Which are already pretty fed up. And I feel that the moment has come when it is time to extract new canned food from the strategic reserves of theory. Is it possible to expand the function into a series in some other way? For example, to express a straight line segment in terms of sines and cosines? It seems incredible, but such seemingly distant functions lend themselves to
"reunion". In addition to the familiar degrees in theory and practice, there are other approaches to expanding a function into a series.
In this lesson, we will get acquainted with the trigonometric Fourier series, touch on the issue of its convergence and sum, and, of course, we will analyze numerous examples for expanding functions into a Fourier series. I sincerely wanted to call the article “Fourier Series for Dummies”, but this would be cunning, since solving problems will require knowledge of other sections of mathematical analysis and some practical experience. Therefore, the preamble will resemble the training of astronauts =)
First, the study of the page materials should be approached in excellent shape. Sleepy, rested and sober. Without strong emotions about the broken paw of a hamster and obsessive thoughts about the hardships of the life of aquarium fish. The Fourier series is not difficult from the point of view of understanding, however, practical tasks simply require an increased concentration of attention - ideally, one should completely abandon external stimuli. The situation is aggravated by the fact that there is no easy way to check the solution and the answer. Thus, if your health is below average, then it is better to do something simpler. Truth.
Secondly, before flying into space, it is necessary to study the instrument panel of the spacecraft. Let's start with the values of the functions that should be clicked on the machine:
For any natural value:
one) . And in fact, the sinusoid "flashes" the x-axis through each "pi":
. In the case of negative values of the argument, the result, of course, will be the same: .
2). But not everyone knew this. The cosine "pi en" is the equivalent of a "flashing light":
A negative argument does not change the case: 
.
Perhaps enough.
And thirdly, dear cosmonaut corps, you need to be able to ... integrate.
In particular, sure bring a function under a differential sign, integrate by parts and be on good terms with Newton-Leibniz formula. Let's start the important pre-flight exercises. I strongly do not recommend skipping it, so that later you don’t flatten in zero gravity:
Example 1
Calculate definite integrals
where takes natural values.
Decision: integration is carried out over the variable "x" and at this stage the discrete variable "en" is considered a constant. In all integrals bring the function under the sign of the differential:
A short version of the solution, which would be good to shoot at, looks like this:
Getting used to:
The four remaining points are on their own. Try to treat the task conscientiously and arrange the integrals in a short way. Sample solutions at the end of the lesson.
After a QUALITY exercise, we put on spacesuits
and getting ready to start!
Expansion of a function in a Fourier series on the interval
Let's consider a function that determined at least on the interval (and, possibly, on a larger interval). If this function is integrable on the segment , then it can be expanded into a trigonometric Fourier series:
, where are the so-called Fourier coefficients.
In this case, the number is called decomposition period, and the number is half-life decomposition.
Obviously, in the general case, the Fourier series consists of sines and cosines: 
Indeed, let's write it in detail:
The zero term of the series is usually written as .
Fourier coefficients are calculated using the following formulas: 
I understand perfectly well that new terms are still obscure for beginners to study the topic: decomposition period, half cycle, Fourier coefficients and others. Don't panic, it's not comparable to the excitement before a spacewalk. Let's figure everything out in the nearest example, before executing which it is logical to ask pressing practical questions:
What do you need to do in the following tasks?
Expand the function into a Fourier series. Additionally, it is often required to draw a graph of a function, a graph of the sum of a series, a partial sum, and in the case of sophisticated professorial fantasies, do something else.
How to expand a function into a Fourier series?
Essentially, you need to find Fourier coefficients, that is, compose and compute three definite integrals.
Please copy the general form of the Fourier series and the three working formulas in your notebook. I am very glad that some of the site visitors have a childhood dream of becoming an astronaut coming true right in front of my eyes =)
Example 2
Expand the function into a Fourier series on the interval . Build a graph, a graph of the sum of a series and a partial sum.
Decision: the first part of the task is to expand the function into a Fourier series.
The beginning is standard, be sure to write down that:
In this problem, the expansion period , half-period .
