-lanin!x²
n=0
where an, is determined according to the given value of g)(0). Mirkil (1956)
gave a proof for the n-dimensional case.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 4 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![College Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![College Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning