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SIXTH WORKSHOP ON FOURIER ANALYSIS AND RELATED FIELDS

Conference Booklet

^ . У Sixth Workshop on Fourier Analysis and Related Fields

Conference Booklet

Pecs, 2017 * List of participants

Roman R. Akopyan Marina Deikalova Ural Federal University Ural Federal University Ekaterinburg, Russia Ekaterinburg, Russia [email protected] marina. deikalova@urfu. ru

Aleksandr Babenko Ti'mea Eisner IMM UB RAS and Ural Federal Uni- University of Pecs versity Pecs, Hungary Ekaterinburg, Russia [email protected] babenko@imm .uran.ru

Hans Feichtinger Ekaterina Berestova University of Vienna Ural Federal University Vienna, Austria Ekaterinburg, Russia hans. feicht inger@univie. ac. at [email protected]

Vanda Fiilop Istvan Berkes University of Szeged Renyi Mathematical Institute and Szeged, Hungary TU Graz fulopv@math. u-szeged. hu Budapest, Hungary/Graz, Austria [email protected] Marcell Gaal University of Szeged Gautami Bhowmik Szeged, Hungary Universite de Lille 1 [email protected] Lille, France [email protected] Gyorgy Gat Istvan Blahota University of Debrecen Debrecen, Hungary University of Nyfregyhaza [email protected] Nyfregyhaza, Hungary [email protected] r Polina Glazyrina Aline Bonami Ural Federal University Universite d'Orleans Ekaterinburg, Russia Orleans, France [email protected] [email protected]

Maxim Gridnev Alberto Debernardi Krasovskii Institute of Mathematics Centre de Recerca Matematica and Mechanics Bellaterra-Barcelona, Spain Ekaterinburg, Russia [email protected] [email protected]

6 Dmitry Gorbachev Balazs Kiraly Tula State University University of Pecs Tula, Russia Pecs, Hungary dvgmail@mail. ru [email protected]

Andi Kivinukk Katalin Gyarmati Tallinn University Eotvos Lorand University Tallinn, Estonia Budapest, Hungary [email protected] gykati@cs. elte.hu

Yurii Kolomoitsev Norbert Hegyvari University of Liibeck Eotvos Lorand University Liibeck, Germany Budapest, Hungary kolomoitsev@math. uni-luebeck .de [email protected] Mihalis Kolountzakis University of Crete Titus Hilberdink Heraklion, Greece University of Reading [email protected] Reading, UK [email protected] Vilmos Komornik University of Strasbourg Strasbourg, France Agota Horvath [email protected] Budapest University of Technology and Economics Budapest, Hungary Stamatis Koumandos [email protected] University of Cyprus Nicosia, Cyprus [email protected] Daniyal Israfilov Balikesir University Bahkesir, Turkey Nikolai Kuklin mdaniyal@balikesir .edu.tr RAS and Ural Federal University Ekaterinburg, Russia [email protected]

Valery Ivanov Tula State University Jozsef Laczko Tula, Russia University of Pecs; HAS Wigner R.C [email protected] Pecs/Budapest, Hungary [email protected]

Turgay H. Kaptanoglu Miklos Laczkovich Bilkent University Eotvos Lorand University Ankara, Turkey Budapest, Hungary kapt an@fen. bilkent. edu. tr [email protected]

7 Partha Sarathi Patra ИТ Hyderabad Mate Matolcsi Hyderabad, India Renyi Mathematical Institute [email protected] Budapest, Hungary [email protected]

Akos Pilgermajer Nacima Memic University of Pecs University of Sarajevo Pecs, Hungary Sarajevo, Bosnia and Herzegovina [email protected] [email protected]

Istvan Prause Ferenc Moricz University of Helsinki University of Szeged Helsinki, Finland Szeged, Hungary istvan. prause@helsinki. fi moricz@math. u-szeged. hu

Szilard Revesz Bela Nagy University of Pecs; Renyi Mathe- Hung. Acad, of Sci. and Univ. 6f matical Institute Szeged Budapest, Hungary Szeged, Hungary revesz@r eny i. hu nbela@math. u-szeged. hu

