[BOŞ_SES] Hello. In the previous section, we saw algebraic spaces. So these were the elements in the vector space. These spaces were finite size. Now, using the same principle, infinite dimensional They may be interested in the function space. Elements of functions here; and most important of these spaces the application of Fourier series. Fourier series digital revolution we are in today, we say, so digital communications, digital recording of DVDs and CDs, In the laboratory, the vibration after an earthquake Saving the vibration of molecules in the laboratory, television and voice communications, the digital revolution, Thanks to the digital revolution is happening. We will see this and will also be an important tool. Taylor series up to now we have seen and what the Taylor series How useful it is because as we observe exponential functions, Beyond that, such as algebraic and trigonometric functions It helped us to understand functions. Fourier series, but in a slightly different context, The function may seem complicated at first, many still How to tell be interpreted. The program is as follows: How the plane After starting in three-dimensional space we saw in n-dimensional space, We see the process from there, vectors, So these impressions component of this vector in space We have learned to identify where we will see similar processes. Sine and cosine functions for our business in the Fourier series, functions form the basis of this work. Similarly, the sine and cosine functions with complex Euler We see it as we can produce exponential function. After reviewing them, how they produced the series with them After seeing the concept of fast Fourier series, we will see that this is very important. This digital revolution we call today the most important It is an essential element of the great revolution in communication tools. Then, as we always do a variety of applications We show them how to do a little more of the account understand and calculate things such as the end to see what you get. Yet a series with some subtleties, such functions as may be things regardless of discontinuities, which series which function as this series infinite series There are series, we'll see how they are closer to them. We saw a bit of the Taylor series, as I said before, I hope that everyone knows how to use it and be very useful. Comparing them with Fourier series Both series also believe that we can better understand. Now let's move on to our topic. We knew the plane, we take two vectors when the vector x the vector y I collect kenal line of work, we were getting a vector to the edge of our rules. This n-dimensional space to be easily able to generalize in terms of components. The sum of the function likewise when given two functions x function 1f, 1g we know that y is a function of how they collected. Moreover, we have seen can generate a linear space of these functions in Chapter 4. Ie the combination of permanent partial discontinuities may be functions form a linear space space. Again, be a linear, secondary mania may be space There was a process that is to hit a car with any number of vectors. This is equivalent to a function in any number of function space with a car multiply and obtain a new function. As you can see there is exactly one to one correlation. Yet in algebraic space, We define the inner product in an n-dimensional space. Inner product of the vector inner product of x and y, each of the opposing It consisted of the sum of the products of that component. So we have a product and there is also a collection; and also the x and y internal by multiplying the inner product of x and y, we know that there are equal. Function spaces in the same situation. Here we show a colon and a parentheses process we show the point. This inner product of functions f and g, the inner product. This is followed by a product and the same here as in a collection, but in vector algebra, algebraic elements in these indicators, j index of one, two, three, four separate indexes. However, the function of x can be interpreted as an indication of the fact that, the only difference being that x continuously. Collection becomes an integral time we know that E is continuous. So again as a concept and a product collection, product and here but because it is constantly collecting x an integral collection here. Still it does not change the result by changing the sequence f g's. These are all real precious, precious complex We will see the function and importance. Again we define the length of a vector in n-dimensional space. Of course the plane and space you measure the length of a ruler in the sense of genuine may have a length, the length of the vector, of course, mean size generally n-dimensional a space because we can not talk about the length of this vector in the sense that, within the meaning of the plane, because we have the size, We do not live in a big three-dimensional space. We describe here the length of the vector with the inner product. I highly recommend you look at the Hilbert space section, If your mind right away in the previous section. Length of a function in the function we need to say it in quotation marks, the length of the function is not something to be measured in millimeters to centimeters, but that, As a measure of this inner product again with it, the inner product of its own Instead you get here for the game, we see that the integral of f frame. As you can see in the plane or on the