Trigonometric Fourier series. trigonometric series. Fourier series. Application of the Finite Difference Method

In a number of cases, by investigating the coefficients of series of the form (C) or it can be established that these series converge (perhaps excepting individual points) and are Fourier series for their sums (see, for example, the previous n°), but in all these cases, the question naturally arises

how to find the sums of these series or, more precisely, how to express them in the final form in terms of elementary functions, if they are expressed in such a form at all. Even Euler (and also Lagrange) successfully used analytic functions of a complex variable to sum up trigonometric series in a final form. The idea behind the Euler method is as follows.

Let us assume that, for a certain set of coefficients, the series (C) and converge to functions everywhere in the interval, excluding only individual points. Consider now a power series with the same coefficients, arranged in powers of a complex variable

On the circumference of the unit circle, i.e., at , this series converges by assumption, excluding individual points:

In this case, according to the well-known property of power series, series (5) certainly converges at ie inside the unit circle, defining there a certain function of a complex variable. Using known to us [see. § 5 of Chapter XII] of the expansion of elementary functions of a complex variable, it is often possible to reduce the function to them. Then for we have:

and by the Abel theorem, as soon as the series (6) converges, its sum is obtained as a limit

Usually this limit is simply equal to which allows us to calculate the function in the final form

Let, for example, the series

The statements proved in the previous paragraph lead to the conclusion that both these series converge (the first one, excluding the points 0 and

serve as Fourier series for the functions they define. But what are these functions? To answer this question, we make a series

By similarity with the logarithmic series, its sum is easily established:

Consequently,

Now an easy calculation gives:

so the modulus of this expression is , and the argument is .

and thus finally

These results are familiar to us and were even once obtained with the help of "complex" considerations; but in the first case, we started from the functions and , and in the second - from the analytic function. Here, for the first time, the series themselves served as a starting point. The reader will find further examples of this kind in the next section.

We emphasize once again that one must be sure in advance of the convergence and series (C) and in order to have the right to determine their sums using the limiting equality (7). The mere existence of a limit on the right-hand side of this equality does not yet allow us to conclude that the mentioned series converge. To show this with an example, consider the series

Standard methods, but reached a dead end with another example.

What is the difficulty and where can there be a snag? Let's put aside the soapy rope, calmly analyze the reasons and get acquainted with the practical methods of solution.

First and most important: in the vast majority of cases, to study the convergence of a series, it is necessary to apply some familiar method, but the common term of the series is filled with such tricky stuffing that it is not at all obvious what to do with it. And you go around in circles: the first sign does not work, the second does not work, the third, fourth, fifth method does not work, then the drafts are thrown aside and everything starts anew. This is usually due to a lack of experience or gaps in other sections of calculus. In particular, if running sequence limits and superficially disassembled function limits, then it will be difficult.

In other words, a person simply does not see the necessary solution due to a lack of knowledge or experience.

Sometimes “eclipse” is also to blame, when, for example, the necessary criterion for the convergence of a series is simply not fulfilled, but due to ignorance, inattention or negligence, this falls out of sight. And it turns out like in that bike where the professor of mathematics solved a children's problem with the help of wild recurrent sequences and number series =)

In the best traditions, immediately living examples: rows and their relatives - diverge, since in theory it is proved sequence limits. Most likely, in the first semester you will be beaten out of your soul for a proof of 1-2-3 pages, but for now it is quite enough to show that the necessary condition for the convergence of the series is not met, referring to known facts. Famous? If the student does not know that the root of the nth degree is an extremely powerful thing, then, say, the series put him in a rut. Although the solution is like two and two: , i.e. for obvious reasons, both series diverge. A modest comment “these limits have been proven in theory” (or even its absence at all) is quite enough for offset, after all, the calculations are quite heavy and they definitely do not belong to the section of numerical series.

And after studying the next examples, you will only be surprised at the brevity and transparency of many solutions:

Example 1

Investigate the convergence of a series

Solution: first of all, check the execution necessary criterion for convergence. This is not a formality, but a great chance to deal with the example of "little bloodshed".

