On every day. Fourier series: the history and influence of the mathematical mechanism on the development of science The constant component e of the Fourier series is equal to

Fourier series is a representation of an arbitrarily taken function with a specific period as a series. In general terms, this solution is called the decomposition of an element in an orthogonal basis. The expansion of functions in a Fourier series is a fairly powerful tool for solving various problems due to the properties of this transformation when integrating, differentiating, as well as shifting an expression in an argument and convolution.

A person who is not familiar with higher mathematics, as well as with the works of the French scientist Fourier, most likely will not understand what these “series” are and what they are for. Meanwhile, this transformation has become quite dense in our lives. It is used not only by mathematicians, but also by physicists, chemists, physicians, astronomers, seismologists, oceanographers and many others. Let us also take a closer look at the works of the great French scientist, who made a discovery ahead of his time.

Man and the Fourier Transform

Fourier series is one of the methods (along with analysis and others) This process occurs every time a person hears any sound. Our ear automatically transforms elementary particles in an elastic medium, they are decomposed into rows (along the spectrum) of successive values ​​of the volume level for tones of different heights. Next, the brain turns this data into sounds familiar to us. All this happens in addition to our desire or consciousness, by itself, but in order to understand these processes, it will take several years to study higher mathematics.

More on the Fourier Transform

The Fourier transform can be carried out by analytical, numerical and other methods. Fourier series refer to the numeral way of decomposing any oscillatory processes - from ocean tides and light waves to cycles of solar (and other astronomical objects) activity. Using these mathematical techniques, it is possible to analyze functions, representing any oscillatory processes as a series of sinusoidal components that go from minimum to maximum and vice versa. The Fourier transform is a function that describes the phase and amplitude of sinusoids corresponding to a specific frequency. This process can be used to solve very complex equations that describe dynamic processes that occur under the influence of thermal, light or electrical energy. Also, Fourier series make it possible to isolate the constant components in complex oscillatory signals, which made it possible to correctly interpret the obtained experimental observations in medicine, chemistry and astronomy.

History reference

The founding father of this theory is the French mathematician Jean Baptiste Joseph Fourier. This transformation was subsequently named after him. Initially, the scientist applied his method to study and explain the mechanisms of heat conduction - the spread of heat in solids. Fourier suggested that the original irregular distribution can be decomposed into the simplest sinusoids, each of which will have its own temperature minimum and maximum, as well as its own phase. In this case, each such component will be measured from minimum to maximum and vice versa. The mathematical function that describes the upper and lower peaks of the curve, as well as the phase of each of the harmonics, is called the Fourier transform of the temperature distribution expression. The author of the theory reduced the general distribution function, which is difficult to describe mathematically, to a very convenient series of cosine and sine, which sum up to give the original distribution.

The principle of transformation and the views of contemporaries

The scientist's contemporaries - the leading mathematicians of the early nineteenth century - did not accept this theory. The main objection was Fourier's assertion that a discontinuous function describing a straight line or a discontinuous curve can be represented as a sum of sinusoidal expressions that are continuous. As an example, consider Heaviside's "step": its value is zero to the left of the gap and one to the right. This function describes the dependence of the electric current on the time variable when the circuit is closed. The contemporaries of the theory at that time had never encountered such a situation, when a discontinuous expression would be described by a combination of continuous, ordinary functions, such as exponential, sinusoid, linear or quadratic.

What confused French mathematicians in Fourier theory?

After all, if the mathematician was right in his statements, then by summing the infinite trigonometric Fourier series, one can obtain an exact representation of the stepwise expression even if it has many similar steps. At the beginning of the nineteenth century, such a statement seemed absurd. But despite all the doubts, many mathematicians have expanded the scope of the study of this phenomenon, taking it beyond the scope of studies of thermal conductivity. However, most scientists continued to be tormented by the question: "Can the sum of the sinusoidal series converge to the exact value of the discontinuous function?"

Fourier Series Convergence: An Example

The question of convergence is raised whenever it is necessary to sum infinite series of numbers. To understand this phenomenon, consider a classic example. Can you ever reach the wall if each successive step is half the size of the previous one? Suppose you are two meters from the goal, the first step brings you closer to the halfway point, the next one to the three-quarters mark, and after the fifth you will overcome almost 97 percent of the way. However, no matter how many steps you take, you will not achieve the intended goal in a strict mathematical sense. Using numerical calculations, it can be shown that in the end it is possible to approach an arbitrarily small given distance. This proof is equivalent to demonstrating that the total value of one-half, one-fourth, etc. will tend to one.