We expand the function in a Fourier series on the interval: 
Using the appropriate formulas, we find Fourier coefficients. Now we need to compose and calculate three definite integrals. For convenience, I will number the points:
1) The first integral is the simplest, however, it already requires an eye and an eye: 
2) We use the second formula:
This integral is well known and he takes it piecemeal: 
When found used method of bringing a function under a differential sign.
In the task under consideration, it is more convenient to immediately use formula for integration by parts in a definite integral 
:
A couple of technical notes. First, after applying the formula the entire expression must be enclosed in large brackets, since there is a constant in front of the original integral. Let's not lose it! Parentheses can be opened at any further step, I did it at the very last turn. In the first "piece" 
we show extreme accuracy in substitution, as you can see, the constant is out of business, and the limits of integration are substituted into the product. This action is marked with square brackets. Well, the integral of the second "piece" of the formula is well known to you from the training task ;-)
And most importantly - the ultimate concentration of attention!
3) We are looking for the third Fourier coefficient:
A relative of the previous integral is obtained, which is also integrated by parts:
This instance is a little more complicated, I will comment out the further steps step by step: 
(1) The entire expression is enclosed in large brackets.. I did not want to seem like a bore, they lose the constant too often.
(2) In this case, I immediately expanded those big brackets. Special attention we devote to the first “piece”: the constant smokes on the sidelines and does not participate in substituting the limits of integration ( and ) into the product . In view of the clutter of the record, it is again advisable to highlight this action in square brackets. With the second "piece" 
everything is simpler: here the fraction appeared after opening large brackets, and the constant - as a result of integrating the familiar integral ;-)
(3) In square brackets, we carry out transformations, and in the right integral, we substitute the limits of integration.
(4) We take out the “flasher” from the square brackets: , after which we open the inner brackets: .
(5) We cancel 1 and -1 in parentheses, we make final simplifications.
Finally found all three Fourier coefficients: 
Substitute them into the formula 
:
Don't forget to split in half. At the last step, the constant ("minus two"), which does not depend on "en", is taken out of the sum.
Thus, we have obtained the expansion of the function in a Fourier series on the interval : 
Let us study the question of the convergence of the Fourier series. I will explain the theory in particular Dirichlet theorem, literally "on the fingers", so if you need strict formulations, please refer to a textbook on calculus (for example, the 2nd volume of Bohan; or the 3rd volume of Fichtenholtz, but it is more difficult in it).
In the second part of the task, it is required to draw a graph, a series sum graph and a partial sum graph.
The graph of the function is the usual straight line on the plane, which is drawn with a black dotted line: 
We deal with the sum of the series. As you know, functional series converge to functions. In our case, the constructed Fourier series 
for any value of "x" converges to the function shown in red. This function is subject to breaks of the 1st kind in points , but also defined in them (red dots in the drawing)
Thus: 
. It is easy to see that it differs markedly from the original function , which is why in the notation 
a tilde is used instead of an equals sign.
Let us study an algorithm by which it is convenient to construct the sum of a series.
On the central interval, the Fourier series converges to the function itself (the central red segment coincides with the black dotted line of the linear function).
Now let's talk a little about the nature of the considered trigonometric expansion. Fourier series 
includes only periodic functions (constant, sines and cosines), so the sum of the series 
is also a periodic function.
What does this mean in our particular example? And this means that the sum of the series 
–necessarily periodic and the red segment of the interval must be infinitely repeated on the left and right.
I think that now the meaning of the phrase "period of decomposition" has finally become clear. Simply put, every time the situation repeats itself again and again.
In practice, it is usually sufficient to depict three decomposition periods, as is done in the drawing. Well, and more "stumps" of neighboring periods - to make it clear that the chart continues.
Of particular interest are discontinuity points of the 1st kind. At such points, the Fourier series converges to isolated values, which are located exactly in the middle of the discontinuity "jump" (red dots in the drawing). How to find the ordinate of these points? First, let's find the ordinate of the "upper floor": for this, we calculate the value of the function at the rightmost point of the central expansion period: . To calculate the ordinate of the “lower floor”, the easiest way is to take the leftmost value of the same period: 
. The ordinate of the mean value is the arithmetic mean of the sum of the "top and bottom": . Nice is the fact that when building a drawing, you will immediately see whether the middle is correctly or incorrectly calculated.