size, function in space with algebraic space of finite size There are quite literal equivalence. We show how these vectors now? Let's start again from three dimensions. We say M three, three-dimensional Euclidean space. Here we take the floor ijk unit vectors. A vector of just anyone can show these terms. This is the vector v1, v2, v3 to say, how do we do? V If it crossed i'yl to have good addition here i v1, The size of length i i i i'yl because this unit is multiplied dimensional It will be an interest; but i and j are perpendicular to each other, we know that, here come here zero zero income, thus multiplying the basis vectors of the vector v v1. Similar to the third jar this time multiplied by v2 v, it is obtained by multiplying a K component. It is completely context, equivalent but infinite-dimensional concept any function space that i, j, k of the function It can be generalized to the space and the infinite number of f we multiply by j When we stood on the same vector as V1 have to see where the base functions, CJ is going on here, we call this the equivalent of v1, v2, but this forever so he can go; Coming equivalent multiplied by the base function. This c j those v1, v2, v3 scope. plug on the i, j, k, but it was an expansion of the scope. Of course, there are infinitely many of them. as well as the plug of the car. Again, the two vectors If the zero vector inner product, they were saying to each other. We know this feature in fact, j k, j to hit the i'yl to zero, j'yl to hit the zero k, k i hit the space because zero I know that the length of the j-long intervening times the cosine of the angle. E i, j, k triple perpendicular to each other. The difference cosine angle of 90 degrees to that, The cosine of 90 degrees, we know this feature because it is zero. The same definition applies function in space. So the function f k, f j function, these different functions We will see concrete examples; When we receive the inner product, here it means the inner product integration. We call these two functions is zero anal Or, other partners. Of course, i, j, k truly geometrically as concrete, We see 90 degrees. If we go this the size he disappears. Here infinite size, he lost the concept of this 90 degrees, but we accept the definition that is still upright. V How the first base component with a bit now We have created the multiplication of vectors; f, components with p j are that there is going components such as we stood on the base function. Let them now what? We stood there with this function in bringing the base vector e. Here likewise, summation over j, multiply any fi k, Let the inner product. Take the inner product of, in this collection You can enter for the product, as in the same vector. Hani vectors inner product axioms in it, If you remember propositions, change of this order is always possible. So do the collection after multiplication here before, Instead of taking the inner product before taking an inner product then get hit with a car collection. So the only difference is the order of the transactions between the two sides. Already the inner products, linear space The result of this change in the order of importance. So it is going to return the following: a van vector, v1 v2, v3 components, i, j, k we obtain when we gather hit with v. When we say let's find these components, multiply by i'm after. Here V1 have to, I'll be fine with the product, will be multiplied by j i, I will be multiplied by k. These k is multiplied by i, j multiplied by i is zero, multiplied by i this resets reduced because one and only remained V1 have. The same process is applicable here. If this f j, f team just fine, j, k, if it is a steep as in jewelry, all these domestic products, j how people, i k is multiplied by zero if they will be giving zero. Now it's just now that is equal representation for s j f j k as we hit this f'yl, wherein just one of these product will be different from zero, because they are perpendicular to each other. Just when k, j equal Or, j is equal to k, where the term remains, the others always goes zero zero, j multiplied by how where i, j and k is multiplied by zero. Now I earn here is very important. There was also the vector in space, three unknown directly down to the right and now she is an unknown unknown unraveling v1. Here, too, the unknown infinite, s1, s2, even minus infinity to plus infinity can go up, the infinite unknown, only one remains unknown. Here comes the important feature of this exceptional here. This infinite instead of solving the unknown, when they have other remains unknown. We solve k scene immediately. All this for only stayed this term is zero, k'yl fi fi k when we divide the left side of the inner product, we find k c. If we write it slightly differently, the base, the lower inner product, a norm, The fi c function; norm, criterion, length, If you use what terms; It shows that the frame. But this i, j, k and length of these functions as if one is, see how, exactly as it is here, how straightforward. Here, because the length a i, This term comes as a more simple and direct v1 equal to the inner product. Here, too, the same ck; because it is the equivalent of v1, the components thereof, The components of the function; This can be achieved with internal slammed. They are more open we wrote above, an integral inner product. An integrally defined between a and b. By multiplying the plug at the base of their own, so integral to the frame. If the