Expand this function into a Fourier series. Fourier series: history and influence of the mathematical mechanism on the development of science

Functions, decomposing them into components. Alternating currents and voltages, displacements, speed and acceleration of crank mechanisms and acoustic waves are typical practical examples of the use of periodic functions in engineering calculations.

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

Standard (=ordinary) 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 using 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 next to Fourier, corresponding to the function f(x). For series (1), the term (a 1 cosx+b 1 sinx) is called the first or fundamental harmonic,

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 =(a 1 2 +b 1 2) 1/2, c n =(a n 2 +b n 2) 1/2 are the amplitudes of the various components, and is equal to a n =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 fundamental harmonic,(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 typically requires an infinite number of terms. 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 into Fourier series.

If the function f(x) is non-periodic, it means that it cannot be expanded into 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, a new function can be constructed by selecting values ​​of f(x) within a certain range and repeating them outside that range at 2π intervals. Since the new function is periodic with period 2π, it can be expanded into 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 in the interval from o to 2π, then outside this interval a periodic function with a period of 2π is constructed (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 a 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 2π range, the same formula of 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 mirror images). Two examples of even functions: y=x2 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., no sine terms) and may include a constant term. Hence,

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 (that is, it does not contain terms with cosines).

Hence,

where are the coefficients of the Fourier series,

Fourier series at half cycle.

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

If you want to get the decomposition Half-cycle Fourier by cosines functions f(x) in the range from 0 to π, then it is necessary to construct an even periodic function. In 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 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 looks like this: in Fig. below. Since we need 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 construct an odd periodic function. In Fig. Below is the function f(x)=x, built on the interval from x=0 to x=π. Since the odd function is symmetrical about the origin, we construct the line CD, as shown in Fig. If we assume that outside the considered interval the resulting sawtooth signal is periodic with a period of 2π, then the final graph has the form shown in Fig. Since we need to obtain the Fourier expansion of 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 a period of 2π to functions with a period of 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 relative 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,

However, more often the above formula results in a dependence on x. Since u=2πx/L, it means du=(2π/L)dx, and the limits of integration are from -L/2 to L/2 instead of - π to π. Consequently, 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,

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

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

For the substitution u=πх/L, the interval from x=0 to x=L corresponds to the interval from u=0 to u=π. Consequently, the function can be expanded into a series only in cosines or only in sines, i.e. V Fourier series at half cycle.

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

Fourier series of periodic functions with period 2π.

The Fourier series allows us 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 examples of the use of periodic functions in engineering calculations.

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

Standard (=ordinary) 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 using 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 next to Fourier, corresponding to the function f(x). For series (1), the term (a 1 cosx+b 1 sinx) is called the first or fundamental harmonic,

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 =(a 1 2 +b 1 2) 1/2, c n =(a n 2 +b n 2) 1/2 are the amplitudes of the various components, and is equal to a n =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 fundamental harmonic,(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 typically requires an infinite number of terms. 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.

If the function f(x) is non-periodic, it means that it cannot be expanded into 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, a new function can be constructed by selecting values ​​of f(x) within a certain range and repeating them outside that range at 2π intervals. Since the new function is periodic with period 2π, it can be expanded into 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 in the interval from o to 2π, then outside this interval a periodic function with a period of 2π is constructed (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 a 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 2π range, the same formula of 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 mirror images). Two examples of even functions: y=x2 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., no sine terms) and may include a constant term. Hence,

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 (that is, it does not contain terms with cosines).

Hence,

where are the coefficients of the Fourier series,

Fourier series at half cycle.

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

If you want to get the decomposition Half-cycle Fourier by cosines functions f(x) in the range from 0 to π, then it is necessary to construct an even periodic function. In 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 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 looks like this: in Fig. below. Since we need to obtain the Fourier expansion in cosines, as before, we calculate the Fourier coefficients a o and a n

If you need to get Fourier half-cycle sine expansion functions f(x) in the range from 0 to π, then it is necessary to construct an odd periodic function. In Fig. Below is the function f(x)=x, built on the interval from x=0 to x=π. Since the odd function is symmetrical about the origin, we construct the line CD, as shown in Fig. If we assume that outside the considered interval the resulting sawtooth signal is periodic with a period of 2π, then the final graph has the form shown in Fig. Since we need to obtain the Fourier expansion of 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 a period of 2π to functions with a period of 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 relative 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

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

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

For the substitution u=πх/L, the interval from x=0 to x=L corresponds to the interval from u=0 to u=π. Consequently, the function can be expanded into a series only in cosines or only in sines, i.e. V Fourier series at half cycle.