"Inspection of the scene" suggests a divergent series (the case of a generalized harmonic series), but again the question arises, how to take into account the logarithm in the numerator?

Approximate examples of tasks at the end of the lesson.

It is not uncommon when you have to carry out a two-way (or even three-way) reasoning:

Example 6

Investigate the convergence of a series

Solution: first, carefully deal with the gibberish of the numerator. The sequence is limited: . Then:

Let's compare our series with the series . By virtue of the double inequality just obtained, for all "en" it will be true:

Now let's compare the series with the divergent harmonic series.

Fraction denominator less the denominator of the fraction, so the fraction itself – more fractions (write down the first few terms, if not clear). Thus, for any "en":

So, by comparison, the series diverges along with the harmonic series.

If we change the denominator a little: , then the first part of the reasoning will be similar: . But to prove the divergence of the series, only the limit test of comparison is already applicable, since the inequality is false.

The situation with converging series is “mirror”, that is, for example, both comparison criteria can be used for a series (the inequality is true), and for a series, only the limiting criterion (the inequality is false).

We continue our safari through the wild, where a herd of graceful and succulent antelopes loomed on the horizon:

Example 7

Investigate the convergence of a series

Solution: the necessary convergence criterion is satisfied, and we again ask the classic question: what to do? Before us is something resembling a convergent series, however, there is no clear rule here - such associations are often deceptive.

Often, but not this time. By using Limit comparison criterion Let's compare our series with the convergent series . When calculating the limit, we use wonderful limit , where as infinitesimal stands:

converges together with next to .

Instead of using the standard artificial technique of multiplication and division by a "three", it was possible to initially compare with a convergent series.
But here a caveat is desirable that the constant-multiplier of the general term does not affect the convergence of the series. And just in this style the solution of the following example is designed:

Example 8

Investigate the convergence of a series

Sample at the end of the lesson.

Example 9

Investigate the convergence of a series

Solution: in the previous examples, we used the boundedness of the sine, but now this property is out of play. The denominator of a fraction of a higher order of growth than the numerator, so when the sine argument and the entire common term infinitely small. The necessary condition for convergence, as you understand, is satisfied, which does not allow us to shirk from work.

We will conduct reconnaissance: in accordance with remarkable equivalence , mentally discard the sine and get a series. Well, something like that….

Making a decision:

Let us compare the series under study with the divergent series . We use the limit comparison criterion:

Let us replace the infinitesimal with the equivalent one: for .

A finite number other than zero is obtained, which means that the series under study diverges along with the harmonic series.

Example 10

Investigate the convergence of a series

This is a do-it-yourself example.

For planning further actions in such examples, the mental rejection of the sine, arcsine, tangent, arctangent helps a lot. But remember, this possibility exists only when infinitesimal argument, not so long ago I came across a provocative series:

Example 11

Investigate the convergence of a series
.

Solution: it is useless to use the limitedness of the arc tangent here, and the equivalence does not work either. The output is surprisingly simple:


Study Series diverges, since the necessary criterion for the convergence of the series is not satisfied.

The second reason"Gag on the job" consists in a decent sophistication of the common member, which causes difficulties of a technical nature. Roughly speaking, if the series discussed above belong to the category of “figures you guess”, then these ones belong to the category of “you decide”. Actually, this is called complexity in the "usual" sense. Not everyone will correctly resolve several factorials, degrees, roots and other inhabitants of the savannah. Of course, factorials cause the most problems:

Example 12

Investigate the convergence of a series

How to raise a factorial to a power? Easily. According to the rule of operations with powers, it is necessary to raise each factor of the product to a power:

And, of course, attention and once again attention, the d'Alembert sign itself works traditionally:

Thus, the series under study converges.

I remind you of a rational technique for eliminating uncertainty: when it is clear order of growth numerator and denominator - it is not at all necessary to suffer and open the brackets.

Example 13

Investigate the convergence of a series

The beast is very rare, but it is found, and it would be unfair to bypass it with a camera lens.

What is double exclamation point factorial? The factorial "winds" the product of positive even numbers:

Similarly, the factorial “winds up” the product of positive odd numbers:

Analyze what is the difference between

Example 14

Investigate the convergence of a series

And in this task, try not to get confused with the degrees, wonderful equivalences and wonderful limits.