A Question of Convergence: The Second Coming, or Lord Kelvin's Appliance

This question was raised again at the end of the nineteenth century, when Fourier series were tried to be used to predict the intensity of ebb and flow. At this time, Lord Kelvin invented a device, which is an analog computing device that allowed sailors of the military and merchant fleet to track this natural phenomenon. This mechanism determined the sets of phases and amplitudes from a table of tide heights and their corresponding time moments, carefully measured in a given harbor during the year. Each parameter was a sinusoidal component of the tide height expression and was one of the regular components. The results of the measurements were entered into Lord Kelvin's calculator, which synthesized a curve that predicted the height of the water as a function of time for the next year. Very soon similar curves were drawn up for all the harbors of the world.

And if the process is broken by a discontinuous function?

At that time, it seemed obvious that a tidal wave predictor with a large number of counting elements could calculate a large number of phases and amplitudes and thus provide more accurate predictions. Nevertheless, it turned out that this regularity is not observed in those cases when the tidal expression, which should be synthesized, contained a sharp jump, that is, it was discontinuous. In the event that data is entered into the device from the table of time moments, then it calculates several Fourier coefficients. The original function is restored thanks to the sinusoidal components (according to the found coefficients). The discrepancy between the original and restored expression can be measured at any point. When carrying out repeated calculations and comparisons, it can be seen that the value of the largest error does not decrease. However, they are localized in the region corresponding to the discontinuity point, and tend to zero at any other point. In 1899, this result was theoretically confirmed by Joshua Willard Gibbs of Yale University.

Convergence of Fourier series and the development of mathematics in general

Fourier analysis is not applicable to expressions containing an infinite number of bursts in a certain interval. In general, Fourier series, if the original function is the result of a real physical measurement, always converge. Questions of the convergence of this process for specific classes of functions have led to the emergence of new sections in mathematics, for example, the theory of generalized functions. It is associated with such names as L. Schwartz, J. Mikusinsky and J. Temple. Within the framework of this theory, a clear and precise theoretical basis was created for such expressions as the Dirac delta function (it describes an area of ​​​​a single area concentrated in an infinitely small neighborhood of a point) and the Heaviside “step”. Thanks to this work, Fourier series became applicable to solving equations and problems in which intuitive concepts appear: a point charge, a point mass, magnetic dipoles, and also a concentrated load on a beam.

Fourier method

Fourier series, in accordance with the principles of interference, begin with the decomposition of complex forms into simpler ones. For example, a change in heat flow is explained by its passage through various obstacles made of irregularly shaped heat-insulating material or a change in the surface of the earth - an earthquake, a change in the orbit of a celestial body - the influence of planets. As a rule, similar equations describing simple classical systems are elementarily solved for each individual wave. Fourier showed that simple solutions can also be summed to give solutions to more complex problems. Expressed in the language of mathematics, Fourier series is a technique for representing an expression as the sum of harmonics - cosine and sinusoids. Therefore, this analysis is also known as "harmonic analysis".

Fourier series - the ideal technique before the "computer age"

Before the creation of computer technology, the Fourier technique was the best weapon in the arsenal of scientists when working with the wave nature of our world. The Fourier series in a complex form allows solving not only simple problems that can be directly applied to the laws of Newton's mechanics, but also fundamental equations. Most of the discoveries of Newtonian science in the nineteenth century were made possible only by Fourier's technique.

Fourier series today

With the development of computers, Fourier transforms have risen to a qualitatively new level. This technique is firmly entrenched in almost all areas of science and technology. An example is a digital audio and video signal. Its realization became possible only thanks to the theory developed by a French mathematician at the beginning of the nineteenth century. Thus, the Fourier series in a complex form made it possible to make a breakthrough in the study of outer space. In addition, this influenced the study of the physics of semiconductor materials and plasma, microwave acoustics, oceanography, radar, and seismology.