integral of the square be anyway if taking such a simplification, It is only obtained with a domestic hit. This completely, vectors in the plane, you see, or vector in n-dimensional space, but the finite size by the processes in the vector is the same as the operation of the components present. ck equivalent of a vector component, function components. We have seen them, if these functions are complex valued inner product is defined as follows: the first function complex You stood conjugate with the second function. Here, too, when you change the order of the inner product It appears to be equivalent. We got to be so mixing in our minds, we want to calculate the length of time because otherwise, if you have not received the full equivalent f we were to leave the square, but for a complex function f frame if there will be a complex function, This time length, will be a complex value. But when we take this complex conjugate, When we found the length of f, f'll get the f multiplied by the complex conjugate, that we will achieve integration of this product. This means the square of the absolute value. Absolute value of a real number, thus defining in this way, norm, comes the equivalent of removing the length of a real number. If a complex function or function is enabled, I want to say sorry Conversely, if the base is a complex function, f there may be cases where f is complex, but the real acceptance Let's face it; may even complex does not change these properties. fi k'yl domestic product of all these things we say is true more What we have said before for the real fun, the only difference here this base The function is to multiply the complex conjugate. Or all the same everything. Complex in the denominator again The plug will be multiplied with the conjugate to itself, the square of the absolute value, so there will be a floating-point number and before that will arise as a generalization we find the real function. The opening series of complex functions with an important job and really this technology revolution can be provided in this way, it could be achieved. Now let's move on to more concrete. Now before the sine and cosine functions in the previous section We have seen in the application, that I gözlemleyel to each other. We are doing this as follows: a cosine k x, k multiplied by a cosine integral of the base, to an A to B. P, wherein the length of motion cycle or 2p, This function is repeated for the release outgoing functions, There is a period; We say it's semi-p, The range of call per half. However, in motion, periodicity of 2p. At trigonometric needed feature: Cosine alfa'yl cosine When you hit beta, it becomes collection you can see in since high school. Why are we doing this? In order to account because it is integral. We know what it is but the product of the integration of their individual integration We do not know yourself. Sine in a similar equivalent, have identity. Multiplied by the cosine of the sinus. These properties using these identities, cf. sinüs'l to see the sine of the product, but there's k'yl k base. Now I want to talk about a particular situation. The following terms: Let's make the simplest a'yl to b. Said a merger between minus. Then we do the random general a and b. In this case b minus a, A minus is a minus pin turns out to be one of the two will be for two. Hence our terms simpler. We are here P'l freed from pI term. Let's state a more simple, then we look at the general said. Here, multiplied by the sinus sine, cosine we know that the difference of. We know the sum of the cosine Kosinüsl the cosine of the product. We know that the sine from cosine total of sinuses. We are using a previous features here. Now that the integration can be done. Even if we do not make this distinction never left. But we know that the cosine of the sine integral. Cosine of the integral sine here. There minus k k base in one here because it sounded negative alpha beta If you look at the previous formula; or alpha plus beta sounded. That's where all this alpha k p x, p x beta k base. k and k prime numbers different from each other. As you can see M, k k minus the base, one for each integer is a whole number. That is exactly the number pi. If you think that the sine function on a circle, sinus floors of pi is zero. Therefore, the upper limit of the zero in the zero lower bound. Similarly, this k is a positive integer k base. It is also the upper limit, the lower limit is zero. Similarly, the cosine is happening here is similar. There's a small difference, but it turns out again zero results. Because the first term alone is not zero. There was zero at each end of each term. Cosine of a minus or a plus can, K plus K or k, depending on whether single or double base of the old k base. But minus the lower boundary value minus the value of the same value again ... e upper boundary value minus value, Similar sub-zero marginal value data here. Now we say concretely that such a cosine and sine We see that such characteristics of the function. This will come in handy. How could i, j, k is crucial to be upright in space, these functions the function and infinite space in them to each other in this way that, Let's say experimentally, we have observed happening. If we exceed the general situation, b, and where that is not necessarily a negative and it's also, a and b are randomly selected, may also show the same features here. I do not want to get into the details, here are the