Sample solutions and answers at the end of the lesson.

But the student gets to feed not only tigers - cunning leopards also track down their prey:

Example 15

Investigate the convergence of a series

Solution: the necessary criterion of convergence, the limiting criterion, the d'Alembert and Cauchy criteria disappear almost instantly. But worst of all, the feature with inequalities, which has repeatedly rescued us, is powerless. Indeed, comparison with a divergent series is impossible, since the inequality incorrect - the multiplier-logarithm only increases the denominator, reducing the fraction itself in relation to the fraction. And another global question: why are we initially sure that our series is bound to diverge and must be compared with some divergent series? Does he fit in at all?

Integral feature? Improper integral evokes a mournful mood. Now, if we had a row … then yes. Stop! This is how ideas are born. We make a decision in two steps:

1) First, we study the convergence of the series . We use integral feature:

Integrand continuous on the

Thus, a number diverges together with the corresponding improper integral.

2) Compare our series with the divergent series . We use the limit comparison criterion:

A finite number other than zero is obtained, which means that the series under study diverges along with side by side .

And there is nothing unusual or creative in such a decision - that's how it should be decided!

I propose to independently draw up the following two-move:

Example 16

Investigate the convergence of a series

A student with some experience in most cases immediately sees whether the series converges or diverges, but it happens that a predator cleverly disguises itself in the bushes:

Example 17

Investigate the convergence of a series

Solution: at first glance, it is not at all clear how this series behaves. And if we have fog in front of us, then it is logical to start with a rough check of the necessary condition for the convergence of the series. In order to eliminate uncertainty, we use an unsinkable multiplication and division method by adjoint expression:

The necessary sign of convergence did not work, but brought our Tambov comrade to light. As a result of the performed transformations, an equivalent series was obtained , which in turn strongly resembles a convergent series .

We write a clean solution:

Compare this series with the convergent series . We use the limit comparison criterion:

Multiply and divide by the adjoint expression:

A finite number other than zero is obtained, which means that the series under study converges together with next to .

Perhaps some have a question, where did the wolves come from on our African safari? Don't know. They probably brought it. You will get the following trophy skin:

Example 18

Investigate the convergence of a series

An example solution at the end of the lesson

And, finally, one more thought that visits many students in despair: instead of whether to use a rarer criterion for the convergence of the series? Sign of Raabe, sign of Abel, sign of Gauss, sign of Dirichlet and other unknown animals. The idea is working, but in real examples it is implemented very rarely. Personally, in all the years of practice, I have only 2-3 times resorted to sign of Raabe when nothing really helped from the standard arsenal. I reproduce the course of my extreme quest in full:

Example 19

Investigate the convergence of a series

Solution: Without any doubt a sign of d'Alembert. In the course of calculations, I actively use the properties of degrees, as well as second wonderful limit:

Here's one for you. D'Alembert's sign did not give an answer, although nothing foreshadowed such an outcome.

After going through the manual, I found a little-known limit proven in theory and applied a stronger radical Cauchy criterion:

Here's two for you. And, most importantly, it is not at all clear whether the series converges or diverges (an extremely rare situation for me). Necessary sign of comparison? Without much hope - even if in an unthinkable way I figure out the order of growth of the numerator and denominator, this still does not guarantee a reward.

A complete d'Alembert, but the worst thing is that the series needs to be solved. Need. After all, this will be the first time that I give up. And then I remembered that there seemed to be some more powerful signs. Before me was no longer a wolf, not a leopard and not a tiger. It was a huge elephant waving a big trunk. I had to pick up a grenade launcher:

Sign of Raabe

Consider a positive number series.
If there is a limit , then:
a) At a row diverges. Moreover, the resulting value can be zero or negative.
b) At a row converges. In particular, the series converges for .
c) When Raabe's sign does not give an answer.