Trigonometric Fourier series

In mathematics, a Fourier series is a way of representing arbitrary complex functions as a sum of simpler ones. In general cases, the number of such expressions can be infinite. Moreover, the more their number is taken into account in the calculation, the more accurate the final result is. Most often, the trigonometric functions of cosine or sine are used as the simplest ones. In this case, the Fourier series are called trigonometric, and the solution of such expressions is called the expansion of the harmonic. This method plays an important role in mathematics. First of all, the trigonometric series provides a means for the image, as well as the study of functions, it is the main apparatus of the theory. In addition, it allows solving a number of problems of mathematical physics. Finally, this theory contributed to the development and brought to life a number of very important sections of mathematical science (the theory of integrals, the theory of periodic functions). In addition, it served as a starting point for the development of the following functions of a real variable, and also marked the beginning of harmonic analysis.

Fourier series of periodic functions with period 2π.

The Fourier series allows you to study periodic functions by decomposing them into components. Alternating currents and voltages, displacements, speed and acceleration of crank mechanisms, and acoustic waves are typical practical applications of periodic functions in engineering calculations.

The Fourier series expansion is based on the assumption that all functions of practical importance in the interval -π ≤ x ≤ π can be expressed as convergent trigonometric series (a series is considered convergent if the sequence of partial sums made up of its terms converges):

Standard (=usual) notation through the sum of sinx and cosx

f(x)=a o + a 1 cosx+a 2 cos2x+a 3 cos3x+...+b 1 sinx+b 2 sin2x+b 3 sin3x+...,

where a o , a 1 ,a 2 ,...,b 1 ,b 2 ,.. are real constants, i.e.

Where, for the range from -π to π, the coefficients of the Fourier series are calculated by the formulas:

The coefficients a o ,a n and b n are called Fourier coefficients, and if they can be found, then series (1) is called near Fourier, corresponding to the function f(x). For series (1), the term (a 1 cosx+b 1 sinx) is called the first or main harmonica,

Another way to write a series is to use the relation acosx+bsinx=csin(x+α)

f(x)=a o +c 1 sin(x+α 1)+c 2 sin(2x+α 2)+...+c n sin(nx+α n)

Where a o is a constant, c 1 \u003d (a 1 2 +b 1 2) 1/2, c n \u003d (a n 2 +b n 2) 1/2 are the amplitudes of the various components, and is equal to a n \u003d arctg a n /b n.

For series (1), the term (a 1 cosx + b 1 sinx) or c 1 sin (x + α 1) is called the first or main harmonica,(a 2 cos2x+b 2 sin2x) or c 2 sin(2x+α 2) is called second harmonic and so on.

To accurately represent a complex signal, an infinite number of terms is usually required. However, in many practical problems it is sufficient to consider only the first few terms.

Fourier series of non-periodic functions with period 2π.

Expansion of non-periodic functions in a Fourier series.

If the function f(x) is non-periodic, then it cannot be expanded in a Fourier series for all values ​​of x. However, it is possible to define a Fourier series representing a function over any range of width 2π.

Given a non-periodic function, one can compose a new function by choosing f(x) values ​​within a certain range and repeating them outside this range at 2π intervals. Since the new function is periodic with a period of 2π, it can be expanded in a Fourier series for all values ​​of x. For example, the function f(x)=x is not periodic. However, if it is necessary to expand it into a Fourier series on the interval from 0 to 2π, then a periodic function with a period of 2π is constructed outside this interval (as shown in the figure below).

For non-periodic functions such as f(x)=x, the sum of the Fourier series is equal to the value of f(x) at all points in the given range, but it is not equal to f(x) for points outside the range. To find the Fourier series of a non-periodic function in the range 2π, the same formula of the Fourier coefficients is used.

Even and odd functions.

They say the function y=f(x) even if f(-x)=f(x) for all values ​​of x. Graphs of even functions are always symmetrical about the y-axis (that is, they are mirrored). Two examples of even functions: y=x 2 and y=cosx.

They say that the function y=f(x) odd, if f(-x)=-f(x) for all values ​​of x. Graphs of odd functions are always symmetrical about the origin.

Many functions are neither even nor odd.

Fourier series expansion in cosines.

The Fourier series of an even periodic function f(x) with period 2π contains only cosine terms (i.e., does not contain sine terms) and may include a constant term. Consequently,

where are the coefficients of the Fourier series,

The Fourier series of an odd periodic function f(x) with period 2π contains only terms with sines (i.e., does not contain terms with cosines).