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

Fourier series are a representation of an arbitrary function with a specific period in the form of a series. In general, this solution is called the decomposition of an element along an orthogonal basis. Expansion of functions into Fourier series is a fairly powerful tool for solving various problems due to the properties of this transformation during integration, differentiation, as well as shifting expressions by 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 needed for. Meanwhile, this transformation has become quite integrated into our lives. It is used not only by mathematicians, but also by physicists, chemists, doctors, 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 that was ahead of its time.

Man and the Fourier transform

Fourier series are one of the methods (along with analysis and others). This process occurs every time a person hears a sound. Our ear automatically transforms elementary particles in an elastic medium into rows (along the spectrum) of successive volume levels for tones of different heights. Next, the brain turns this data into sounds that are familiar to us. All this happens without our desire or consciousness, on its own, but in order to understand these processes, it will take several years to study higher mathematics.

More about Fourier transform

The Fourier transform can be carried out using analytical, numerical and other methods. Fourier series refer to the numerical method of decomposing any oscillatory processes - from ocean tides and light waves to cycles of solar (and other astronomical objects) activity. Using these mathematical techniques, you can analyze functions, representing any oscillatory processes as a series of sinusoidal components that move from minimum to maximum and back. 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 arising under the influence of thermal, light or electrical energy. Also, Fourier series make it possible to isolate constant components in complex oscillatory signals, making it possible to correctly interpret the experimental observations obtained in medicine, chemistry and astronomy.

Historical 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 used his method to study and explain the mechanisms of thermal conductivity - the spread of heat in solids. Fourier suggested that the initial irregular distribution can be decomposed into simple 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 back. 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 together give the original distribution.

The principle of transformation and the views of contemporaries

The scientist's contemporaries - 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 the Heaviside step: its value is zero to the left of the discontinuity and one to the right. This function describes the dependence of the electric current on a temporary variable when the circuit is closed. Contemporaries of the theory at that time had never encountered a similar situation where a discontinuous expression would be described by a combination of continuous, ordinary functions such as exponential, sine, linear or quadratic.

What confused French mathematicians about Fourier's theory?

After all, if the mathematician was right in his statements, then by summing the infinite trigonometric Fourier series, one can obtain an accurate representation of the step 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 expanded the scope of study of this phenomenon, taking it beyond the study of thermal conductivity. However, most scientists continued to be tormented by the question: “Can the sum of a sinusoidal series converge to the exact value of the discontinuous function?”

Convergence of Fourier series: an example

The question of convergence arises whenever it is necessary to sum infinite series of numbers. To understand this phenomenon, consider a classic example. Will you ever be able to reach the wall if each subsequent step is half the size of the previous one? Let's say you're two meters from your target, the first step takes you to the halfway mark, the next one takes you to the three-quarters mark, and after the fifth you'll have covered almost 97 percent of the way. However, no matter how many steps you take, you will not achieve your intended goal in a strict mathematical sense. Using numerical calculations, it can be proven that it is eventually possible to get as close as a given distance. This proof is equivalent to demonstrating that the sum of one-half, one-fourth, etc. will tend to unity.

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

This issue was raised again at the end of the nineteenth century, when they tried to use Fourier series to predict the intensity of tides. At this time, Lord Kelvin invented an instrument, an analog computing device that allowed military and merchant marine sailors to monitor this natural phenomenon. This mechanism determined sets of phases and amplitudes from a table of tide heights and corresponding time points, carefully measured in a given harbor throughout the year. Each parameter was a sinusoidal component of the tide height expression and was one of the regular components. The measurements were fed into Lord Kelvin's calculating instrument, which synthesized a curve that predicted the height of the water as a function of time for the following year. Very soon similar curves were drawn up for all the harbors of the world.