We compose the limit and carefully simplify the fraction:


Yes, the picture is, to put it mildly, unpleasant, but I was no longer surprised. lopital rules, and the first thought, as it turned out later, turned out to be correct. But first, for about an hour, I twisted and turned the limit using “usual” methods, but the uncertainty did not want to be eliminated. And walking in circles, as experience suggests, is a typical sign that the wrong way of solving has been chosen.

I had to turn to Russian folk wisdom: "If nothing helps, read the instructions." And when I opened the 2nd volume of Fichtenholtz, to my great joy I found a study of an identical series. And then the solution went according to the model.

The Navier solution is suitable only for the calculation of plates hinged along the contour. More general is Levy's solution. It allows you to calculate a plate hinged on two parallel sides, with arbitrary boundary conditions on each of the other two sides.

In the rectangular plate shown in Fig. 5.11, (a), hinged edges are those parallel to the axis y. The boundary conditions at these edges have the form

Rice. 5.11

It is obvious that each term of the infinite trigonometric series

https://pandia.ru/text/78/068/images/image004_89.gif" width="99" height="49">; second partial derivatives of the deflection function

(5.45)

at x = 0 and x = a are also zero because they contain https://pandia.ru/text/78/068/images/image006_60.gif" width="279" height="201 src="> (5.46)

Substituting (5.46) into (5.18) gives

Multiplying both sides of the resulting equation by , integrating from 0 to a and remembering that

,

we get to define the function Ym such a linear differential equation with constant coefficients

. (5.48)

If, to shorten the notation, denote

equation (5.48) takes the form

. (5.50)

The general solution of the inhomogeneous equation (5.50), as is known from the course of differential equations, has the form

Ym(y) = jm (y)+ fm(y), (5.51)

where jm (y) is a particular solution of the inhomogeneous equation (5.50); its form depends on the right side of equation (5.50), i.e., in fact, on the type of load q (x, y);

fm(y)= Am shamy + Bmchamy+y(cm shamy + Dmchamy), (5.52)

general solution of the homogeneous equation

Four arbitrary constants Am,ATm ,Cm and Dm must be determined from the four conditions for fixing the edges of the plate, parallel to the axis , applied to the plate constant q (x, y) = q the right side of equation (5.50) takes the form

https://pandia.ru/text/78/068/images/image014_29.gif" width="324" height="55 src=">. (5.55)

Since the right side of the equation (5.55) is constant, its left side is also constant; so all derivatives jm (y) are zero, and

, (5.56)

, (5.57)

where indicated: .

Consider a plate pinched along edges parallel to the axis X(Fig. 5.11, (c)).

Boundary conditions at the edges y = ± b/2

. (5.59)

Due to the symmetry of the deflection of the plate about the axis Ox, in the general solution (5.52) only terms containing even functions should be retained. Because sh amy is an odd function, and сh am y- even and, with the adopted position of the axis Oh, y sh amy- even, in at ch am y is odd, then the general integral (5.51) in the case under consideration can be represented as

. (5.60)

Since in (5.44) does not depend on the value of the argument y, the second pair of boundary conditions (5.58), (5.59) can be written as:

Ym = 0, (5.61)

Y¢ m = = 0. (5.62)

Y¢ m = ambm sh amy + cm sh amy + y cmam ch amy=

ambm sh amy + cm(sh amy+yam ch amy)

From (5.60) - (5.63) it follows

https://pandia.ru/text/78/068/images/image025_20.gif" width="364" height="55 src=">. (5.65)

Multiplying equation (5.64) by , and equation (5..gif" width="191" height="79 src=">. (5.66)

Substituting (5.66) into equation (5.64) allows us to obtain bm

https://pandia.ru/text/78/068/images/image030_13.gif" width="511" height="103">. (5.68)

With this function expression Ym. , formula (5.44) for determining the deflection function takes the form

(5.69)

Series (5.69) converges quickly. For example, for a square plate in its center, i.e. at x=a/2, y = 0

(5.70)

Keeping in (5.70) only one term of the series, i.e., taking , we obtain a deflection value overestimated by less than 2.47%. Bearing in mind that p 5 = 306.02, find Variation" href="/text/category/variatciya/" rel="bookmark"> V..Ritz's variational method is based on Lagrange's variational principle formulated in Section 2.

