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Chapter 1: Problem 1

Let \(\mathcal{D}=\left\\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2}<1\right\\}\) denote the open unit disk in \(\mathbb{R}^{2}\). Let \(C(\mathcal{D})\) be the collection of all complex-valued continuous functions \(f\) : \(\mathcal{D} \rightarrow \mathbb{C}\) defined on \(\mathcal{D}\). For each \(f, g \in C(\mathcal{D})\) set $$ \langle f, g\rangle=\iint_{\mathcal{D}} f(x, y) \overline{g(x, y)} d x d y . $$ For each \(n \in \mathbb{N}\) we define $$ f_{n}(x, y)=(x+i y)^{n} . $$ (a) Prove that \(\langle f, g\rangle\) is an inner product on \(C(\mathcal{D})\). (b) Prove that the sequence of functions \(\left\\{f_{n}\right\\}_{n=0}^{\infty}\) is an infinite orthogonal system on \(C(\mathcal{D})\).

Short Answer

Expert verified
(a) Verify linearity, conjugate symmetry and positivity properties. (b) Functions \(f_n(x, y) = (x + iy)^n\) are orthogonal over the unit disk because integrals of mixed terms vanish.

Step by step solution

01

Verify Linearity of First Argument

To show that \(\langle f, g \rangle\) is an inner product, we begin by verifying linearity in the first argument. Consider \( f_1, f_2 \in C(\mathcal{D}) \) and scalar \( \alpha \in \mathbb{C} \). We need to show that \( \langle \alpha f_1 + f_2, g \rangle = \alpha \langle f_1, g \rangle + \langle f_2, g \rangle \). By definition,\[ \langle \alpha f_1 + f_2, g \rangle = \iint_{\mathcal{D}} (\alpha f_1(x, y) + f_2(x, y)) \overline{g(x, y)} \, dx \, dy \]. By linearity of integration, this equals\[ \alpha \iint_{\mathcal{D}} f_1(x, y) \overline{g(x, y)} \, dx \, dy + \iint_{\mathcal{D}} f_2(x, y) \overline{g(x, y)} \, dx \, dy = \alpha \langle f_1, g \rangle + \langle f_2, g \rangle \].
02

Verify Conjugate Symmetry

We must show that \( \langle f, g \rangle = \overline{\langle g, f \rangle} \) for all \( f, g \in C(\mathcal{D}) \). Start with:\[ \langle f, g \rangle = \iint_{\mathcal{D}} f(x, y) \overline{g(x, y)} \, dx \, dy \]. Since \( \overline{\langle g, f \rangle} = \overline{\iint_{\mathcal{D}} g(x, y) \overline{f(x, y)} \, dx \, dy} \), by properties of complex conjugation and integration, this equals\[ \iint_{\mathcal{D}} \overline{g(x, y)} f(x, y) \, dx \, dy = \langle f, g \rangle \].
03

Verify Positivity

We need to prove that \( \langle f, f \rangle \geq 0 \) for any \( f \in C(\mathcal{D}) \), and \( \langle f, f \rangle = 0 \) if and only if \( f = 0 \). For positivity, notice that \[ \langle f, f \rangle = \iint_{\mathcal{D}} |f(x, y)|^2 \, dx \, dy \]. The expression \( |f(x, y)|^2 \) is non-negative, so the integral is non-negative. If \( \langle f, f \rangle = 0 \), then \( |f(x, y)|^2 = 0 \) almost everywhere on \( \mathcal{D} \), implying \( f(x, y) = 0 \) everywhere due to continuity.
04

Show Orthogonality of \(f_n\)