What if the process is disrupted 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. However, it turned out that this pattern is not observed in cases where the tidal expression that should be synthesized contained a sharp jump, that is, it was discontinuous. If data from a table of time moments is entered into the device, it calculates several Fourier coefficients. The original function is restored thanks to the sinusoidal components (in accordance with the found coefficients). The discrepancy between the original and reconstructed expression can be measured at any point. When carrying out repeated calculations and comparisons, it is clear that the value of the largest error does not decrease. However, they are localized in the region corresponding to the discontinuity point, and at any other point they tend to zero. 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 spikes over a certain interval. In general, Fourier series, if the original function is represented by the result of a real physical measurement, always converge. Questions about the convergence of this process for specific classes of functions led to the emergence of new branches in mathematics, for example, the theory of generalized functions. She is associated with such names as L. Schwartz, J. Mikusinski 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 a region of a single area concentrated in an infinitesimal neighborhood of a point) and the Heaviside “step”. Thanks to this work, Fourier series became applicable to solving equations and problems involving intuitive concepts: point charge, point mass, magnetic dipoles, and 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 heat-insulating material of irregular shape 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, such equations that describe simple classical systems can be easily solved for each individual wave. Fourier showed that simple solutions can also be summed to produce solutions to more complex problems. In mathematical terms, Fourier series are a technique for representing an expression as a sum of harmonics - cosine and sine. Therefore, this analysis is also known as “harmonic analysis”.

Fourier series - an 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 complex form makes it possible to solve not only simple problems that are amenable to the direct application of Newton's laws of 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 established in almost all areas of science and technology. An example is digital audio and video. Its implementation became possible only thanks to a 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, it 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, trigonometric functions of cosine or sine are used as the simplest ones. In this case, Fourier series are called trigonometric, and the solution of such expressions is called harmonic expansion. This method plays an important role in mathematics. First of all, the trigonometric series provides a means for depicting and also studying functions; it is the main apparatus of the theory. In addition, it allows you to solve a number of problems in mathematical physics. Finally, this theory contributed to the development of a number of very important branches of mathematical science (the theory of integrals, the theory of periodic functions). In addition, it served as the starting point for the development of the following functions of a real variable, and also laid the foundation for harmonic analysis.

Fourier series expansion of even and odd functions expansion of a function given on an interval into a series in 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 Minimal property of Fourier coefficients Bessel’s inequality Equality Parseval Closed systems Completeness and closedness of systems