Consequently,

where are the coefficients of the Fourier series,

Fourier series on a half-cycle.

If a function is defined over a range, say 0 to π, and not just 0 to 2π, it can be expanded into a series only in terms of sines or only in terms of cosines. The resulting Fourier series is called near Fourier on a half cycle.

If you want to get a decomposition Fourier on a half-cycle in cosines functions f(x) in the range from 0 to π, then it is necessary to compose an even periodic function. On fig. below is the function f(x)=x built on the interval from x=0 to x=π. Since the even function is symmetrical about the f(x) axis, we draw the line AB, as shown in Fig. below. If we assume that outside the considered interval, the resulting triangular shape is periodic with a period of 2π, then the final graph has the form, display. in fig. below. Since it is required to obtain the Fourier expansion in cosines, as before, we calculate the Fourier coefficients a o and a n

If you want to get functions f (x) in the range from 0 to π, then you need to compose an odd periodic function. On fig. below is the function f(x)=x built on the interval from x=0 to x=π. Since the odd function is symmetric with respect to the origin, we construct the line CD, as shown in Fig. If we assume that outside the considered interval, the received sawtooth signal is periodic with a period of 2π, then the final graph has the form shown in Fig. Since it is required to obtain the Fourier expansion on the half-cycle in terms of sines, as before, we calculate the Fourier coefficient. b

Fourier series for an arbitrary interval.

Expansion of a periodic function with period L.

The periodic function f(x) repeats as x increases by L, i.e. f(x+L)=f(x). The transition from the previously considered functions with period 2π to functions with period L is quite simple, since it can be done using a change of variable.

To find the Fourier series of the function f(x) in the range -L/2≤x≤L/2, we introduce a new variable u so that the function f(x) has a period of 2π with respect to u. If u=2πx/L, then x=-L/2 for u=-π and x=L/2 for u=π. Also let f(x)=f(Lu/2π)=F(u). The Fourier series F(u) has the form

Where are the coefficients of the Fourier series,

More often, however, the above formula leads to dependence on x. Since u=2πх/L, then du=(2π/L)dx, and the limits of integration are from -L/2 to L/2 instead of -π to π. Therefore, the Fourier series for the dependence on x has the form

where in the range from -L/2 to L/2 are the coefficients of the Fourier series,

(Integration limits can be replaced by any interval of length L, for example, from 0 to L)

Fourier series on a half-cycle for functions given in the interval L≠2π.

For the substitution u=πx/L, the interval from x=0 to x=L corresponds to the interval from u=0 to u=π. Therefore, the function can be expanded into a series only in terms of cosines or only in terms of sines, i.e. in Fourier series on a half cycle.

The expansion in cosines in the range from 0 to L has the form

Function defined for all values x called periodical, if there is such a number T (T≠ 0), that for any value x equality f(x + T) = f(x). Number T in this case is the period of the function.

Properties of periodic functions:

1) Sum, difference, product and quotient of periodic period functions T is a periodic function of the period T.

2) If the function f(x) has a period T, then the function f(ax) has a period

Indeed, for any argument X:

(multiplying the argument by a number means squeezing or stretching the graph of this function along the axis OH)

For example, a function has a period , the period of a function is

3) If f(x) period periodic function T, then any two integrals of this function are equal, taken over the interval of length T(it is assumed that these integrals exist).

Fourier series for a function with period T= .

A trigonometric series is a series of the form:

or, in short,

Where , , , , , … , , , … are real numbers, called coefficients of the series.

Each term of the trigonometric series is a periodic function of the period (because - has any

period, and the period () is , and hence ). Each term (), with n= 1,2,3… is an analytical expression of a simple harmonic oscillation , where A- amplitude,

initial phase. Given the above, we get: if the trigonometric series converges on a segment of the length of the period, then it converges on the entire numerical axis and its sum is a periodic function of the period.

Let the trigonometric series converge uniformly on a segment (and therefore on any segment) and its sum is equal to . To determine the coefficients of this series, we use the following equalities:

We also use the following properties.

1) As is known, the sum of a series composed of continuous functions uniformly convergent on a certain segment is itself a continuous function on this segment. Given this, we get that the sum of a trigonometric series uniformly converging on a segment is a continuous function on the entire real axis.

2) The uniform convergence of the series on a segment will not be violated if all terms of the series are multiplied by a function that is continuous on this segment.

