Problem 22 - Advanced Engineering Mathematics

Question

The Gram-Schmidt process for constructing an orthogonal set that was discussed in Section \(7.7\) carries over to a linearly independent set \(\left\\{f_{0}(x), f_{1}(x), f_{2}(x), \ldots\right\\}\) of real-valued functions continuous on an interval \([a, b]\). With the inner product \(\left(f_{n}, \phi_{n}\right)=\int_{a}^{b} f_{n}(x) \phi_{n}(x) d x\), define the functions in the set \(B^{\prime}=\left\\{\phi_{0}(x), \phi_{1}(x), \phi_{2}(x), \ldots\right\\}\) to be $$ \begin{aligned} &\phi_{0}(x)=f_{0}(x) \\ &\phi_{1}(x)=f_{1}(x)-\frac{\left(f_{1}, \phi_{0}\right)}{\left(\phi_{0}, \phi_{0}\right)} \phi_{0}(x) \\ &\phi_{2}(x)=f_{2}(x)-\frac{\left(f_{2}, \phi_{0}\right)}{\left(\phi_{0}, \phi_{0}\right)} \phi_{0}(x)-\frac{\left(f_{2}, \phi_{1}\right)}{\left(\phi_{1}, \phi_{1}\right)} \phi_{1}(x) \end{aligned} $$ and so on. (a) Write out \(\phi_{3}(x)\) in the set. (b) By construction, the set \(B^{\prime}=\left\\{\phi_{0}(x), \phi_{1}(x), \phi_{2}(x), \ldots\right\\}\) is orthogonal on \([a, b] .\) Demonstrate that \(\phi_{0}(x), \phi_{1}(x)\), and \(\phi_{2}(x)\) are mutually orthogonal.

Step-by-Step Solution

Answer

(a) \(\phi_3(x) = f_3(x) - \frac{(f_3, \phi_0)}{(\phi_0, \phi_0)} \phi_0(x) - \frac{(f_3, \phi_1)}{(\phi_1, \phi_1)} \phi_1(x) - \frac{(f_3, \phi_2)}{(\phi_2, \phi_2)} \phi_2(x)\). (b) \(\phi_0(x), \phi_1(x),\) and \(\phi_2(x)\) are orthogonal by construction.

Step 1: Define \(\phi_3(x)\)

The function \(\phi_3(x)\) is derived from the functions \(f_0(x), f_1(x), f_2(x),\) and \(f_3(x)\) using the Gram-Schmidt process.\[\phi_3(x) = f_3(x) - \frac{(f_3, \phi_0)}{(\phi_0, \phi_0)} \phi_0(x) - \frac{(f_3, \phi_1)}{(\phi_1, \phi_1)} \phi_1(x) - \frac{(f_3, \phi_2)}{(\phi_2, \phi_2)} \phi_2(x)\]Calculate each inner product term to obtain \(\phi_3(x)\).

Step 2: Show Orthogonality of \(\phi_0(x)\) and \(\phi_1(x)\)

To show \(\phi_0(x)\) and \(\phi_1(x)\) are orthogonal, calculate their inner product:\[(\phi_0, \phi_1) = \int_a^b \phi_0(x) \left( f_1(x) - \frac{(f_1, \phi_0)}{(\phi_0, \phi_0)} \phi_0(x) \right) dx \]Given \(\phi_0(x) = f_0(x)\), this simplifies as:\[(\phi_0, \phi_1) = (f_0, f_1) - \frac{(f_1, \phi_0)}{(\phi_0, \phi_0)} (\phi_0, \phi_0) = (f_0, f_1) - (f_1, \phi_0)\]Since \(\phi_1\) is constructed to be orthogonal to \(\phi_0\), this equals zero.

Step 3: Show Orthogonality of \(\phi_0(x)\) and \(\phi_2(x)\)

Now, calculate the inner product \((\phi_0, \phi_2)\):\[(\phi_0, \phi_2) = \int_a^b \phi_0(x) \left( f_2(x) - \frac{(f_2, \phi_0)}{(\phi_0, \phi_0)} \phi_0(x) - \frac{(f_2, \phi_1)}{(\phi_1, \phi_1)} \phi_1(x) \right) dx\]Simplifies to:\[(\phi_0, \phi_2) = (f_0, f_2) - (f_2, \phi_0) - \frac{(f_2, \phi_1)}{(\phi_1, \phi_1)}(\phi_0, \phi_1) = 0\]Both terms cancel due to how \(\phi_2\) is constructed, confirming orthogonality.