Let us construct a partial sum of the series and at the same time repeat the meaning of the term "convergence". The motive is known from the lesson about the sum of the number series. Let's describe our wealth in detail:
To make a partial sum, you need to write down zero + two more terms of the series. I.e,
In the drawing, the graph of the function is shown in green, and, as you can see, it wraps around the total sum quite tightly. If we consider a partial sum of five terms of the series, then the graph of this function will approximate the red lines even more accurately, if there are a hundred terms, then the “green serpent” will actually completely merge with the red segments, etc. Thus, the Fourier series converges to its sum.
It is interesting to note that any partial sum is continuous function, but the total sum of the series is still discontinuous.
In practice, it is not uncommon to build a partial sum graph. How to do it? In our case, it is necessary to consider the function on the segment, calculate its values at the ends of the segment and at intermediate points (the more points you consider, the more accurate the graph will be). Then you should mark these points on the drawing and carefully draw a graph on the period, and then “replicate” it into adjacent intervals. How else? After all, approximation is also a periodic function ... ... its graph somehow reminds me of an even heart rhythm on the display of a medical device.
Of course, it is not very convenient to carry out the construction, since you have to be extremely careful, maintaining an accuracy of no less than half a millimeter. However, I will please readers who are at odds with drawing - in a "real" task, it is far from always necessary to perform a drawing, somewhere in 50% of cases it is required to expand the function into a Fourier series and that's it.
After completing the drawing, we complete the task:
Answer: 
In many tasks, the function suffers rupture of the 1st kind right on the decomposition period:
Example 3
Expand in a Fourier series the function given on the interval . Draw a graph of the function and the total sum of the series.
The proposed function is given piecewise (and, mind you, only on the segment) and endure rupture of the 1st kind at point . Is it possible to calculate the Fourier coefficients? No problem. Both the left and right parts of the function are integrable on their intervals, so the integrals in each of the three formulas should be represented as the sum of two integrals. Let's see, for example, how this is done for a zero coefficient:
The second integral turned out to be equal to zero, which reduced the work, but this is not always the case.
Two other Fourier coefficients are written similarly.
How to display the sum of a series? On the left interval we draw a straight line segment , and on the interval - a straight line segment (highlight the axis section in bold-bold). That is, on the expansion interval, the sum of the series coincides with the function everywhere, except for three "bad" points. At the discontinuity point of the function, the Fourier series converges to an isolated value, which is located exactly in the middle of the “jump” of the discontinuity. It is not difficult to see it orally: left-hand limit:, right-hand limit: 
and, obviously, the ordinate of the midpoint is 0.5.
Due to the periodicity of the sum , the picture must be “multiplied” into neighboring periods, in particular, depict the same thing on the intervals and . In this case, at the points, the Fourier series converges to the median values.
In fact, there is nothing new here.
Try to solve this problem on your own. An approximate sample of fine design and drawing at the end of the lesson.
Expansion of a function in a Fourier series on an arbitrary period
For an arbitrary expansion period, where "el" is any positive number, the formulas for the Fourier series and Fourier coefficients differ in a slightly complicated sine and cosine argument:
If , then we get the formulas for the interval with which we started.
The algorithm and principles for solving the problem are completely preserved, but the technical complexity of the calculations increases:
Example 4
Expand the function into a Fourier series and plot the sum. 
Decision: in fact, an analogue of Example No. 3 with rupture of the 1st kind at point . In this problem, the expansion period , half-period . The function is defined only on the half-interval , but this does not change things - it is important that both parts of the function are integrable.
Let's expand the function into a Fourier series:
Since the function is discontinuous at the origin, each Fourier coefficient should obviously be written as the sum of two integrals:
1) I will write the first integral as detailed as possible:
2) Carefully peer into the surface of the moon:
Second integral take in parts:
What should you pay close attention to after we open the continuation of the solution with an asterisk?
