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In my research, I've come across some results which can be found in the aforementioned paper by Levitan (see here). Several authors simply paraphrase the work in the paper, particularly the results about the Bessel translation operator; however I am not interested just in the result but also in the proofs contained therein. Unfortunately, I cannot read Russian to tell what exactly it says. I can put pieces here or there together, but, for the most part, it's inaccessible. I was wondering if there was an English translation anywhere. For such a highly cited paper (110+ times!), I would think that an English translation would be floating around somewhere but I can't seem to find one. Or alternatively if there is an English reference which proves a lot of the results contained therein (not necessarily a translation), that would work as well. If worse comes to worst, I might try to recreate the proofs myself by piecing together the equations. If it comes to that, I'll throw it up on arXiv for ease of accessibility.

Cameron Williams
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  • I'm a native Russian speaker so I can translate a needed part of paper (the whole paper is quite long). The only problem is that my English is quite poor) – user144765 Mar 23 '15 at 20:19
  • @user144765 Your English is fine! Would you mind translating Section 7 of the paper? It's what I'm most interested in for my purposes. If you do, I'll award you a nice bounty as it's quite a lot to ask :) – Cameron Williams Mar 23 '15 at 20:32
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    Here is the table of contents:
    1. Expansion in Fourier-Bessel Series
    2. Expansion in Fourier-Hankel Integral
    3. Riemann Method
    4. Euler-Poisson Equation
    5. Poisson Formula
    6. Taylor-Delsarte Formula
    7. Properties of $T^y$ Operators
    8. Sonin's and Poisson's Operators
    9. Relation between Sonin's and Poisson's Operators
    10. Fourier-Stieltjes-Bessel integrals
    11. Generalized Positive-Definite Functions
    12. Integral Representation of Generalized Positive-Definite Functions
     – user144765 Mar 23 '15 at 20:49
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    OK, I'll translate the section 7. Just give me one or two days. I'll omit the formulas in text (I don't know TeX) and replace them with only its numerals (e.g (7.11))? – user144765 Mar 23 '15 at 20:57
  • That works great for me. I can fill in the $\LaTeX$ as I go. Thanks a ton! – Cameron Williams Mar 23 '15 at 21:03
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    I had to use LaTeX because of not all formulas in the text had numbers. So as it was my attempt to use it here is the PDF file with translation but text in it can't be copied). The original LaTeX file is also there. I tried to translate as closer to the original as possible almost word-to-word. The formulas which initially had numbers in text are omitted. If you need any clarification just let me know. Hope it will help. – user144765 Mar 24 '15 at 12:05
  • This is great work! Why don't you make your comment an answer and I'll accept it/give you a nice bounty for it. I appreciate it a lot. – Cameron Williams Mar 24 '15 at 18:41
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    Thanks, but bounty is not required. I'm glad it can be helpful. BTW, you should change any "with respect to" in translation with "because of" or "due to" - I misunderstood the meaning of that idiom. In the original text it always means consequence, literally "in the strength of...", which relates to English "because of" or "due to". – user144765 Mar 24 '15 at 19:55

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