Fourier series expansion of even and odd functions A function f(x), defined on the interval \-1, where I > 0, is called even if the graph of the even function is symmetrical about the ordinate axis. A 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 interval |-jt, jt), since for all x e b) The function is odd, since Fourier series expansion of even and odd functions is expansion of a function given on an interval into a series in sines or cosines Fourier series for a function with an arbitrary period Complex representation of the Fourier series Fourier series for general orthogonal systems of functions Fourier series for an orthogonal system Minimal property of Fourier coefficients Bessel’s inequality Parseval’s equality Closed systems Completeness and closedness of systems c) Function f(x)=x2-x, where does not belong neither to even nor to odd functions, since Let the function f(x), satisfying the conditions of Theorem 1, be even on the interval x|. Then for everyone i.e. /(x) cos nx is an even function, and f(x) sinnx is an odd one. Therefore, the Fourier coefficients of an even function f(x) will be equal. Therefore, the Fourier series of an even function has the form f(x) sin х - an even function. Therefore, we will have Thus, the Fourier series of an odd function has the form Example 1. Expand the function 4 into a Fourier series on the interval -x ^ x ^ n Since this function is even and satisfies the conditions of Theorem 1, then its Fourier series has the form Find the Fourier coefficients. We have Applying integration by parts twice, we obtain that So, 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 sum of the resulting series are given in Fig. Comment. This Fourier series allows us to find the sum of one of the convergent numerical series, namely, for x = 0 we obtain that Example 2. Expand the function /(x) = x into a Fourier series on the interval. 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 B at points x - ±t the sum of the Fourier series does not coincide with the values ​​of the function /(x) = x, since it is equal to. Outside the interval [-*, i-] the sum of the series is a periodic continuation of the function /(x) = x; its graph is shown in Fig. 6. § 6. Expansion of a function given on an interval into a series in sines or cosines Let a bounded piecewise monotonic function / be given on the interval. The values ​​of this function on the interval 0| can be further defined in various ways. For example, you can define a function / on the segment tc] so that /. In this case they say that) “is extended to the segment 0] in an even manner”; its Fourier series will contain only cosines. If the function /(x) is defined on the interval [-l-, mc] so that /(, then the result is an odd function, and then they say that / is “extended to the interval [-*, 0] in an odd way”; in this In this case, the Fourier series will contain only sines. Thus, each bounded piecewise monotonic function /(x) defined on the interval can be expanded into a Fourier series in both sines and cosines. Example 1. Expand the function into a Fourier series: a) by cosines; b) by sines. M This function, with its even and odd continuations into the segment |-x,0) will be bounded and piecewise monotonic. a) Extend /(z) into the segment 0) a) Extend j\x) into the segment (-π,0| in an even manner (Fig. 7), then its Fourier series i will have the form Π = 1 where the Fourier coefficients are equal, respectively for Therefore, b) Let us extend /(z) into 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 period and it can be expanded on the segment into a Fourier series. Returning to the variable x, i.e., setting, we obtain All theorems valid for Fourier series of periodic functions with period 2π , remain valid for periodic functions with an arbitrary period 21. In particular, a sufficient criterion for the decomposability of a function in a Fourier series also remains valid. Example 1. Expand into a Fourier series a periodic function with a period of 21, given on the interval [-/,/] 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 Let us note one important property of periodic functions. Theorem 5. If a function has period T and is integrable, then for any number a the equality m holds. that is, the integral of a segment whose length is equal to the period T has the same value regardless of the position of this segment on the number axis. In fact, 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 Expansion into a Fourier series of even and odd functions, expansion of a function given on an interval into a series in sines or cosines Fourier series for a function with an arbitrary period Complex notation of the Fourier series Fourier series in general orthogonal systems functions Fourier series in an orthogonal system Minimal property of Fourier coefficients Bessel’s inequality Parseval’s equality Closed systems Completeness and closedness of systems Example 2. The function x is periodic with a period Due to the oddness of this function, without calculating integrals, we can state that for any The proven property, in particular, shows 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 into a Fourier series a function given on an interval with a period of 2x (Fig. 11). 4 Let's 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 recording of the Fourier series This section uses some elements of complex analysis (see Chapter XXX, where all actions 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 segment x] it can be represented by a series of the form Using Euler’s formulas Substituting these expressions into series (1) instead of cos πx and sin φx we will have We introduce the following notation Then series (2) will take the form Thus, the Fourier series (1) is represented in complex form (3). Let's find expressions for the coefficients through integrals. We have Similarly, we find The final formulas for с„, с_п and с can be written as follows: . . The coefficients с„ are called the complex Fourier coefficients of the function. For a periodic function with a period), the complex form of the Fourier series will take the form where the coefficients Cn are calculated using the formulas. The convergence of series (3) and (4) is understood as follows: series (3) and (4) are called convergent for a given values ​​if there are limits Example. Expand the period function into a complex Fourier series. This function satisfies sufficient conditions for expansion into a Fourier series. Let us 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 for general orthogonal systems of functions 9.1. Orthogonal systems of functions Let us denote by the set of all (real) functions defined and integrable on the interval [a, 6] with a square, i.e., those for which an integral exists. In particular, all functions f(x) 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. A 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 zero. The integral is understood in the Lebesgue sense. and we call the quantity the norm of the function. If in an orthogonal system for any n we have, then the system of functions is called orthonormal. If the system (y>„(x)) is orthogonal, then the system Example 1. The 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 are orthonormal. Let us introduce the notation that they are orthogonal on the interval (0, f|, but not orthonormal (for I Ф- 2). Since their norms are COS Example 3. Polynomials defined by equality are called Legendre polynomials (polynomials). For n = 0 we have It can be proven , that the functions form an orthonormal system of functions on the interval. Let us show, for example, the orthogonality of the Legendre polynomials. 