First, we do not lose the first integral 
, where we immediately execute bringing under the sign of the differential. Secondly, do not forget the ill-fated constant before the big brackets and don't get confused by signs when using the formula 
. Large brackets, after all, it is more convenient to open immediately in the next step.
The rest is a matter of technique, only insufficient experience in solving integrals can cause difficulties.
Yes, it was not in vain that the eminent colleagues of the French mathematician Fourier were indignant - how did he dare to decompose functions into trigonometric series ?! =) By the way, probably everyone is interested in the practical meaning of the task in question. Fourier himself worked on a mathematical model of heat conduction, and subsequently the series named after him began to be used to study many periodic processes, which are apparently invisible in the outside world. Now, by the way, I caught myself thinking that it was no coincidence that I compared the graph of the second example with a periodic heart rhythm. Those interested can get acquainted with the practical application Fourier transforms from third party sources. ... Although it’s better not to - it will be remembered as First Love =)
3) Given the repeatedly mentioned weak links, we deal with the third coefficient:
Integrating by parts: 
We substitute the found Fourier coefficients into the formula 
, not forgetting to divide the zero coefficient in half:
Let's plot the sum of the series. Let us briefly repeat the procedure: on the interval we build a line, and on the interval - a line. With a zero value of "x", we put a point in the middle of the "jump" of the gap and "replicate" the chart for neighboring periods: 
At the "junctions" of the periods, the sum will also be equal to the midpoints of the "jump" of the gap.
Ready. I remind you that the function itself is conditionally defined only on the half-interval and, obviously, coincides with the sum of the series on the intervals
Answer:
Sometimes a piecewise given function is also continuous on the expansion period. The simplest example: 
. Decision (See Bohan Volume 2) is the same as in the two previous examples: despite function continuity at the point , each Fourier coefficient is expressed as the sum of two integrals.
In the breakup interval discontinuity points of the 1st kind and / or "junction" points of the graph may be more (two, three, and in general any final amount). If a function is integrable on every part, then it is also expandable in a Fourier series. But from practical experience, I don’t remember such a tin. Nevertheless, there are more difficult tasks than just considered, and at the end of the article for everyone there are links to Fourier series of increased complexity.
In the meantime, let's relax, leaning back in our chairs and contemplating the endless expanses of stars:
Example 5
Expand the function into a Fourier series on the interval and plot the sum of the series.
In this task, the function continuous on the decomposition half-interval, which simplifies the solution. Everything is very similar to Example #2. You can't get away from the spaceship - you'll have to decide =) Sample design at the end of the lesson, the schedule is attached.
Fourier series expansion of even and odd functions
With even and odd functions, the process of solving the problem is noticeably simplified. And that's why. Let's return to the expansion of the function in a Fourier series on a period of "two pi" 
and arbitrary period "two ales" 
.
Let's assume that our function is even. The general term of the series, as you can see, contains even cosines and odd sines. And if we decompose an EVEN function, then why do we need odd sines?! Let's reset the unnecessary coefficient: .
Thus, an even function expands into a Fourier series only in cosines:
Insofar as integrals of even functions over a segment of integration symmetric with respect to zero can be doubled, then the rest of the Fourier coefficients are also simplified.
For span: 
For an arbitrary interval: 
Textbook examples that are found in almost any calculus textbook include expansions of even functions 
. In addition, they have repeatedly met in my personal practice:
Example 6
Given a function. Required:
1) expand the function into a Fourier series with period , where is an arbitrary positive number;
2) write down the expansion on the interval , build a function and graph the total sum of the series .
Decision: in the first paragraph, it is proposed to solve the problem in a general way, and this is very convenient! There will be a need - just substitute your value.
1) In this problem, the expansion period , half-period . In the course of further actions, in particular during integration, "el" is considered a constant
The function is even, which means that it expands into a Fourier series only in cosines: 
.
Fourier coefficients are sought by the formulas 
. Pay attention to their absolute advantages. First, the integration is carried out over the positive segment of the expansion, which means that we safely get rid of the module 
, considering only "x" from two pieces. And, secondly, integration is noticeably simplified.
