HW3. Math 499. Spring 2007 Independent studies course Supervisor: Dr Angel R. Pineda, Assistant Professor Mathematics Department California State University, Fullerton

Student: Nasser Abbasi

Problem 1

question: By setting the derivative to zero, find the value of $b_{1}$ that minimizes

MATH

compare with the Fourier coefficient $b_{1}$

Answer:

First I thought it might be a good idea to refresh myself with Fourier series and how it comes about from geometrical perspective. Understanding how a function can be represented using Fourier series can be made easier by making an analogy of how a vector is represented using vector basis.

We know from basic Euclidean geometry, that a vector in the standard 3 dimensional space is written as the sum of its projections on the 3 basis vectors. When we write MATH, then in this case, $a$ is the projection of the vector $\vec{v}$ onto the direction of the base vector $\vec{i}$, and similarly for the numbers $b$ and $c$. The numbers $a,b,c$ are called the coordinates of the vector $\vec{v}$ in this particular coordinate system. The same vector $\vec{v}$ can then have different coordinates values depending on which coordinates system we are making our measurements against, yet it the same exact vector. Hence a vector is invariant under coordinate transformation, but its representation (the coordinates) will change. This diagram below illustrates the above


d2.png

Now that we know how a vector is represented by adding its projections along the direction of each base vector, we are ready to make the switch to a new and exciting world, where vectors become functions and the number of basis instead of being fixed at 3 become very large, in fact, it become infinitely large. This new vector space is called the Hilbert space.

Our goal is to express, or represent a function such as $f\left( x\right) $ using as basis the functions $\sin$ and $\cos$. This will lead us to fourier series representation of a function. One of the issues we need to consider right away, is what basis to use now. There are many families of basis to select. Here we select the sin and cos functions as the basis. As long as each base is orthogonal to each other (using a new definition of what it means to have 2 functions orthogonal to each others). Hence by selecting MATH, and MATH. I.e. MATH for $n$ over all the integers from $0\cdots\infty$. These basis work since any 2 different basis have zero as their dot product using the following definition of dot product, hence they are orthogonal to each others. In Hilbert space, 2 functions are orthogonal to each others if the dot product is zero, defined as follows between the function MATH

MATH

So, now when we are given a function $f\left( x\right) $ and asked for its representation with respect to the coordinate system called the fourier coordinates system, we follow the same idea as with normal vectors, and write

MATH

The above is the same as we did with Euclidean space. We now need to know how to find a projection of a function such as $f\left( x\right) $ onto a base which is itself a function as such MATH. This diagram shows how to do find one such projection of $f\left( x\right) $ onto one base function MATH


d1.png

So the above tells us that the coordinate of $f\left( x\right) $ along MATH is given by MATH

So, let us express $f\left( x\right) $ using the first few coordinates. The first base is MATH, the second base is MATH the third base is MATH, etc... and now for the $\sin$ basis, again we use MATH. Hence we have

MATH

MATH

The standard is to write the above in the order of increasing the frequency of each base, hence we write

MATH

MATH

MATH

MATH

Now, the coordinates are given standard names, the first one is called $a_{0}$, the second is called $a_{1}$ the third is called $b_{1}$, etc.. i.e. the coordinates of $f\left( x\right) $ on the cos basis are called $a_{0},a_{1},\cdots$ and the coordinates of $f\left( x\right) $ on the sin basis are called $b_{1},b_{2},\cdots$. Notice that $b_{0}$ does not exist, since MATH.

So, we write the above as

MATH

MATH

MATH

Now, using the above definition of an inner product, we know how to calculate each of the coordinates $a_{n},b_{n}:~$

MATH

MATH

MATH

Hence we see that

MATH

MATH $\ \ \ \ \ \ n>1$

Similarly for the $b_{n}$ coordinates, we obtain

MATH

etc... Hence we obtain

MATH

We know how to measure the norm of a vector in our standard Euclidean space, so we need to decide how to measure the norm of a function in Hilbert space. For this we use the following definition MATH

I used the above range of integration because for fourier series, the basis used are the MATH.

Now that we have reviewed the fourier series expansion, let us try to answer the actual question.

First, use calculus to answer the question itself:

MATH

Hence for minimum,

$b_{1}=0$

Now the question is asking to compare this to the fourier coefficient $b_{1}$, i.e. with the coordinate $b_{1}$ of the function being expanded. The question did not tell us what is $f\left( x\right) $ itself. But from geormetary we deduce that the problem is minimized the distance between the function $f\left( x\right) $ and the basis, which is MATH in this case. Hence $b_{1}\sin x-\cos x$ is the vector between the function being expressed and the basis MATH Hence MATH in this example, as shown in this diagram


d3.png

Hence, we now need to find $b_{1}$ given that $f\left( x\right) $ is MATH in this example:

MATH

Hence confirmed to be the same.

Problem 2

Show that the complex exponential MATH Note_1 are eigen functions of the convolution operator

MATH for MATH and how representing $f\left( x\right) $ as a linear combination of complex exponential greatly simplifies this equation

Answer: We need to show that by applying the convolution operator on MATH, we obtain a scaled version of MATH, i.e need to show that

MATH

Where $\lambda$ is a scalar.

From the above definition, we obtain

MATH

Using the commutative property of convolution, where MATH, we can write the above as

MATH

But MATH is the fourier transform of the function $k\left( x\right) $, Call this fourier transform MATH

Hence MATH

Where I called MATH as the parameter $\lambda$ since MATH does not depend on $x$ but depends on $n,$ I.e. given the function MATH, we can determine its fourier transform for the specific $n$ provided, and this fourier transform integral, which will evaluate to some value, is then multiplied with $a_{0}$ to obtain the scaling factor by which we scale $e^{inx}$ which is MATH with. Hence we showed that MATH is an eigen function of $g\left( x\right) .$

Now for the second part. If $f\left( x\right) $ can be written as linear combination of complex exponential functions as in MATH, then we write

MATH

But MATH is the fourier transform of MATH, call it $X\left( n\right) ,$ hence the above becomes

MATH

Hence we have replaced the integration operation with a summation operation and we have simplified this equation.

Problem 3

The transpose of a matrix can be defined as the matrix $A^{T}$ such that MATH

This definition generalizes to function operators like the fourier transform MATH

Find the adjoint MATH using the definition above.

Answer:

First, a gemoertric view of a matrix transpose can be illustrated in this diagram


d4.png

Now let us try to apply the above diagram to find the adjoint operator we need. Instead of using the Matrix notation of $A$ and $A^{T}$, we now use the notation of $L$ and $L^{\ast}$ , where here $L^{\ast}$ is the adjoint operator of $L$. Hence we seek to find an operator $L^{\ast}$ such that MATH

We are given what $L$ is, it is the fourier transform, it takes the function $f\left( x\right) $ and generates MATH according to this operation

MATH

For the inner product operation on the space of complex functions over the infinite domain, I will use the following definition

MATH

Hence, applying MATH

MATH

I think now I can write as follwoing and exhange the order of integration

MATH

Hence we see that

MATH

So, the adjoint operator is the inverse fourier transform.