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. A system of functions (pn(x)) is called orthogonal on the interval (a, b) by an overhang p(x) if: 1) for all n = 1,2,... there are integrals. 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. Having performed 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 zeros of the Bessel function Example 5. Consider the Chebyshev-Hermite polynomials, which can be defined using the equality. Fourier series on the orthogonal system Let there be 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): Multiplying both sides of the last equality by - fixed) and integrating over x from a to 6, in Due to 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 for us now it is the formal interpretation that is important. So, let a function be given. Let us form the numbers c* according to 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(i)). The numbers Cn are called the Fourier coefficients of the function f(x) with respect to this system. The sign ~ in formula (6) only means that the numbers Cn are related to the function f(x) by formula (5) (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. Convergence on average Definition. A sequence converges to the element ] on average if the norm is in the space Theorem 6. If a sequence ) converges uniformly, then it converges on average. M Let the sequence ()) converge uniformly on the interval [a, b] to the function /(x). This means that for everyone, for all sufficiently large n, we have Therefore, from which our statement follows. The converse is not true: the sequence () may converge on average to /(x), but not be uniformly convergent. Example. 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 cosines Fourier series for a function with an arbitrary period Complex representation of the Fourier series Fourier series for general orthogonal systems of functions Fourier series for an orthogonal system Minimal property of Fourier coefficients Bessel’s inequality Parseval’s equality Closed systems Completeness and closedness of systems and let We denote by c* the Fourier coefficients of the function /(x ) by an orthonormal system b Consider a linear combination where n ^ 1 is a fixed integer, and find the values ​​of the constants at which the integral takes a 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 non-negative. Therefore, the integral (*) takes a minimum value at ak = sk. The integral is called the mean square approximation of the function /(x) by a linear combination of Tn(x). Thus, the root mean square approximation of the function /\ takes a minimum value when. when Tn(x) is the 71st partial sum of the Fourier series of the function /(x) over the system (. Setting ak = sk, 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 am here arbitrarily, 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 interval [-x, m], then inequality (10) translated into the usual notation of the trigonometric Fourier series gives the relation do that is valid for any function /(x) with an integrable square. If f2(x) is integrable, then, due to 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 (^„(x)), the inequality sign in formula (10) can be replaced (for all functions f(x) 6 ×) by an equal sign. The resulting equality is called the Parseval-Steklov equality (completeness condition). Bessel's identity (9) allows us to write condition (12) in an equivalent form. Thus, the fulfillment of the completeness condition means that the partial sums Sn(x) of the Fourier series of the function /(x) converge to the function /(x) on average, i.e. according to the norm of space 6]. Definition. An orthonormal system ( is called complete in b2[аy b] if every function can be approximated with any accuracy on average by a linear combination of the form with a sufficiently large number of terms, i.e. if for any function /(x) ∈ b2[a, b\ and for any e > 0 there is a natural number nq and numbers a\, a2y..., such that No From the above reasoning follows Theorem 7. If by orthonormalization the system ) is complete in space, the Fourier series of any function / in this system converges to f( x) on average, i.e. according to the norm. It can be shown that the trigonometric system is complete in space. This implies the statement. Theorem 8. If a function /o its trigonometric Fourier series converges to it in 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 nonzero 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 2 into a Fourier series in the interval (-i-, x) 2. Expand the function into a Fourier series in the interval (-tr, tr) 3. Expand the function 4 into a Fourier series in the interval (-tr, tr) into the Fourier series in the interval (-jt, tr) function 5. Expand the function f(x) = x + x into a Fourier series in the interval (-tr, tr). 6. Expand the function n into a Fourier series in the interval (-jt, tr) 7. Expand the function /(x) = sin2 x into a Fourier series in the interval (-tr, x). 8. Expand the function f(x) = y into a Fourier series in the interval (-tr, jt) 9. Expand the function f(x) = | sin x|. 10. Expand the function f(x) = § into a Fourier series in the interval (-π-, π). 11. Expand the function f(x) = sin § into a Fourier series in the interval (-tr, tr). 12. Expand the function f(x) = n -2x, given in the interval (0, x), into a Fourier series, extending it into the interval (-x, 0): a) in an even manner; b) in an odd way. 13. Expand the function /(x) = x2, given in the interval (0, x), into a Fourier series in sines. 14. Expand the function /(x) = 3, given in the interval (-2,2), into a Fourier series. 15. Expand the function f(x) = |x|, given in the interval (-1,1), into a Fourier series. 16. Expand the function f(x) = 2x, specified in the interval (0,1), into a Fourier series in sines.