In particular, uniform convergence on a segment of a given trigonometric series will not be violated if all members of the series are multiplied by or by .

By condition

As a result of term-by-term integration of the uniformly convergent series (4.2) and taking into account the above equalities (4.1) (the orthogonality of trigonometric functions), we obtain:

Therefore, the coefficient

Multiplying equality (4.2) by , integrating this equality within the range from to and, taking into account the above expressions (4.1), we obtain:

Therefore, the coefficient

Similarly, multiplying equality (4.2) by and integrating it within the limits from to , taking equalities (4.1) into account, we have:

Therefore, the coefficient

Thus, the following expressions for the coefficients of the Fourier series are obtained:

Sufficient criteria for the expansion of a function into a Fourier series. Recall that the point x o function break f(x) is called a discontinuity point of the first kind if there are finite limits on the right and left of the function f(x) in the vicinity of the point.

Limit on the right

Left limit.

Theorem (Dirichlet). If the function f(x) has a period and is continuous on the segment or has a finite number of discontinuity points of the first kind and, in addition, the segment can be divided into a finite number of segments so that inside each of them f(x) is monotonic, then the Fourier series for the function f(x) converges for all values x. Moreover, at the points of continuity of the function f(x) its sum is f(x), and at the discontinuity points of the function f(x) its sum is , i.e. the arithmetic mean of the limit values ​​on the left and right. In addition, the Fourier series for the function f(x) converges uniformly on any segment that, together with its ends, belongs to the interval of continuity of the function f(x).

Example: expand the function in a Fourier series

Satisfying the condition.

Solution. Function f(x) satisfies the Fourier expansion conditions, so we can write:

In accordance with formulas (4.3), one can obtain the following values ​​of the coefficients of the Fourier series:

When calculating the coefficients of the Fourier series, the formula "integration by parts" was used.

And therefore

Fourier series for even and odd functions with period T = .

We use the following property of the integral over a symmetric with respect to x=0 span:

If a f(x)- odd function,

if f(x) is an even function.

Note that the product of two even or two odd functions is an even function, and the product of an even function and an odd function is an odd function. Let now f(x)- even periodic function with period , which satisfies the conditions of expansion into a Fourier series. Then, using the above property of integrals, we get:

Thus, the Fourier series for an even function contains only even functions - cosines and is written as follows:

and the coefficients bn = 0.

Arguing similarly, we get that if f(x) - an odd periodic function that satisfies the conditions of expansion into a Fourier series, then, therefore, the Fourier series for an odd function contains only odd functions - sines and is written as follows:

wherein an=0 at n=0, 1,…

Example: expand in a Fourier series a periodic function

Since the given odd function f(x) satisfies the Fourier expansion conditions, then

or, which is the same,

And the Fourier series for this function f(x) can be written like this:

Fourier series for functions of any period T=2 l.

Let f(x)- periodic function of any period T=2l(l- half-period), piecewise-smooth or piecewise-monotone on the interval [ -l,l]. Assuming x=at, get the function f(at) argument t, whose period is . Let's pick a so that the period of the function f(at) was equal to , i.e. T = 2l

Solution. Function f(x)- odd, satisfying the conditions of expansion into a Fourier series, therefore, based on formulas (4.12) and (4.13), we have:

(when calculating the integral, the formula "integration by parts" was used).

Fourier series expansion of even and odd functions expansion of a function given on a segment into a series in terms of sines or cosines Fourier series for a function with an arbitrary period Complex representation of the Fourier series Fourier series in general orthogonal systems of functions Fourier series in an orthogonal system Minimum property of Fourier coefficients Bessel's inequality Equality Parseval Closed systems Completeness and closedness of systems