details. Because K and K base will be a whole number, k and k positive integer base and it will be this operations yapınca again, likewise the product of this sinus sinus, The kosinüsl cosine, sine the indicators where the cosine of the product, The coefficients k and k are different from each other when the base zero We prove that, in the general case. a and b may be any number. Now we thought it was different until the k'yl k base. But K and K may be equal to the base. And may be zero, it may be nonzero. Zero is a special case, let it be zero, but they are equal. See then here are also integral sinus For the sine of the same variables that are sine square. Here it will be cosine square. Here's the thing, so no equivalent to it. Trigonometric cosine square again cosine of an angle in terms plus double as identity, We know that's a minus cosine sine square. Indeed, we know that we collect both a cosine squared plus sine-squared is. See here plus cosine term missing two alfayl to destroy each other. A half here, half there also have collected as income. So this is going to provide a simple. This identity now, k p x over cosine squared If we return, because when the base where k and k are equal, cosine and sine-squared squares sounded the same argument, k p divided by p times x. Here's one very easy integration. b minus a cosine is a full overthrown in this area hence providing added value comes equal to the negative value. If you want something so integral to the cosine sine he wrote it, As you can show more detail here. But it is also able to find a hunch it would be zero, we see. Because veyahut cosine sine, cosine when you get here, plus it takes a period of negative space takes another. Period where you get taken and will take. Therefore, integration of the fall term, here minus cosine of both falls and a divide b minus two air data. We also define two pin this period he also income equal to half the period. Latest particular case is not equal to k k base but on top If we look at the situation as a zero, this is a special case thoroughly. K we put zero in the first integral above, It is the integral of the cosine of the one because it is a zero. B until one of the integral of a. M, which of course is very simple as x will, integration of one. x will be given b, will give a, two difference of 2p was also a B minus. Now here's the important one, the thing to be aware of cosine square pan while the sine square, while a pin in this most special cases, small base equal, here equal to the small base; but also equal to zero. Because around here as to whether the release was constant No part of this function takes each other, this gives 2p. Now we have seen that happening in that summary. We discovered a treasure. So much so that this function, endless one, all of them to each other. So i, j, k instead of three, we have a very rich, There are infinitely many functions and function space. That will give us an important indication possibility. And cosine, sine While there, When we combine these complex It becomes exponential. We know that e to the i theta cosine of theta plus i sine of theta times. Thus the features that need to be in the exponential function to kosinüsl. We feel that with foreboding. But this complex function. I minus the square root of one here. Virtual unit numbers. This means taking the inner product functions for the complex, eşleneg get the first of the complex. I am writing here kbps, instead I'm writing kbps j. Coming in for a jar. Instead we get the first good in the complex are putting eşleneg minus. But as you can see there's a lot in common. p x split pin joint, ie the joint. When we bring together the IPX split the pie j k minus is coming. j be an integer, an integer k. This is integral to get easier, you can look to them as a theta. Again, the same tactics, comes with a coefficient in front. We reckon that a and b. Again, it's the value of a and b, the difference between thinking that the 2p, thinking the pie in the denominator, here he was given details, it turns out that it also be zero. So we're now gained a new treasure it as an advantage because There: Yes, you can safely go back because of the complexity. But we see with a single business function. However, in both sine and cosine were needed earlier. Now that you have identified them, we see that they are in each other. Here You've probably encountered this so-called fast Fourier transformation, Fast Fourier Transform is called in English. FFT is defined as the letters from its initials. Here in these CDs, DVDs, high-resolution television communications, telephone, digital phone, such as audio and video in car This complex Fourier representation, This complex will provide great advantages base, we saw it. For now, we did quite a process. Now, let's break here. We have seen that until now, cosine and sine orthogonal to each other. And then j k equivalent in function space. Now we cut here today. So we talked to quite a few above but supremely simple generalization and it should remain in our minds, the cosine sine complex exponential function and form a base assembly with each other. Our message is extremely short and simple. Of course we got to see their application. Take a break for it. Both will have the opportunity to spend a reviewer, as well as natural a part replaced. Bye now. I hope in sight until you have looked a bit more. Because there is a lot of hard work. Maybe some new issues for some of you.