Now, consider the functions \( f_n(x, y) = (x + iy)^n \). To prove orthogonality, compute \( \langle f_n, f_m \rangle \) with \( n eq m \).\[ \langle f_n, f_m \rangle = \iint_{\mathcal{D}} (x + iy)^n \overline{(x + iy)^m} \, dx \, dy = \iint_{\mathcal{D}} (x + iy)^{n} (x - iy)^m \, dx \, dy = \iint_{\mathcal{D}} (x^2 + y^2)^n (x^2 + y^2)^{m/2} e^{i(n-m)\theta} \, dx \, dy \].Rewriting in polar coordinates \( x = r \cos \theta, \) and \( y = r \sin \theta \), the modulus \( r^{n+m} \) integrates out over the unit disk, and \( \int_{0}^{2\pi} e^{i(n-m)\theta} d\theta = 0 \) if \( n eq m \). Hence \( \langle f_n, f_m \rangle = 0 \).
05

Establish Orthogonality Basis Extension

To verify the orthogonal system into an orthonormal system, check if \( \|f_n\| = \sqrt{\langle f_n, f_n \rangle} \). In polar form, \[ \langle f_n, f_n \rangle = \iint_0^{2\pi} \int_0^{1} r^{2n} r \cdot e^{i\cdot0} \cdot r \, dr \, d\theta = \int_0^{2\pi} d\theta \int_0^{1} r^{2n+1} \, dr \]. Evaluate the radial part: \[ \int_0^{1} r^{2n+1} \, dr = \frac{1}{2n+2} \]. The angular part contributes \( 2\pi \), showing \( \|f_n\|^2 = \frac{2\pi}{2n+2} \). Divide by norm to normalize \(f_n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Functions
Complex functions are a fundamental component of Fourier Analysis, and they come into play significantly when dealing with continuous functions on the unit disk, \( \mathcal{D} \). A complex function is a function that takes a complex number as input and outputs another complex number. In our context, we look at functions \( f: \mathcal{D} \rightarrow \mathbb{C} \), meaning that they are defined on the unit disk in the real plane and output values in the complex plane.
  • The notation \( (x, y) \rightarrow f(x, y) \) transforms a two-dimensional real input into a complex output.
  • Complex conjugation often appears in calculations to maintain certain algebraic properties, such as symmetry. For any complex number \( z = a + ib \), its conjugate is \( \overline{z} = a - ib \).
  • In integrations, conjugation ensures the symmetry holds true, which is crucial in verifying properties of inner products.
Understanding complex functions can seem daunting, but by visualizing them as transformations from one plane to another, they become more tangible. The core advantage of using complex functions in Fourier analysis is their ability to handle oscillatory behaviors efficiently, making them invaluable in signal processing and physics.
Inner Product Space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This structure allows for a generalized version of the dot product of vectors, and it can introduce concepts like length and angles into the space.
  • For our exercise, the inner product is defined over the functions on the unit disk \( \mathcal{D} \) as \( \langle f, g \rangle = \iint_{\mathcal{D}} f(x, y) \overline{g(x, y)} dx \, dy \).
  • This definition satisfies several properties crucial for it to be considered an inner product: linearity in the first argument, conjugate symmetry, and positivity.
  • Linearity ensures that the operation respects addition and scalar multiplication, a necessary trait for vector-like behavior.
  • Conjugate symmetry ensures that swapping the functions involves taking the conjugate of the result, maintaining the overall structure and behavior of the inner product.
  • Lastly, positivity guarantees that the inner product of any function with itself is non-negative, providing a sense of `magnitude` or `norm` for functions.
Understanding the inner product space forms the backbone for discussions of orthogonality and other advanced topics.
Orthogonal Functions
Orthogonal functions serve as crucial tools in functions spaces, providing a basis upon which functions can be decomposed similarly to how vectors can be described with orthogonal axes.
  • In the unit disk \( \mathcal{D} \), the sequence \( \{f_n\} = (x + iy)^n \) is proven to be orthogonal via the inner product definition.
  • Orthogonality implies \( \langle f_n, f_m \rangle = 0 \) for any \( n eq m \), essentially meaning these functions do not ``overcome" one another, maintaining independence.
  • This property is valuable because it allows us to write any function usually in the space as a sum of these basis functions without redundancy.
  • A system of orthogonal functions can even be extended to be orthonormal when each function has a norm of one, further simplifying expansions in practical computations.
Orthogonal functions greatly simplify problems in physics and engineering, turning complex problems manageable by breaking them into simpler, independent parts.
Unit Disk in \( \mathbb{R}^2 \)
The unit disk, \( \mathcal{D} \), refers to a set of points \((x, y)\) in the real plane \( \mathbb{R}^2 \) where \( x^2 + y^2 < 1 \). This is essentially the interior of a circle with radius one centered at the origin.
  • Being an open set means it doesn't include the boundary where \( x^2 + y^2 = 1 \).
  • Defining functions over \( \mathcal{D} \) allows for a natural environment for studying complex functions since \( \mathbb{R}^2 \) easily correlates to the complex plane via \( x + iy \).
  • Integrals over \( \mathcal{D} \) are often best handled in polar coordinates, transforming the Cartesian system \( (x, y) \) into \( (r, \theta) \), simplifying many calculations, especially those involving radial symmetry.
  • This geometric viewpoint helps us grasp the nature of convergence and behavior of functions within the disk.
The unit disk is thus a simple yet powerful concept, providing a foundational backdrop for more advanced mathematical and physical explorations involving complex functions.

