The question related is reduced, as is known in mathematical physics, to the definition of a function ϕ (x, y, t), depending on the point (x, y) and on time t, which satisfies the following conditions (see article). It is absolutely clear, as M. Vergue notes in his interesting memoir, that the expansion theorem also holds for waves occurring under the influence of impulsive forces applied on the surface of the liquid (ondes par impulsion) - the only difference is that sin is replaced by cos in the formula.
To solve the question of whether the remainder Rn of the Taylor series tends to zerowhen lim n = ∞, they try, as is known, to present the remainder in various forms, because often for this one form is more convenient than the other, as can be seen for example. at least in the log (1 + x) expansion. This method can obviously be generalized to find various forms of the remainder of interpolation formulas.
In view of the simplicity of analytical tools for proof and the absence of the need to study a large literature on the issue, the problem of finding various forms of the remainder of the Lagrange and Newton interpolation formulas could be of particular interest to those starting mathematical research for the first time; It is therefore offered to these individuals as a topic for independent work, not of a compilative nature.
Для индивидуального определения коэффициентов полученного уравнения (см. статью) полученное выражение тождественно с тем, которое получается примененим метода Kitz’a, а потому, в силу результата нашей выше цитированной статьи, можем утверждать, что процесс Boussinesq'a в рассматриваемом частность случай дает точное решение задачи. В общем случай вопрос остается открытым, хотя некоторые соображения заставляют предполагать, что рассмотренный выше частный случай не будет единственным случаем „сходимости" процесса Boussinesq’a.
In various mathematical studies related to the question of the existence of a minimum, it is often useful, as we will try to show below, the application of that basic relationship in the theory of trigonometric series, which was named by Prof. V. A. Steklov as a closed equation and generalized by him for many other systems of orthogonal functions found in analysis and mathematical physics.
T. J. Stieltjes was one of the first mathematicians who posed from a very general point of view the question of the convergence of formulas of the so-called mechanical quadratures. Recently, a very simple and elegant proof of Stieltjes' result for the case of continuous functions was given by J. V. Uspenskii, who uses Weierstrass's main theorem regarding the approximation of continuous functions using polynomials. However, in view of the importance of the question and some of the consequences that follow from this, it seems not uninteresting, in our opinion, to interpret the question without the help of the above-mentioned Weierstrass theorem and without a preliminary study, a quite complex one, which will be the subject of the first paragraph of this work.
The question of the existence of so-called “fundamental functions” for differential equations of higher orders has been the subject of research in a number of works, but further processing is undoubtedly possible in the sense of applying various methods to its solution. This article represents an attempt to generalize the method of the American geometer Max Mason, which he outlined for 2nd order differential equations, to the case of 4th order differential equations, which, as is known, leads to the question of vibrations of an elastic inhomogeneous rod. Kolganovka August 6-8, 1915.
Integrable functions also include functions whose discontinuities can be contained in intervals, and the sum of the extension of the latter is an arbitrarily small value; in fact, by enclosing the discontinuity points in intervals, we can represent the difference (b - a) as the sum of two parts of which the first refers to intervals that, in turn, lie inside the intervals enclosing the discontinuity points (see article). It is enough to show that the difference (b - a) can be made arbitrarily small only under one, well-defined division law, then we can take division points that coincide with the ends of the intervals containing discontinuities in the function f and then the difference (b - a) will actually be as samll as you can imagine. From here we conclude that integrable functions include, among others, functions that have a finite number of discontinuities, as well as functions whose discontinuity points, being infinite in number, have a finite number of so-called limit points.
A very simple proof was given by Prof. V.A. Steklov, expressed by the formula, but ше still differs significantly from various proofs of the same theorem given by prof. V. A. Steklov and, rather in terms of ideas, approaches the proof of Prof. Hurwitz (based on the Cesaro-Fejer method of ariemetic averages), having, it seems to me, the advantage of greater simplicity, since it is based on the application of the theorem on term-by-term integration of trigonometric series, which itself is a special case of the closedness theorem.
The solution to one of the main problems of mathematical physics, namely the Dirichlet problem for a sphere, is reduced, as is known, to the question of expanding the so-called “arbitrary” function of two angles into a series arranged according to the spherical Laplace functions. The possibility of expansion for a function that has two first derivatives has been proven and reasoning similar to that given in our article: “On the theory of trigonometric series”. It is possible to establish the possibility of expansion for a function that satisfies Lipchitz’s condition.
When expanding “arbitrary” functions into series using the method of least squares, the coefficients of the series, as is known, are formed according to a very specific law, namely, the law of formation of the coefficient is the one that would take place in the case of uniform convergence of the series, i.e., in other words, the coefficients acquire the form of the so-called Fourier coefficients and the series will be the so-called Fourier series of the expandable functions, in the special case of expansion in trigonometric functions of the form (see article).
In this short essay, devoted primarily to the scientific and pedagogical activities of the late professor, I would first of all like to share with readers the memories of the unforgettable, indelible impression that was made on me, as probably on almost all of his former listeners, in his time lectures.