Two: 
Integrating by parts:
Thus:
, while the constant , which does not depend on "en", is taken out of the sum.
Answer: 
2) Let's write the expansion on the interval, for this we substitute the desired value of the half-period into the general formula:
a series in 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 a sufficiently arbitrary curve 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, permutation of limits, infinite systems of equations, interpolation of functions by polynomials, etc.
And in the future, as in this initial period, the theory of T. r. served as a source of new ideas in mathematics. The question that led to controversy among mathematicians in the 18th century was resolved in 1807 by J. Fourier, who indicated 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. The modern view of the 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, the measure of 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. derivative of function F:
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 (the Cantor-Lebesgue theorem). 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 functions of T. R. by Abel-Poisson and Riemann summable almost everywhere methods, D. E. Men’shov proved (1940) the following theorem, which relates to the most important case when the representation of the function f is understood as the convergence of 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 the function 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 a continuous function 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 where convergence almost everywhere is replaced by a weaker requirement, convergence in measure. Convergence in measure to functions that can take on infinite values is defined as follows: a sequence of partial sums 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, the empty set, as well as any finite set, are U-sets. An arbitrary countable set 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 zero 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.
- - the final trigonometric sum, - an expression of the form with real coefficients a 0, and k, bk, k=l, . . ., n; number n called. order T. 0)...
Mathematical Encyclopedia
- - a series in cosines and sines of multiple arcs, i.e. a series of the form or in complex form where ak, bk or, respectively, ck are called. coefficients of T. r. For the first time T. r. meet at L. Euler ...
Mathematical Encyclopedia
- - triangulation point, - geodetic point, the position of which on the earth's surface is determined by the method of triangulation ...
Big encyclopedic polytechnic dictionary
- - see Triangulation...
Encyclopedic Dictionary of Brockhaus and Euphron
- - in geodesy, a structure installed on the ground at trigonometric points. T. h. consists of two parts - external and underground...
Great Soviet Encyclopedia
- - a functional series of the form, that is, a series located along the sines and cosines of multiple arcs. Often T. r. written in complex form.
Let a trigonometric series be given
To find out if it converges, it is natural to consider the number series
(2)
the majorizing, as they say, series (1). Its members exceed, respectively, the absolute values of the members of series (1):
.
It follows that if the series (2) converges, then the series (1) converges for all and, moreover, absolutely and uniformly (see our book Higher Mathematics. Differential and Integral Calculus, § 9.8, Theorem 1). But series (1) can converge without series (2) converging. After all, its terms for each change sign (oscillate) an infinite number of times when changing, and it can turn out to converge due to the compensation of positive terms by negative ones. In the general theory of series, there are signs of convergence of similar series. Such tests are the Dirichlet and Abel tests (see § 9.9, Theorems 3 and 4 of the same book), which are well adapted to the study of trigonometric series.
One way or another, if it is established that the series (1) converges uniformly, then from the fact that its terms are continuous functions of the period , it follows that its sum
(3)
is a continuous period function (see § 9.8, Theorem 2 and § 9.9, Theorem 2 of the same book) and the series (3) can be integrated term by term.
Series (3) can be formally differentiated by:
(4)
and compose its majorizing series
(5)
Again, if series (5) converges, then series (4) converges uniformly. Moreover, based on the well-known theorem from the theory of uniformly convergent series, then the sum of the series (4) is the derivative of the sum of the series (3), i.e.
.
In general, if a series
converges for some natural number, then the series (3) can legally be differentiated term by term.
However, we must remember that it is possible that the series (3) can legitimately be differentiated one more time (i.e., times).
Example. Find out how many times a series can be differentiated term by term
Numbers a n, b n or c n are called the coefficients of T. r.