Let us consider this method as applied to the problem of plate bending. Imagine the curved surface of the plate as a row

, (5.71)

where fi(x, y) continuous coordinate functions, each of which must satisfy kinematic boundary conditions; Ci are unknown parameters determined from the Lagrange equation. This equation

(5.72)

leads to a system of n algebraic equations with respect to parameters Ci.

In the general case, the deformation energy of the plate consists of bending U and membrane U m parts

, (5.73)

, (5.74)

where Mh.,My. ,Mxy– bending forces; NX., Ny. , Nxy– membrane forces. The part of the energy corresponding to the transverse forces is small and can be neglected.

If a u, v and w are the components of the actual displacement, px. , py and pz are the components of the surface load intensity, Ri- concentrated force, D i the corresponding linear displacement, Mj- focused moment qj- the angle of rotation corresponding to it (Fig. 5.12), then the potential energy of external forces can be represented as follows:

If the edges of the plate allow movement, then the edge forces vn. , mn. , mnt(Fig. 5.12, (a)) increase the potential of external forces

Rice. 5.12

Here n and t– normal and tangent to edge element ds.

In Cartesian coordinates, taking into account known expressions for forces and curvatures

, (5.78)

total potential energy E of a rectangular plate of size a ´ b, under the action of only vertical load pz

(5.79)

As an example, consider a rectangular plate with an aspect ratio of 2 a´ 2 b(Fig. 5.13).

The plate is clamped along the contour and loaded with a uniform load

pz = q = const. In this case, expression (5.79) for the energy E is simplified

. (5.80)

Accept for w(x, y) row

which satisfies the contour conditions

Rice. 5.13

Keep only the first member of the series

.

Then according to (5.80)

.

Minimizing the energy E according to (5..gif" width="273 height=57" height="57">.

.

Deflection of the center of a square plate size 2 a´ 2 a

,

which is 2.5% more than the exact solution 0.0202 qa 4/D. Note that the deflection of the center of the plate supported on four sides is 3.22 times greater.

This example illustrates the advantages of the method: simplicity and the possibility of obtaining a good result. The plate can have different outlines, variable thickness. Difficulties in this method, as, indeed, in other energy methods, arise when choosing suitable coordinate functions.

5.8. Orthogonalization method

The orthogonalization method proposed by and is based on the following property of orthogonal functions ji. , jj

. (5.82)

An example of orthogonal functions on the interval ( – p, p) can serve as trigonometric functions cos nx and sin nx for which

If one of the functions, for example the function ji (x) is identically equal to zero, then condition (5.82) is satisfied for an arbitrary function jj (x).

To solve the problem of plate bending, the equation is

can be imagined like this

, (5.83)

where F is the area bounded by the contour of the plate; jij are functions specified so that they satisfy the kinematic and force boundary conditions of the problem.

Let us represent the approximate solution of the plate bending equation (5.18) in the form of a series

. (5.84)

If solution (5.84) were exact, then equation (5.83) would hold identically for any system of coordinate functions jij. , because in this case D c2c2 wn – q = 0. We require that the equation D c2c2 wn – q was orthogonal to the family of functions jij, and we use this requirement to determine the coefficients Cij. . Substituting (5.84) into (5.83) we get

. (5.85)

After performing some transformations, we obtain the following system of algebraic equations for determining Cij

, (5.86)

and hij = hji.

The Bubnov-Galerkin method can be given the following interpretation. Function D c2c2 wn – q = 0 is essentially an equilibrium equation and is a projection of external and internal forces acting on a small element of the plate in the direction of the vertical axis z. Deflection function wn is a movement in the direction of the same axis, and the functions jij can be considered possible movements. Therefore, equation (5.83) approximately expresses the equality to zero of the work of all external and internal forces on possible displacements jij. . Thus, the Bubnov-Galerkin method is essentially variational.

As an example, consider a rectangular plate clamped along the contour and loaded with a uniformly distributed load. The dimensions of the plate and the location of the coordinate axes are the same as in Fig. 5.6.

Border conditions

at x = 0, x= a: w = 0, ,

at y = 0, y = b: w = 0, .

