Доклады Академии наукДоклады Академии наук0869-5652The Russian Academy of Sciences1282610.31857/S0869-56524852149-152Research ArticleInterpolation by sums of series of exponentials and global Cauchy problem for convolution operatorsMerzlyakovS. G.msg2000@mail.ruPopenovS. V.spopenov@gmail.comUfa Branch of the Russian Academy of Sciences2005201948521491522205201922052019Copyright © 2019, Russian academy of sciences2019<p>The study is made of the problem of multiple interpolation on an infinite nodes set by the sums of absolutely convergent series of exponentials whose exponents are from a given set. For entire function conditions on nodes and exponents are obtained that give solubility of the problem. A new approach is demonstrated that enable us, for the case of holomorphic function in a domain, to obtain criteria of solubility of the problem for some class of exponents set and for a special class of nodes set. Moreover the necessity of the conditions is proved in great generality namely for arbitrary nodes sets and in the setting of interpolation by functions that are represented as the Laplace transforms of the Radon measures over the exponents set. Solubility is obtained of the global Cauchy problem for convolution operator with data on the nodes set in domain, in the form of the series of exponentials whose exponents belong to a sparse subset of zero set of characteristic function of the operator. The results substantially strengthen the known results on the theme.</p>series of exponentialsmultiple interpolationconvolution operatorряд экспоненткратная интерполяцияоператор свёртки[Леонтьев А. Ф. Последовательности полиномов из экспонент. М.: Наука, 1980.][Кривошеев А. С. // Изв. РАН. Сер. мат. 2004. Т. 68. № 2. С. 71-136.][Кривошеева О. А., Кривошеев А. С. // Функцион. анализ и прил. 2012. Т. 46. № 4. С. 14-30.][Кривошеев А. С., Кривошеева О. А. // Уфим. мат. журн. 2013. Т. 5. № 3. С. 96-120.][Беллман Р., Кук К. Дифференциально-разностные уравнения. М.: Мир, 1967.][Напалков В. В., Попенов С. В. // ДАН. 2001. Т. 381. № 2. С. 164-166.][Напалков В. В., Нуятов А. А. // Мат. сб. 2012. Т. 203. № 2. С. 77-86.][Напалков В. В., Нуятов А. А. // ТМФ. 2014. Т. 180. № 2. С. 264-271.][Зименс К. Р., Напалков В. В. // ДАН. 2014. Т. 458. № 4. С. 387-389.][Мерзляков С. Г., Попенов С. В. // Уфим. мат. журн. 2013. Т. 5. № 3. С. 130-143.][Мерзляков С. Г., Попенов С. В. // Уфим. мат. журн. 2015. Т. 7. № 1. С. 46-58.][Мерзляков С. Г., Попенов С. В. Математический анализ. Итоги науки и техн. Сер. Соврем. математика и ее прил. М.: ВИНИТИ РАН, 2017. Т. 143. С. 4862.][Мерзляков С. Г. // Уфим. мат. журн. 2011. Т. 3. № 2. С. 57-80.]