Fourier series expansion of even and odd functions The function f(x), defined on the segment \-1, where I > 0, is called even if the Graph of the even function is symmetrical about the y-axis. The function f(x) defined on the segment J, where I > 0, is called odd if the Graph of the odd function is symmetrical with respect to the origin. Example. a) The function is even on the segment |-jt, jt), since for all x e b) The function is odd, since the Fourier series expansion of even and odd functions is the expansion of a function given on the segment in a series of sines or cosines Fourier series for a function with an arbitrary period Complex notation of the Fourier series Fourier series in general orthogonal systems of functions Fourier series in an orthogonal system Minimum property of Fourier coefficients Bessel inequality Parseval equality Closed systems Completeness and closedness of systems neither to even nor to odd functions, since Let the function f(x) satisfying the conditions of Theorem 1 be even on the segment x|. Then for all i.e. /(g) cos nx is an even function, and f(x)sinnx is an odd one. Therefore, the Fourier coefficients of an even function /(x) will be equal. Therefore, the Fourier series of an even function has the form f(x) sin nx is an even function. Therefore, we will have Thus, the Fourier series of an odd function has the form We have Applying integration by parts twice, we get that Hence, the Fourier series of this function looks like this: or, in expanded form, This equality is valid for any x €, since at the points x = ±ir the sum of the series coincides with the values ​​of the function f(x ) = x2, since the graphs of the function f(x) = x and the sums of the resulting series are given in fig. Comment. This Fourier series allows you to find the sum of one of the convergent numerical series, namely, for x \u003d 0, we get that The function /(x) satisfies the conditions of Theorem 1, therefore it can be expanded into a Fourier series, which, due to the oddness of this function, will have the form Integrating by parts, we find the Fourier coefficients Therefore, the Fourier series of this function has the form This equality holds for all x В points x - ±tg the sum of the Fourier series does not coincide with the values ​​of the function / (x) \u003d x, since it is equal. its graph is shown in Fig. 6. § 6. Expansion of a function given on an interval into a series in terms of sines or cosines. Let a bounded piecewise monotonic function / be given on an interval . The values ​​of this function on the interval 0| can be defined in various ways. For example, it is possible to define the function / on the segment mc] in such a way that /. In this case it is said that) "is extended to the segment 0] in an even way"; its Fourier series will contain only cosines. If, however, the function /(x) is defined on the segment [-x, mc] so that /(, then we get an odd function, and then we say that / “is extended to the segment [-*, 0] in an odd way”; in this In this case, the Fourier series will contain only sines.So, each bounded piecewise-monotone function /(x), defined on the segment , can be expanded into a Fourier series both in terms of sines and cosines.Example 1. The function can be expanded in a Fourier series: a) by cosines; b) along the sines. M This function, with its even and odd extensions to the segment |-x, 0) will be bounded and piecewise monotonic. a) We continue / (z) into the segment 0) a) We continue j \ x) into the segment (-m, 0 | in an even way (Fig. 7), then its Fourier series i will have the form P \u003d 1 where the Fourier coefficients are equal, respectively for Therefore, b) Let's continue /(z) in the segment [-x,0] in an odd way (Fig. 8). Then its Fourier series §7. Fourier Series for a Function with an Arbitrary Period Let the function fix) be periodic with a period of 21.1 ^ 0. To expand it into a Fourier series on the interval where I > 0, we make a change of variable by setting x = jt. Then the function F(t) = / ^tj will be a periodic function of the argument t with a period and it can be expanded on a segment in a Fourier series Returning to the variable x, i.e., setting, we obtain , remain valid for periodic functions with an arbitrary period 21. In particular, the sufficient criterion for the expansion of a function into a Fourier series also remains valid. Example 1. Expand in a Fourier series a periodic function with a period of 21, given on the segment [-/,/] by the formula (Fig. 9). Since this function is even, its Fourier series has the form Substituting the found values ​​of the Fourier coefficients into the Fourier series, we obtain We note one important property of periodic functions. Theorem 5. If a function has a period T and is integrable, then for any number a the equality m holds. That is, the integral no segment, the length of which is equal to the period T, has the same value regardless of the position of this segment on the real axis. Indeed, We make a change of variable in the second integral, assuming This gives and therefore, Geometrically, this property means that in the case of the area shaded in Fig. 10 areas are equal to each other. In particular, for a function f(x) with a period, we obtain at the Fourier series expansion of even and odd functions the expansion of a function given on a segment into a series in terms of sines or cosines Fourier series for a function with an arbitrary period Complex representation of the Fourier series Fourier series in general orthogonal systems functions Fourier series in an orthogonal system Minimal property of Fourier coefficients Bessel inequality Parseval equality Closed-loop systems Completeness and closedness of systems that the Fourier coefficients of a periodic function f(x) with a period of 21 can be calculated using the formulas where a is an arbitrary real number (note that the functions cos - and sin have a period of 2/). Example 3. Expand in a Fourier series a function given on an interval with a period of 2x (Fig. 11). 