We choose an approximate expression for the deflection function in the form of a series (5.84) where the function jij

satisfies the boundary conditions; Cij are the desired coefficients. Limited to one member of the series

we get the following equation

After integration

Where can we calculate the coefficient FROM 11

,

which fully corresponds to the coefficient FROM 11. obtained by the method

V. Ritz -.

As a first approximation, the deflection function is as follows

.

Maximum deflection at the center of a square plate a ´ a

.

5.9. Application of the Finite Difference Method

Let us consider the application of the finite difference method for rectangular plates with complex contour conditions. The difference operator is an analogue of the differential equation of the curved surface of the plate (5.18), for a square grid, for D x = D y = D takes the form (3.54)

20 wi, j + 8 (wi, j+ 1 + wi, j– 1 + wi– 1, j + wi+ 1, j) + 2 (wi– 1, j– 1 + wi– 1, j+ 1 +

Rice. 5.14

Taking into account the presence of three axes of symmetry of loading and deformations of the plate, we can restrict ourselves to considering its eighth and determine the values ​​of deflections only at nodes 1 ... 10 (Fig. 5.14, (b)). On fig. 5.14, (b) shows the grid and node numbering (D = a/4).

Since the edges of the plate are pinched, then by writing the contour conditions (5.25), (5.26) in finite differences

By cosines and sines of multiple arcs, i.e. a series of the form

or in complex form

where a k,b k or, respectively, c k called coefficients of T. r.
For the first time T. r. meet at L. Euler (L. Euler, 1744). He got expansions

All R. 18th century In connection with the study of the problem of the free vibration of a string, the question arose of the possibility of representing the function characterizing the initial position of the string as a sum of T. r. This question caused a heated debate that lasted for several decades, the best analysts of that time - D. Bernoulli, J. D "Alembert, J. Lagrange, L. Euler ( L. Euler). Disputes related to the content of the concept of function. At that time, functions were usually associated with their analytics. assignment, which led to the consideration of only analytic or piecewise analytic functions. And here it became necessary for a function whose graph is sufficiently arbitrary to construct a T. r. representing this function. But the significance of these disputes is greater. In fact, they discussed or arose in connection with questions related to many fundamentally important concepts and ideas of mathematics. analysis in general - the representation of functions by Taylor series and analytical. continuation of functions, use of divergent series, limits, infinite systems of equations, functions by polynomials, etc.
And in the future, as in this initial one, the theory of T. r. served as a source of new ideas in mathematics. Fourier integral, almost periodic functions, general orthogonal series, abstract . Researches on T. river. served as a starting point for the creation of set theory. T. r. are a powerful tool for representing and exploring features.
The question that led to controversy among 18th-century mathematicians was resolved in 1807 by J. Fourier, who provided formulas for calculating the coefficients of T. R. (1), which must. represent on the function f(x):

and applied them in solving heat conduction problems. Formulas (2) are called Fourier formulas, although they were encountered earlier by A. Clairaut (1754), and L. Euler (1777) came to them using term-by-term integration. T. r. (1), the coefficients of which are determined by formulas (2), called. near the Fourier function f, and the numbers a k , b k- Fourier coefficients.
The nature of the results obtained depends on how the representation of a function is understood as a series, how the integral in formulas (2) is understood. Modern theory of T. river. acquired after the appearance of the Lebesgue integral.
The theory of T. r. can be conditionally divided into two large sections - the theory Fourier series, in which it is assumed that the series (1) is the Fourier series of a certain function, and the theory of general T. R., where such an assumption is not made. Below are the main results obtained in the theory of general T. r. (in this case, sets and the measurability of functions are understood according to Lebesgue).
The first systematic research T. r., in which it was not assumed that these series are Fourier series, was the dissertation of V. Riemann (V. Riemann, 1853). Therefore, the theory of general T. r. called sometimes the Riemannian theory of thermodynamics.
To study the properties of arbitrary T. r. (1) with coefficients tending to zero B. Riemann considered the continuous function F(x) , which is the sum of a uniformly convergent series

obtained after two-fold term-by-term integration of series (1). If the series (1) converges at some point x to a number s, then at this point the second symmetric exists and is equal to s. F functions:

