Academic Journal
A complete orthonormal system of divergence
| Τίτλος: | A complete orthonormal system of divergence |
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| Συγγραφείς: | K. S. Kazarian |
| Πηγή: | Comptes Rendus. Mathématique. 337:85-88 |
| Στοιχεία εκδότη: | Cellule MathDoc/Centre Mersenne, 2003. |
| Έτος έκδοσης: | 2003 |
| Θεματικοί όροι: | independent functions, Independent functions, complete orthonormal system, Complete orthonormal system, 01 natural sciences, Representation of functions by series, 0103 physical sciences, representation of functions by series, 0101 mathematics, divergence, Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis, Completeness of sets of functions in nontrigonometric harmonic analysis, divergence almost everywhere, Analysis, Divergence almost everywhere |
| Περιγραφή: | The main theorem reads as follows. There exists a complete orthonormal system (CONS) \(\{\Theta_n:n=1,2,\dots\}\) of bounded functions defined on the closed interval \([0,1]\) such that the orthogonal series \(\sum^\infty_{n=1} a_n\Theta_n\) diverges almost everywhere (a.e.) for every sequence of real numbers \(\{a_n\} \notin\ell^2\). \textit{A. A. Talaljan} [Russ. Math. Surv. 15, No. 5, 75--136 (1960); translation from Usp. Mat. Nauk 15, No. 5(95), 77--141 (1960; Zbl 0098.04203)] proved the following representation theorem: If \(\{f_n:n=1,2,\dots \}\) is a CONS of functions defined on \([0,1]\), then for every measurable function defined on \([0,1]\) there exists a sequence of real numbers \(\{a_n\}\) such that \(\sum^\infty_{n=1}a_nf_n=f\), where the series converges in measure. Analysing the proof it turns out that Talalyan's theorem remains valid if one discards any finite number of the functions from the system \(\{f_n\}\). Thus, it follows that for every CONS there exist nontrivial series \(\sum^\infty_{n=1} b_nf_n\) which converge to zero in measure. The author of the present paper draws the following surprising corollary of his main theorem: Every nontrivial orthogonal series with respect to the system \(\{\Theta_n\}\) which converges to zero in measure diverges a.e. |
| Τύπος εγγράφου: | Article |
| Περιγραφή αρχείου: | application/xml |
| Γλώσσα: | English |
| ISSN: | 1778-3569 |
| DOI: | 10.1016/s1631-073x(03)00286-3 |
| DOI: | 10.1016/j.jfa.2003.11.005 |
| Σύνδεσμος πρόσβασης: | https://zbmath.org/1981382 https://doi.org/10.1016/s1631-073x(03)00286-3 https://www.sciencedirect.com/science/article/abs/pii/S1631073X03002863 https://www.sciencedirect.com/science/article/pii/S1631073X03002863 https://core.ac.uk/display/82208846 http://www.sciencedirect.com/science/article/pii/S0022123603004063 https://www.sciencedirect.com/science/article/pii/S0022123603004063 |
| Rights: | Elsevier Non-Commercial |
| Αριθμός Καταχώρησης: | edsair.doi.dedup.....3dcf545ef7a1068e3cda3d9a7dee376c |
| Βάση Δεδομένων: | OpenAIRE |
| ISSN: | 17783569 |
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| DOI: | 10.1016/s1631-073x(03)00286-3 |