Yauheni Rouba Karoly Nagy Grodno State University University of Nyi'regyhaza Grodno, Belarus Nyiregyhaza, Hungary [email protected] nagy. karoly @nye. hu

Imre Z. Ruzsa Zoltan Nemeth Renyi Mathematical Institute University of Szeged Budapest, Hungary Szeged, Hungary [email protected] znemeth@math. u-szeged. hu

Yurii Shteinikov Serap Oztop SRISA and Steklov Mathematical Istanbul University Institute Istanbul, Turkey Moscow, Russia oztops@ist anbul .edu.tr [email protected]

Margit Pap Ilona Simon University of Pecs University of Pecs Pecs, Hungary Pecs, Hungary [email protected] [email protected]

8 Sivaramakrishnan С Rodolfo Toledo IIT Hyderabad University of Nyi'regyhaza Hyderabad, India .. Nyi'regyhaza, Hungary [email protected] toledo. [email protected]

Laszlo Szekelyhidi Anastasiia Torgashova University of Debrecen Ural Federal University Debrecen, Hungary Ekaterinburg, Russia lszekely hidi@gmail. com [email protected]

Ahmet Testici Bahkesir University Laszlo Toth Bahkesir, Turkey University of Pecs [email protected] Pecs, Hungary [email protected]

Sergey Tikhonov Barcelona Graduate School of Math- Ferenc Weisz ematics Eotvos Lorand University Barcelona, Spain Budapest, Hungary [email protected] [email protected]

9 Rational Fourier - Chebyshev series of some elementary functions

Rouba, Yauheni

In the present report we consider the following system of Chebyshev - Markov rational fractions with real nonnegative parameter a:

( Vl + a2 \ Mn(x) = cosnarccos x . = I, x G [—1,1], n = 0,1,... . V vl + a2x2 J

It is orthogonal on the segment [—1,1] with respect to the weight

p(x, a) — ———. „, -1 < x < 1.

Let / be absolutely integrable with respect to the weight p(x, a) function on the

segment [—1,1]. We associate the Fourier series with respect to the system {Mn} to such function /: +oo

/Or)~| + £c„Mn(x), z n=1 with coefficients: l

= ~ J p(t> a)f{t)Mn(t) dt, n = 0,1,... . -l Theorem 1. For the partial sums of this Fourier series the following equality holds

7Г 1 - a4 [ sin (n + l/2)\(u,v) dv 3n{x;f)= / /(cosv) 2?r sin X(u, v)/2 1 + 2a2 cos 2v + a4'

41 where 4 2 w ^ f 1 - a , vT + a -! A(it,u) = / — 4 dy, Q = . J 1 + 2a2 cos 2y + a a

2 2 n Besides, Sn{x, f) = pn{x)/(1 + a x ) , where pn{x) is an algebraic polynomial of degree n and Sn(x, 1) = 1.

In this work we study the Fourier series with respect to the system {Mn} of the function |x|. We found Fourier coefficients in the explicit form. Let's introduce the following deviations:

x £2n{x, a) = |x| - s2n(M; € [-1,1]; e2n((*) = \\ein{x, a)||c[-i,i]- Theorem 2. The following inequalities hold:

1 + 2a2 cos 2u + a4 1 - t2 . Nn(x,a)| \X2n{t)\dt,x = cosu, (*) + 2t2 cos 2u + t4 l-a2t2

l t2-a2 dt £2n(a) < II 1 - a2*2 1 + t2' Inequality (*) is exaci in i/ie sense that if all the poles have even multiplicity then inequality (*) becomes equality for x = 0 and x = 1.

Theorem 3. If e^n — infa£2n(c0, the following estimates hold n2 1 lim -—e n-Yoo In п 2n 7Г

2 inf |e2„(x,a)| < . , [-1,0) U (0,1], n > n0. a 7Г |X| 71°

Also, we are going to briefly discuss the other results on asymptotic estimates of elementary functions by trigonometric series.