T. r. play a very important role in mathematics and its applications. First of all, T. r. provide means for depicting and studying functions and are therefore one of the main apparatuses of the theory of functions. Further, thermal radiation naturally appears in the solution of a number of problems of mathematical physics, among which we can note the problem of the vibration of a string, the problem of the propagation of heat, and others. Finally, the theory of thermal radiation. contributed to the clarification of the basic concepts of mathematical analysis (function, integral), brought to life a number of important sections of mathematics (the theory of Fourier integrals, the theory of almost periodic functions), served as one of the starting points for the development of set theory, the theory of functions of a real variable and functional analysis, and put beginning of general harmonic analysis.
Euler pointed out the connection between power series and T. R.: if 
c n are real, then
namely:
Lit.: Luzin N. N., Integral and trigonometric series, M. - L., 1951; Barin. K., Trigonometric series, Moscow, 1961; Sigmund A., Trigonometric series, trans. from English, 2nd ed., vol. 1-2, M., 1965.
Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
See what the "Trigonometric Series" is in other dictionaries:
A series of cosines and sines of multiple arcs, i.e. a series of the form or in complex form where ak, bk or, respectively, ck are called. coefficients of T. r. For the first time T. r. meet at L. Euler (L. Euler, 1744). He received expansions in ser. 18th century in connection with…… Mathematical Encyclopedia
A series of the form where the coefficients a0, a1, b1, a2, b2 ... do not depend on the variable x ... Big Encyclopedic Dictionary
In mathematics, a trigonometric series is any series of the form: A trigonometric series is called a Fourier series of a function if the coefficients and are defined as follows ... Wikipedia
A series of the form where the coefficient a0, a1, b1, a2, b2, ... do not depend on the variable x. * * * TRIGONOMETRIC SERIES TRIGONOMETRIC SERIES, a series of the form where the coefficients a0, a1, b1, a2, b2 ... do not depend on the variable x ... encyclopedic Dictionary
The trigonometric Fourier series is a representation of an arbitrary function with a period in the form of a series (1) or, using complex notation, in the form of a series: . Contents ... Wikipedia
infinite trigonometric Fourier series- - Telecommunication topics, basic concepts EN Fourier series ... Technical Translator's Handbook
Series of the type Series of type (1) K. Weierstrass introduced in 1872 a continuous nowhere differentiable function. J. Hadamard in 1892 applied series (1), calling them lacunar, to the study of analytic. function continuation. Systematic … Mathematical Encyclopedia
To series series These series are, respectively, the real and imaginary parts of the series at z=eix. The formula for the partial sums of the function j(x) conjugate to the Fourier series trigonometric. series where is the Dirichlet conjugate kernel. If f(x) is a function of bounded variation... ... Mathematical Encyclopedia
Adding Fourier Series Terms ... Wikipedia
I is an infinite sum, for example, of the form u1 + u2 + u3 + ... + un + ... or, in short, One of the simplest examples of R., already found in elementary mathematics, is an infinitely decreasing sum ... ... Great Soviet Encyclopedia
Introductory remarks
In this section, we will consider the representation of periodic signals using a Fourier series. Fourier series are the basis of the theory of spectral analysis, because, as we will see later, the Fourier transform of a non-periodic signal can be obtained as the limit transition of the Fourier series with an infinite repetition period. As a result, the properties of the Fourier series are also valid for the Fourier transform of non-periodic signals.We will consider the expressions for the Fourier series in trigonometric and complex forms, and also pay attention to the Dirichlet conditions for the convergence of the Fourier series. In addition, we will dwell in detail on the explanation of such a concept as the negative frequency of the signal spectrum, which often causes difficulty when getting acquainted with the theory of spectral analysis.
Periodic signal. Trigonometric Fourier series
Let there be a continuous-time periodic signal , which repeats with a period c, i.e. , where is an arbitrary integer.As an example, Figure 1 shows a sequence of rectangular pulses of duration c, repeating with a period of c.
Figure 1. Periodic sequence
Rectangular pulses
From the course of mathematical analysis it is known that the system of trigonometric functions
with multiple frequencies , where rad/s is an integer, forms an orthonormal basis for the expansion of periodic signals with a period satisfying the Dirichlet conditions .