then this leads to the summation of the series (1) generated by the factors called by the Riemann summation method. Using the function F, the Riemann localization principle is formulated, according to which the behavior of the series (1) at the point x depends only on the behavior of the function F in an arbitrarily small neighborhood of this point.
If T. r. converges on a set of positive measure, then its coefficients tend to zero (Cantor-Lebesgue). Tendency to zero coefficients T. r. also follows from its convergence on a set of the second category (W. Young, W. Young, 1909).
One of the central problems of the theory of general thermodynamics is the problem of representing an arbitrary function T. r. Strengthening the results of N. N. Luzin (1915) on the representation of T. R. functions by Abel-Poisson and Riemann summable methods, D. E. Men'shov proved (1940) the following theorem, which refers to the most important case when the representation of the function f is understood as T. r. to f(x) almost everywhere. For every measurable and finite almost everywhere function f, there exists a T. R. that converges to it almost everywhere (Men'shov's theorem). It should be noted that even if f is integrable, then, generally speaking, one cannot take the Fourier series of the function f as such a series, since there are Fourier series that diverge everywhere.
The above Men'shov theorem admits the following refinement: if a function f is measurable and finite almost everywhere, then there exists such that almost everywhere and the term-by-term differentiated Fourier series of the function j converges to f(x) almost everywhere (N. K. Bari, 1952).
It is not known (1984) whether it is possible to omit the finiteness condition for the function f almost everywhere in Men'shov's theorem. In particular, it is not known (1984) whether T. r. converge almost everywhere
Therefore, the problem of representing functions that can take on infinite values ​​on a set of positive measure was considered for the case when it is replaced by the weaker requirement - . Convergence in measure to functions that can take on infinite values ​​is defined as follows: partial sums of T. p. s n(x) converges in measure to the function f(x) . if where f n(x) converge to / (x) almost everywhere, and the sequence converges to zero in measure. In this setting, the problem of representation of functions has been solved to the end: for every measurable function, there exists a T. R. that converges to it in measure (D. E. Men'shov, 1948).
Much research has been devoted to the problem of the uniqueness of T. r.: Can two different T. diverge to the same function? in a different formulation: if T. r. converges to zero, does it follow that all the coefficients of the series are equal to zero. Here one can mean convergence at all points or at all points outside a certain set. The answer to these questions essentially depends on the properties of the set outside of which convergence is not assumed.
The following terminology has been established. Many names. uniqueness set or U- set if, from the convergence of T. r. to zero everywhere, except, perhaps, for points of the set E, it follows that all the coefficients of this series are equal to zero. Otherwise Enaz. M-set.
As G. Cantor (1872) showed, as well as any finite are U-sets. An arbitrary is also a U-set (W. Jung, 1909). On the other hand, every set of positive measure is an M-set.
The existence of M-sets of measure was established by D. E. Men'shov (1916), who constructed the first example of a perfect set with these properties. This result is of fundamental importance in the problem of uniqueness. It follows from the existence of M-sets of measure zero that, in the representation of functions of T. R. that converge almost everywhere, these series are defined invariably ambiguously.
Perfect sets can also be U-sets (N. K. Bari; A. Rajchman, A. Rajchman, 1921). Very subtle characteristics of sets of measure zero play an essential role in the problem of uniqueness. The general question about the classification of sets of measure zero on M- and U-sets remains (1984) open. It is not solved even for perfect sets.
The following problem is related to the uniqueness problem. If T. r. converges to the function then whether this series must be the Fourier series of the function /. P. Dubois-Reymond (P. Du Bois-Reymond, 1877) gave a positive answer to this question if f is integrable in the sense of Riemann and the series converges to f(x) at all points. From results III. J. Vallee Poussin (Ch. J. La Vallee Poussin, 1912) implies that the answer is positive even if the series converges everywhere except for a countable set of points and its sum is finite.
If a T. p converges absolutely at some point x 0, then the points of convergence of this series, as well as the points of its absolute convergence, are located symmetrically with respect to the point x 0 (P. Fatou, P. Fatou, 1906).
According to Denjoy - Luzin theorem from the absolute convergence of T. r. (1) on a set of positive measure, the series converges and, consequently, the absolute convergence of series (1) for all X. This property is also possessed by sets of the second category, as well as by certain sets of measure zero.
This survey covers only one-dimensional T. r. (one). There are separate results related to general T. p. from several variables. Here in many cases it is still necessary to find natural problem statements.