4 Find the Fourier coefficients of this function. Putting in the formulas we find that for Therefore, the Fourier series will look like this: At the point x = jt (discontinuity point of the first kind) we have §8. Complex notation of the Fourier series In this section, some elements of complex analysis are used (see Chapter XXX, where all the operations performed here with complex expressions are strictly justified). Let the function f(x) satisfy sufficient conditions for expansion into a Fourier series. Then on the interval x] it can be represented by a series of the form Using the Euler formulas Substituting these expressions into the series (1) instead of cos nx and sin xy we will have We introduce the following notation Then the series (2) takes the form Thus, the Fourier series (1) is presented in the complex form (3). Let us find expressions for the coefficients in terms of integrals. We have Similarly, we find Finally, the formulas for с„, с_п and с can be written as follows: . . The coefficients cn are called the complex Fourier coefficients of the function For a periodic function with a period), the complex form of the Fourier series takes the form values ​​w if limits exist Example. Expand the period function into a complex Fourier series This function satisfies sufficient conditions for expansion into a Fourier series. Let Find the complex Fourier coefficients of this function. We have for odd for even n, or, in short. Substituting the values), we finally obtain Note that this series can also be written as follows: Fourier series in general orthogonal systems of functions 9.1. Orthogonal Systems of Functions Denote by the set of all (real) functions that are square-defined and integrable on the interval [a, 6], i.e., those for which there exists an integral. In particular, all functions f(x) that are continuous on the interval [a , 6], belong to 6], and the values ​​of their Lebesgue integrals coincide with the values ​​of the Riemann integrals. Definition. The system of functions, where, is called orthogonal on the interval [a, b\, if Condition (1) assumes, in particular, that none of the functions is identically equal to zero. The integral is understood in the sense of Lebesgue. and we call the quantity the norm of a function. If in an orthogonal system for any n we have, then the system of functions is called orthonormal. If the system (y>n(x)) is orthogonal, then the system Example 1. A trigonometric system is orthogonal on a segment. The system of functions is an orthonormal system of functions on, Example 2. The cosine system and the sine system is orthonormal. Let us introduce the notation that they are orthogonal on the segment (0, f|, but not orthonormal (for I ↦ 2). Since their norms are COS that the functions form an orthonormal system of functions on a segment. Let us show, for example, that the Legendre polynomials are orthogonal. Let m > n. In this case, integrating n times by parts, we find, since for the function t/m = (z2 - I)m, all derivatives up to order m - I inclusive vanish at the ends of the segment [-1,1). Definition. The system of functions (pn(x)) is called orthogonal on the interval (a, b) by overhang p(x) if: 1) there are integrals for all n = 1,2,... Here it is assumed that the weight function p(x) is defined and positive everywhere on the interval (a, b), with the possible exception of a finite number of points where p(x) can vanish. After performing differentiation in formula (3), we find. It can be shown that the Chebyshev-Hermite polynomials are orthogonal on the interval Example 4. The system of Bessel functions (jL(pix)^ is orthogonal on the interval of zeros of the Bessel function system Let an orthogonal system of functions in the interval (a, 6) and let the series (cj = const) converge on this interval to the function f(x): By virtue of the orthogonality of the system, we obtain that This operation has, generally speaking, a purely formal character. However, in some cases, for example, when the series (4) converges uniformly, all functions are continuous and the interval (a, 6) is finite, this operation is legal. But it is the formal interpretation that is important for us now. So let's say a function is given. We form the numbers c * according to the formula (5) and write The series on the right side is called the Fourier series of the function f (x) with respect to the system (^n (n)) - The numbers Cn are called the Fourier coefficients of the function f (x) in this system. The sign ~ in formula (6) only means that the numbers Cn are related to the function f(x) by formula (5) (in this case, it is not assumed that the series on the right converges at all, much less converges to the function f(x)). Therefore, the question naturally arises: what are the properties of this series? In what sense does it "represent" the function f(x)? 