The Dirichlet conditions for the convergence of the Fourier series require that a periodic signal be given on the segment , while satisfying the following conditions:
For example, the periodic function 
does not satisfy the Dirichlet conditions, because the function has discontinuities of the second kind and takes infinite values for , where is an arbitrary integer. So the function cannot be represented by a Fourier series. You can also give an example of a function 
, which is bounded, but also does not satisfy the Dirichlet conditions, since it has an infinite number of extremum points as it approaches zero. Function Graph shown in figure 2.
Figure 2. Graph of the function :
A - two periods of repetition; b - in the neighborhood
Figure 2a shows two repetition periods of the function , and in Figure 2b is the area in the vicinity of . It can be seen that when approaching zero, the oscillation frequency increases infinitely, and such a function cannot be represented by a Fourier series, because it is not piecewise monotonic.
It should be noted that in practice there are no signals with infinite values of current or voltage. Functions with an infinite number of extrema of type are also not found in applied problems. All real periodic signals satisfy the Dirichlet conditions and can be represented by an infinite trigonometric Fourier series of the form:
 |
In expression (2), the coefficient specifies the constant component of the periodic signal .
At all points where the signal is continuous, the Fourier series (2) converges to the values of the given signal, and at discontinuity points of the first kind, to the mean value , where and are the limits to the left and right of the discontinuity point, respectively.
It is also known from the course of mathematical analysis that the use of a truncated Fourier series containing only the first terms instead of an infinite sum leads to an approximate representation of the signal:
which ensures the minimum mean squared error. Figure 3 illustrates the approximation of a periodic square wave train and a periodic sawtooth signal using different numbers of Fourier series terms.
Figure 3. Approximation of signals by a truncated Fourier series:
A - rectangular pulses; b - sawtooth signal
Fourier series in complex form
In the previous paragraph, we considered the trigonometric Fourier series for the expansion of an arbitrary periodic signal that satisfies the Dirichlet conditions. Using the Euler formula, we can show: |
Then the trigonometric Fourier series (2) taking into account (4):
Thus, a periodic signal can be represented by the sum of a DC component and complex exponents rotating at frequencies with coefficients for positive frequencies , and for complex exponents rotating at negative frequencies.
Consider the coefficients for complex exponents rotating with positive frequencies:
Expressions (6) and (7) coincide, in addition, the constant component can also be written in terms of the complex exponential at zero frequency:
Thus, (5), taking into account (6)-(8), can be represented as a single sum when indexed from minus infinity to infinity:
 |
Expression (9) is a Fourier series in complex form. The coefficients of the Fourier series in complex form are related to the coefficients and of the series in trigonometric form, and are defined for both positive and negative frequencies. The index in the frequency notation indicates the number of the discrete harmonic, with negative indices corresponding to negative frequencies.
It follows from expression (2) that for a real signal the coefficients and of series (2) are also real. However, (9) assigns to a real signal , a set of complex conjugate coefficients , related to both positive and negative frequencies .
Some explanations for the Fourier series in complex form
In the previous section, we made the transition from the trigonometric Fourier series (2) to the Fourier series in complex form (9). As a result, instead of expanding periodic signals in the basis of real trigonometric functions, we got an expansion in the basis of complex exponentials, with complex coefficients, and even negative frequencies appeared in the expansion! Since this issue is often misunderstood, it is necessary to give some clarification.First, working with complex exponents is in most cases easier than working with trigonometric functions. For example, when multiplying and dividing complex exponentials, it is enough just to add (subtract) the exponents, while the formulas for multiplying and dividing trigonometric functions are more cumbersome.
Differentiating and integrating exponents, even complex ones, is also easier than trigonometric functions, which are constantly changing when differentiating and integrating (sine becomes cosine and vice versa).
If the signal is periodic and real, then the trigonometric Fourier series (2) seems more illustrative, because all expansion coefficients , and remain real. However, one often has to deal with complex periodic signals (for example, modulation and demodulation use a quadrature representation of the complex envelope). In this case, when using the trigonometric Fourier series, all coefficients , and expansions (2) will become complex, while when using the Fourier series in complex form (9), the same expansion coefficients will be used for both real and complex input signals.