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Most popular questions from this chapter

For each \(f, g \in C[-1,1]\) we define $$ \langle f, g\rangle=\int_{-1}^{1} \frac{f(x) \overline{g(x)}}{\sqrt{1-x^{2}}} d x . $$ (a) Prove that this is an inner product on \(C[-1,1]\). (b) For each non-negative integer \(n\), define the function \(T_{n}\) in \(C[-1,1]\) by $$ T_{n}(x)=\cos (n \arccos x) . $$ The function \(T_{n}\) is called the Chebyshev polynomial of degree \(n\). Prove that the sequence of functions \(\left\\{T_{n}\right\\}_{n=1}^{\infty}\) is an infinite orthonormal system on \(C[-1,1]\). (c) By substituting \(\theta=\arccos x\), prove that for each \(n, T_{n}(x)\) is in fact a polynomial of degree \(n\) in \(x\).

Let \(V=C^{2}[-\pi, \pi]\) be the space of real-valued twice continuously differentiable functions defined on the interval \([-\pi, \pi]\). Set $$ \langle f, g\rangle=f(-\pi) g(-\pi)+\int_{-\pi}^{\pi} f^{\prime \prime}(x) g^{\prime \prime}(x) d x . $$ Is this an inner product on \(V\) ?

Find a sequence of functions in the space \(C[0, \infty)\) which converges uniformly to the zero function on the interval \([0, \infty)\), but does not converge to the zero function in the norm $$ \|f\|=\left(\int_{0}^{\infty}|f(x)|^{2} d x\right)^{\frac{1}{2}} $$

Let \(V\) be an inner product space, \(u, v \in V, u, v \neq \overrightarrow{0}\). (a) Show that \(\langle u, v\rangle=\|u\| \cdot\|v\|\) if and only if \(u=a v\) for some \(a \in \mathbb{C}\). (b) Show that \(\|u+v\|=\|u\|+\|v\|\) if and only if \(u=a v\) for some \(a \geq 0\).

Let \(N\) be a natural number. For each \(m=0,1, \ldots, N-1\), we define a vector in \(\mathbb{C}^{N}\) by $$ u_{m}=\left(1, e^{\frac{-2 \pi i m}{N}}, e^{\frac{-4 \pi i m}{N}}, \ldots, e^{\frac{-2(N-1) \pi i m}{N}}\right) . $$ Prove that: (a) The system of vectors \(\left\\{u_{m}\right\\}_{m=0}^{N-1}\) is an orthogonal system in \(\mathbb{C}^{N}\) with the standard inner product. (b) For each \(m=0,1, \ldots, N-1,\left\|u_{m}\right\|^{2}=N\). (c) For every \(u \in \mathbb{C}^{N}\), we have \(u=\frac{1}{N} \sum_{m=0}^{N-1}\left\langle u, u_{m}\right\rangle u_{m}\).
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