9.3. Average Convergence Definition. A sequence converges to an element ] on average if the norm is in space Theorem 6. If a sequence ) converges uniformly, then it also converges on average. M Let the sequence ()) converge uniformly on the segment [a, b] to the function f(x). This means that for any, for all sufficiently large n, we have Hence, from which our assertion follows. The converse is not true: the sequence () can converge on average to /(x), but not be uniformly convergent. Example. Let us consider the sequence nx It is easy to see that But this convergence is not uniform: there exists e, for example, such that no matter how large n is, on the interval Fourier series for a function with an arbitrary period Complex notation of the Fourier series Fourier series in general orthogonal systems of functions Fourier series in an orthogonal system Minimal property of Fourier coefficients Bessel inequality Parseval equality Closed systems Completeness and closedness of systems and let ) in the orthonormal system b Consider a linear combination where n ^ 1 is a fixed integer, and find the values ​​of the constants for which the integral takes its minimum value. Let us write it in more detail Integrating term by term, due to the orthonormality of the system, we obtain The first two terms on the right side of equality (7) are independent, and the third term is nonnegative. Therefore, the integral (*) takes on a minimum value at ak = sk. The integral is called the root-mean-square approximation of the function f(x) as a linear combination of Tn(x). Thus, the root-mean-square approximation of the function /\ takes on a minimum value when. when Tn(x) is the 71st partial sum of the Fourier series of the function /(x) in the system (. Setting ak = ck, from (7) we obtain Equality (9) is called the Bessel identity. Since its left side is non-negative, then from it Bessel's inequality follows Since i is arbitrary here, Bessel's inequality can be represented in a strengthened form, i.e., for any function /, the series of squared Fourier coefficients of this function in an orthonormal system ) converges. Since the system is orthonormal on the segment [-x, r], then inequality (10) translated into the usual notation of the trigonometric Fourier series gives the relation do valid for any function f(x) with an integrable square. If f2(x) is integrable, then, by virtue of the necessary condition for the convergence of the series on the left side of inequality (11), we obtain that. Parseval's equality For some systems (^n(x)) the inequality sign in formula (10) can be replaced (for all functions f(x) 6 x) by an equals sign. The resulting equality is called the Parseval-Steklov equality (completeness condition). The Bessel identity (9) allows us to write condition (12) in an equivalent form by the space norm 6]. Definition. An orthonormal system ( is called complete in b2[ay b] if any function can be approximated with any accuracy on the average by a linear combination of the form with a sufficiently large number of terms, i.e. if for any function f(x) ∈ b2[a, b\ and for any e > 0 there is a natural number nq and numbers a\, a2y..., such that No. Theorem 7. If the system ) is complete in space by orthonormalization, the Fourier series of any function / in this system converges to f( x) on the average, i.e., according to the norm It can be shown that the trigonometric system is complete in space. This implies the assertion. Theorem 8. If a function /0 its trigonometric Fourier series converges to it on the average. 9.5. closed systems. Completeness and closedness of systems Definition. An orthonormal system of functions \, is called closed if in the space Li\a, b) there is no non-zero function orthogonal to all functions. In the space L2\a, b\ the concepts of completeness and closedness of orthonormal systems coincide. Exercises 1. Expand the function in the Fourier series in the interval (-i-, x) 2. Expand the function in the Fourier series in the interval (-r, r) 3. Expand the function in the Fourier series in the interval (-r, r) 4. Expand in a Fourier series in the interval (-jt, r) function 5. Expand in a Fourier series in the interval (-r, r) the function f(x) = x + x. 6. Expand in a Fourier series in the interval (-jt, r) the function n 7. Expand in a Fourier series in the interval (-r, x) the function / (x) \u003d sin2 x. 8. Expand in a Fourier series in the interval (-m, jt) the function f(x) = y 9. Expand in a Fourier series in the interval (-mm, -k) the function f(x) = | sinx|. 10. Expand in a Fourier series in the interval (-x-, r) the function f(x) = g. 11. Expand in a Fourier series in the interval (-r, r) the function f (x) \u003d sin §. 12. Expand in a Fourier series the function f (x) = n -2x, given in the interval (0, x), continuing it in the interval (-x, 0): a) in an even way; b) in an odd way. 13. Expand in a Fourier series in terms of sines the function / (x) \u003d x2, given in the interval (0, x). 14. Expand in a Fourier series the function / (x) \u003d 3-x, given in the interval (-2,2). 15. Expand in a Fourier series the function f (x) \u003d |x |, given in the interval (-1,1). 16. Expand in a Fourier series in terms of sines the function f (x) \u003d 2x, specified in the interval (0,1).