And finally, it is necessary to dwell on the explanation of the negative frequencies that appeared in (9). This question is often misunderstood. In everyday life, we do not encounter negative frequencies. For example, we never tune our radio to a negative frequency. Let's consider the following analogy from mechanics. Let there be a mechanical spring pendulum that oscillates freely with a certain frequency . Can a pendulum oscillate with a negative frequency? Of course not. Just as there are no radio stations that go on the air at negative frequencies, so the frequency of the pendulum cannot be negative. But a spring pendulum is a one-dimensional object (the pendulum oscillates along one straight line).
We can also give another analogy from mechanics: a wheel rotating at a frequency of . The wheel, unlike the pendulum, rotates, i.e. a point on the surface of the wheel moves in a plane, and does not just oscillate along a single straight line. Therefore, to uniquely set the rotation of the wheel, it is not enough to set the rotation frequency, because it is also necessary to set the direction of rotation. This is exactly what we can use the frequency sign for.
So, if the wheel rotates at a frequency of rad / s counterclockwise, then we consider that the wheel rotates with a positive frequency, and if it rotates in a clockwise direction, then the rotation frequency will be negative. Thus, to specify a rotation, a negative frequency ceases to be nonsense and indicates the direction of rotation.
And now the most important thing that we must understand. The oscillation of a one-dimensional object (for example, a spring pendulum) can be represented as the sum of the rotations of the two vectors shown in Figure 4.
Figure 4. Oscillation of a spring pendulum
As the sum of rotations of two vectors
on the complex plane
The pendulum oscillates along the real axis of the complex plane with a frequency according to the harmonic law. The movement of the pendulum is shown as a horizontal vector. The top vector rotates in the complex plane at a positive frequency (counterclockwise), and the bottom vector rotates at a negative frequency (clockwise). Figure 4 clearly illustrates the well-known relation from the trigonometry course:
Thus, the Fourier series in complex form (9) represents periodic one-dimensional signals as the sum of vectors on the complex plane rotating with positive and negative frequencies. At the same time, we note that in the case of a real signal, according to (9), the expansion coefficients for negative frequencies are complex conjugate to the corresponding coefficients for positive frequencies . In the case of a complex signal, this property of the coefficients does not hold due to the fact that and are also complex.
Spectrum of periodic signals
The Fourier series in complex form is the decomposition of a periodic signal into a sum of complex exponentials rotating with positive and negative frequencies in multiples of rad/s with the corresponding complex coefficients , which determine the spectrum of the signal . The complex coefficients can be represented by the Euler formula as , where is the amplitude spectrum and a is the phase spectrum.Since periodic signals are decomposed into a series only on a fixed frequency grid, the spectrum of periodic signals is line (discrete).
Figure 5. Spectrum of a Periodic Sequence
Rectangular pulses:
A is the amplitude spectrum; b - phase spectrum
Figure 5 shows an example of the amplitude and phase spectrum of a periodic sequence of rectangular pulses (see Figure 1) for c, pulse duration c, and pulse amplitude B.
The amplitude spectrum of the original real signal is symmetric with respect to zero frequency, while the phase spectrum is antisymmetric. At the same time, we note that the values of the phase spectrum and 
correspond to the same point in the complex plane.
It can be concluded that all the expansion coefficients of the reduced signal are purely real, and the phase spectrum 
corresponds to negative coefficients .
Note that the dimension of the amplitude spectrum coincides with the dimension of the signal . If describes the change in voltage over time, measured in volts, then the amplitudes of the harmonics of the spectrum will also have the dimension of volts.
findings
In this section, we consider the representation of periodic signals using the Fourier series. Expressions for the Fourier series in trigonometric and complex forms are given. We paid special attention to the Dirichlet conditions for the convergence of the Fourier series and gave examples of functions for which the Fourier series diverges.We dwelled in detail on the expression of the Fourier series in complex form and showed that periodic signals, both real and complex, are represented by a series of complex exponentials with positive and negative frequencies. In this case, the expansion coefficients are also complex and characterize the amplitude and phase spectrum of a periodic signal.
In the next section, we will consider the properties of the spectra of periodic signals in more detail.
