Timeline for Linear independence of exponential functions: a reference
Current License: CC BY-SA 4.0
25 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 18, 2020 at 0:28 | comment | added | Iosif Pinelis | @AbdelmalekAbdesselam : Thank you for pointing out to this connection. | |
| May 17, 2020 at 18:37 | comment | added | Abdelmalek Abdesselam | somewhat related mathoverflow.net/questions/277655/… | |
| May 17, 2020 at 16:10 | comment | added | Iosif Pinelis | @AndrésE.Caicedo : Thank you for your comment. You are probably referring to Example 7 on page 10 of Section 1.7 of of the book at shorturl.at/cjwxZ . However, the proof there works only for for real $z_k$'s. | |
| May 17, 2020 at 16:06 | answer | added | Todd Trimble | timeline score: 7 | |
| May 17, 2020 at 14:39 | comment | added | Andrés E. Caicedo | Apostol's Calculus book has a proof; I believe it is at the beginning of volume 2. | |
| May 17, 2020 at 13:47 | comment | added | Iosif Pinelis | Previous comment continued: Yet another proof, of a stronger result -- with $t$ in an $n$-set rather than in an interval but only for real $z_k$'s -- is given on p. 10 of "Tchebycheff systems: with applications in analysis and statistics" by Karlin and Studden. That proof uses Rolle's theorem and thus does not seem to work for complex $z_k$'s. | |
| May 17, 2020 at 13:38 | comment | added | Iosif Pinelis | @AlexandreEremenko : I have been unable to find the Pólya--Szego book online; could you please give a link to it? As for complex $z_k$'s and otherwise, my biggest concern is about how ready-to-use the published result is. As I said in the post and a comment, the desired result is obvious and the simplest and most natural proof (for me) is this: "divide by $e^{tz_n}$, differentiate in $t$, and use induction on $n$. $\Box$" -- without using Vandermonde determinants. | |
| May 17, 2020 at 8:31 | comment | added | Liviu Nicolaescu | @IosifPinelis Glad it helps. I'll keep my answer as a comment. My contribution is minimal | |
| May 17, 2020 at 3:28 | comment | added | Alexandre Eremenko | @Iosif Pinelis: It does not matter that $z_k$ are complex: computation of the Vandermonde determinant holds for any field. Frankly speaking I do not understand your difficulty. Polya Szego can be found on line. | |
| May 17, 2020 at 2:43 | comment | added | Iosif Pinelis | @LiviuNicolaescu : Thank you for your comment. This is exactly what I needed except that in Barbu's Lemma 3.2 the linear independence is for the exponential functions defined on $\mathbb R$ rather than on a finite nonzero-length interval. However, the proof (which seems the most natural to me and which I had foremost in mind) will of course work for the finite-interval case as well. Would you like to present your comment as a formal answer? | |
| May 17, 2020 at 2:32 | comment | added | Iosif Pinelis | @AlexandreEremenko : Thank you for your latter comment as well. Unfortunately, I don't have the Pólya--Szego book right now. Do they include the case of complex $z_k$ as well? | |
| May 16, 2020 at 9:16 | comment | added | Liviu Nicolaescu | See Lemma 3.2 p. 92 in the book Differential Equations, Springer Verlag, 2016 by Viorel Barbu. The very elegant proof there is not the usual proof based on Wronskians. He proves a bit more, namely that the exponentials are linearly independent over the field of rational functions with complex coefficients. | |
| May 16, 2020 at 4:24 | answer | added | Alexandre Eremenko | timeline score: 1 | |
| May 16, 2020 at 3:52 | review | Close votes | |||
| May 18, 2020 at 13:45 | |||||
| May 16, 2020 at 3:27 | comment | added | Alexandre Eremenko | @Iosif Pinelis: You may combine probl. 2, and problem 60, of the second volume of Polya Szego, Problems and theorems of Analysis, part 7 "Determinants and Quadratic forms" (The first problem is Vandermonde det, the second is the criterion of linear independence). | |
| May 16, 2020 at 0:53 | comment | added | Pat Devlin | I put this question on my final exam for linear algebra, and I got oodles of cool very different proofs. :-) | |
| May 15, 2020 at 23:01 | comment | added | Iosif Pinelis | @YCor : Thank you for your comment. This may help. | |
| May 15, 2020 at 22:51 | comment | added | Iosif Pinelis | @AlexandreEremenko : Thank you for your comment. A reference to a textbook would be fine. | |
| May 15, 2020 at 22:50 | comment | added | YCor | Publication includes textbooks by definition, so it should definitely appear in several of them... I searched Google books [linear independence exponentials] and immediately got this reference: books.google.fr/… | |
| May 15, 2020 at 22:50 | comment | added | Iosif Pinelis | @yarchik : Thank you for your comment. However, I cannot find this fact in that paper by Ycart. | |
| May 15, 2020 at 22:48 | comment | added | Iosif Pinelis | @ZachTeitler : Thank you for your comment. However, I don't need a proof -- only a reference. | |
| May 15, 2020 at 22:42 | comment | added | Alexandre Eremenko | You will probably find no 20 century publication, except an exercise is some Calculus or Linear algebra book. All these facts were really clarified in 18th century (perhaps by Wronski himself, or Vandermonde), but at that time they did not speak of "linear independence":-) | |
| May 15, 2020 at 22:20 | comment | added | yarchik | Ycart, Bernard (2013), "A case of mathematical eponymy: the Vandermonde determinant", Revue d'Histoire des Mathématiques, 13, arXiv:1204.4716 | |
| May 15, 2020 at 22:20 | comment | added | Zach Teitler | Vandermonde matrix is nonsingular? | |
| May 15, 2020 at 22:03 | history | asked | Iosif Pinelis | CC BY-SA 4.0 |