Step 4: Show Orthogonality of \(\phi_1(x)\) and \(\phi_2(x)\)

Calculate \((\phi_1, \phi_2)\):\[(\phi_1, \phi_2) = \int_a^b \phi_1(x) \left( f_2(x) - \frac{(f_2, \phi_0)}{(\phi_0, \phi_0)} \phi_0(x) - \frac{(f_2, \phi_1)}{(\phi_1, \phi_1)} \phi_1(x) \right) dx\]This simplifies to:\[(\phi_1, \phi_2) = (\phi_1, f_2) - \frac{(f_2, \phi_1)}{(\phi_1, \phi_1)}(\phi_1, \phi_1)\]Since \(\phi_2\) was constructed to be orthogonal to prior \(\phi_n(x)\), this equals zero.

Key Concepts

Orthogonal Functions

In mathematics, orthogonal functions are akin to the notion of perpendicular vectors in geometry. When functions are orthogonal over a specified interval, it means their inner product equals zero. This property of orthogonality ensures that each function provides "unique" information without redundancy.

Features of orthogonal functions include:

• No overlap: Each function does not share components with others in the set, akin to separate directions in space.
• Simplified calculations: Dealing with orthogonal functions often simplifies calculations for series expansions in various applications, including Fourier series.

The Gram-Schmidt process helps in creating orthogonal functions from a set of linearly independent functions. This ensures that given a set of functions, each subsequent function in the sequence is orthogonal to those that preceded it.

Consider functions \(\phi_i(x)\) created from real-valued \(f_i(x)\). If they are orthogonal over the interval \[a, b\], their inner product is computed as follows:\[(\phi_i, \phi_j) = \int_a^b \phi_i(x) \phi_j(x) \, dx = 0 \text{ for } i eq j.\]This criterion guarantees that the functions will not interfere with each other in analytical or numerical processes.

Linear Independence

Linear independence is a critical concept in linear algebra and analysis, pertinent in the context of function spaces like those encountered in the Gram-Schmidt process. A set of functions is linearly independent if no function in the set can be written as a linear combination of the others.

The importance of linear independence can be highlighted in several ways:

• Ensures diversity: It guarantees that each function contributes something new and unique to the span, thereby generating a more comprehensive vector space.
• Basis formation: Only linearly independent function sets can form a basis for a function space, essential for constructing function approximations.

When a set of functions \(\{f_0(x), f_1(x), \dots\}\) is linearly independent, it can serve as a starting point for processes like Gram-Schmidt to create orthogonal functions. If \(c_0, c_1, \ldots, c_n\) are constants with \[c_0 f_0(x) + c_1 f_1(x) + \ldots + c_n f_n(x) = 0,\]for all x in a given domain, then all \(c_i\) must be zero.

This non-triviality condition is a hallmark of linear independence, differentiating these functions from dependent ones that can be expressed in such combinations with non-zero coefficients.

Inner Product

The inner product is a measure of the "dot product" of two functions over a certain interval and is crucial in understanding orthogonality and independence in function spaces. It generalizes the concept of the dot product from finite-dimensional vector spaces to spaces of functions.

Essential characteristics of the inner product are:

• Integral-based calculation: Given functions \(f(x)\) and \(g(x)\) continuous over \[a, b\], their inner product is defined as\[(f, g) = \int_a^b f(x) g(x) \, dx.\]
• Symmetric property: The inner product satisfies \((f, g) = (g, f)\), making it commutative irrespective of the order of functions.
• Linearity: It maintains linearity with respect to both vectors, meaning that distribution and scaling laws hold true within the integration limits.

When applying the Gram-Schmidt process, the inner product plays a crucial role in determining projection factors as functions are orthogonalized. Calculating terms like \((f_k, \phi_j)/(\phi_j, \phi_j)\) ensures each generated function \(\phi(x)\) remains orthogonal to its predecessors, adhering to the properties of orthogonality discussed earlier.