Lit.: Bari N. K., Trigonometric series, M., 1961; Sigmund A., Trigonometric series, trans. from English, vol. 1-2, M., 1965; Luzin N. N., Integral and trigonometric series, M.-L., 1951; Riemann B., Works, trans. from German, M.-L., 1948, p. 225-61.
S. A. Telyakovsky.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

In science and technology, one often has to deal with periodic phenomena, i.e. those that are reproduced after a certain period of time T called the period. The simplest of the periodic functions (except for a constant) is a sinusoidal value: asin(x+ ), harmonic oscillation, where there is a “frequency” related to the period by the ratio: . From such simple periodic functions, more complex ones can be composed. Obviously, the constituent sinusoidal quantities must be of different frequencies, since the addition of sinusoidal quantities of the same frequency results in a sinusoidal quantity of the same frequency. If we add several values ​​of the form

For example, we reproduce here the addition of three sinusoidal quantities: . Consider the graph of this function

This graph is significantly different from a sine wave. This is even more true for the sum of an infinite series composed of terms of this type. Let us pose the question: is it possible for a given periodic function of the period T represent as the sum of a finite or at least an infinite set of sinusoidal quantities? It turns out that with respect to a large class of functions, this question can be answered in the affirmative, but this is only if we include precisely the entire infinite sequence of such terms. Geometrically, this means that the graph of a periodic function is obtained by superimposing a series of sinusoids. If we consider each sinusoidal value as a certain harmonic oscillatory movement, then we can say that this is a complex oscillation characterized by a function or simply by its harmonics (first, second, etc.). The process of decomposition of a periodic function into harmonics is called harmonic analysis.

It is important to note that such expansions often turn out to be useful in the study of functions that are given only in a certain finite interval and are not generated at all by any oscillatory phenomena.

Definition. A trigonometric series is a series of the form:

Or (1).

The real numbers are called the coefficients of the trigonometric series. This series can also be written like this:

If a series of the type presented above converges, then its sum is a periodic function with period 2p.

Definition. The Fourier coefficients of a trigonometric series are called: (2)

(3)

(4)

Definition. Near Fourier for a function f(x) is called a trigonometric series whose coefficients are the Fourier coefficients.

If the Fourier series of the function f(x) converges to it at all its points of continuity, then we say that the function f(x) expands in a Fourier series.

Theorem.(Dirichlet's theorem) If a function has a period of 2p and is continuous on a segment or has a finite number of discontinuity points of the first kind, the segment can be divided into a finite number of segments so that the function is monotonic inside each of them, then the Fourier series for the function converges for all values X, and at the points of continuity of the function, its sum S(x) is equal to , and at the discontinuity points its sum is equal to , i.e. the arithmetic mean of the limit values ​​on the left and right.

In this case, the Fourier series of the function f(x) converges uniformly on any interval that belongs to the interval of continuity of the function .

A function that satisfies the conditions of this theorem is called piecewise smooth on the interval .

Let's consider examples on the expansion of a function in a Fourier series.

Example 1. Expand the function in a Fourier series f(x)=1-x, which has a period 2p and given on the segment .

Solution. Let's plot this function

This function is continuous on the segment , that is, on a segment with a length of a period, therefore it can be expanded into a Fourier series that converges to it at each point of this segment. Using formula (2), we find the coefficient of this series: .

We apply the integration-by-parts formula and find and using formulas (3) and (4), respectively:

Substituting the coefficients into formula (1), we obtain or .

This equality takes place at all points, except for the points and (gluing points of the graphs). At each of these points, the sum of the series is equal to the arithmetic mean of its limit values ​​on the right and left, that is.

Let us present an algorithm for expanding the function in a Fourier series.

The general procedure for solving the problem posed is as follows.