Ministry of General and Vocational Education

Sochi State University of Tourism

and resort business

Pedagogical Institute

Faculty of Mathematics

Department of General Mathematics

GRADUATE WORK

Fourier series and their applications

In mathematical physics.

Completed by: 5th year student

signature of full-time education

Specialty 010100

"Mathematics"

Kasperova N.S.

Student ID No. 95471

Scientific supervisor: associate professor, candidate.

technical signature sciences

Pozin P.A.

Sochi, 2000

1. Introduction.

2. The concept of a Fourier series.

2.1. Determination of Fourier series coefficients.

2.2. Integrals of periodic functions.

3. Signs of convergence of Fourier series.

3.1. Examples of expansion of functions in Fourier series.

4. A note on the Fourier series expansion of a periodic function

5. Fourier series for even and odd functions.

6. Fourier series for functions with period 2 l .

7. Fourier series expansion of a non-periodic function.

Introduction.

Jean Baptiste Joseph Fourier - French mathematician, member of the Paris Academy of Sciences (1817).

Fourier's first works related to algebra. Already in lectures of 1796, he presented a theorem on the number of real roots of an algebraic equation lying between given boundaries (published in 1820), named after him; a complete solution to the number of real roots of an algebraic equation was obtained in 1829 by J.S.F. By assault. In 1818, Fourier investigated the question of the conditions for the applicability of the method of numerical solution of equations developed by Newton, not knowing about similar results obtained in 1768 by the French mathematician J.R. Murailem. The result of Fourier's work on numerical methods for solving equations is “Analysis of Definite Equations,” published posthumously in 1831.

Fourier's main area of ​​study was mathematical physics. In 1807 and 1811, he presented his first discoveries on the theory of heat propagation in solids to the Paris Academy of Sciences, and in 1822 he published the famous work “Analytical Theory of Heat,” which played a major role in the subsequent history of mathematics. This is the mathematical theory of thermal conductivity. Due to the generality of the method, this book became the source of all modern methods of mathematical physics. In this work, Fourier derived the differential equation of thermal conductivity and developed the ideas outlined earlier by D. Bernoulli; he developed a method for separating variables (Fourier's method) to solve the heat equation under certain given boundary conditions, which he applied to a number of special cases ( cube, cylinder, etc.). This method is based on the representation of functions by trigonometric Fourier series.

Fourier series have now become a well-developed tool in the theory of partial differential equations for solving boundary value problems.

1. The concept of a Fourier series.(p. 94, Uvarenkov)

Fourier series play an important role in mathematical physics, elasticity theory, electrical engineering, and especially their special case - trigonometric Fourier series.

A trigonometric series is a series of the form

or, symbolically:

where ω, a 0, a 1, …, a n, …, b 0, b 1, …, b n, … are constant numbers (ω>0).

Historically, certain problems in physics have led to the study of such series, for example, the problem of string vibrations (18th century), the problem of regularities in the phenomena of heat conduction, etc. In applications, consideration of trigonometric series , is primarily associated with the task of representing a given movement, described by the equation y = ƒ(χ), in

Thus, we come to the following problem: to find out whether for a given function ƒ(x) on a given interval there exists a series (1) that would converge on this interval to this function. If this is possible, then they say that on this interval the function ƒ(x) is expanded into a trigonometric series.

Series (1) converges at some point x 0, due to the periodicity of the functions

and therefore

2. Determination of series coefficients using Fourier formulas.

Let a periodic function ƒ(x) with period 2π be such that it is represented by a trigonometric series converging to a given function in the interval (-π, π), i.e., is the sum of this series:

Let us assume that the integral of the function on the left side of this equality is equal to the sum of the integrals of the terms of this series. This will be true if we assume that the number series composed of the coefficients of a given trigonometric series is absolutely convergent, i.e., the positive number series converges

Series (1) is majorizable and can be integrated term by term in the interval (-π, π). Let's integrate both sides of equality (2):

Let us separately evaluate each integral appearing on the right-hand side:

Thus,

Estimation of Fourier coefficients.(Bugrov)

Theorem 1. Let the function ƒ(x) of period 2π have a continuous derivative ƒ ( s) (x) order s, satisfying the inequality on the entire real axis:

│ ƒ (s) (x)│≤ M s ; (5)

then the Fourier coefficients of the function ƒ satisfy the inequality

Proof. Integrating by parts and taking into account that

ƒ(-π) = ƒ(π), we have

Integrating the right-hand side of (7) sequentially, taking into account that the derivatives ƒ ΄, …, ƒ (s-1) are continuous and take the same values ​​at points t = -π and t = π, as well as estimate (5), we obtain the first estimate ( 6).

The second estimate (6) is obtained in a similar way.

Theorem 2. For the Fourier coefficients ƒ(x) the following inequality holds:

Proof. We have