Question

Suppose \(\phi_{1}, \phi_{2}, \ldots, \phi_{m}\) are orthogonal on \([a, b]\) and $$ \int_{a}^{b} \phi_{n}^{2}(x) d x \neq 0, \quad n=1,2, \ldots, m $$ If \(a_{1}, a_{2}, \ldots, a_{m}\) are arbitrary real numbers, define $$ P_{m}=a_{1} \phi_{1}+a_{2} \phi_{2}+\cdots+a_{m} \phi_{m} $$ Let $$ F_{m}=c_{1} \phi_{1}+c_{2} \phi_{2}+\cdots+c_{m} \phi_{m} $$ where $$ c_{n}=\frac{\int_{a}^{b} f(x) \phi_{n}(x) d x}{\int_{a}^{b} \phi_{n}^{2}(x) d x} $$ that is, \(c_{1}, c_{2}, \ldots, c_{m}\) are Fourier coefficients of \(f\). (a) Show that $$ \int_{a}^{b}\left(f(x)-F_{m}(x)\right) \phi_{n}(x) d x=0, \quad n=1,2, \ldots, m $$ (b) Show that $$ \int_{a}^{b}\left(f(x)-F_{m}(x)\right)^{2} d x \leq \int_{a}^{b}\left(f(x)-P_{m}(x)\right)^{2} d x $$ with equality if and only if \(a_{n}=c_{n}, n=1,2, \ldots, m\) (c) Show that $$ \int_{a}^{b}\left(f(x)-F_{m}(x)\right)^{2} d x=\int_{a}^{b} f^{2}(x) d x-\sum_{n=1}^{m} c_{n}^{2} \int_{a}^{b} \phi_{n}^{2} d x $$ (d) Conclude from (c) that $$ \sum_{n=1}^{m} c_{n}^{2} \int_{a}^{b} \phi_{n}^{2}(x) d x \leq \int_{a}^{b} f^{2}(x) d x $$

Short answer

Question: Prove the following properties of orthogonal functions and their Fourier coefficients: (a) Show that the error term \(f(x)-F_m(x)\) is orthogonal to all the orthogonal functions \(\phi_n(x)\). (b) Demonstrate the least square property of the Fourier approximations. (c) Show the relationship between the square error term and the Fourier coefficients. (d) Establish a bound for the sum of the square of the Fourier coefficients. Answer: (a) The orthogonality between the error term and the orthogonal functions was proven by showing that the integral \(\int_{a}^{b}\left(f(x)-F_{m}(x)\right) \phi_{n}(x) d x\) is always zero for all \(n=1,2,\ldots,m\). (b) The least square property of the Fourier approximation was proven by showing that the only way the square error term difference between \(f(x)-F_{m}(x)\) and \(f(x)-P_{m}(x)\) becomes zero is if \(a_n=c_n\) for all \(n=1,2,\ldots,m\). (c) The relationship between the square error term and the Fourier coefficients was shown to be \(\int_{a}^{b}\left(f(x)-F_{m}(x)\right)^2 dx = \int_{a}^{b}f^2(x)dx -\sum_{n=1}^{m} c_n^2 \int_{a}^{b} \phi_n^2(x)dx\) (d) The bound for the sum of the square of the Fourier coefficients was proven to be \(\sum_{n=1}^{m} c_n^2 \int_{a}^{b} \phi_n^2(x) dx \leq \int_{a}^{b} f^2(x) dx\).

Step-by-step solution

(Compute the integral of the product)

To show the orthogonality between \(f(x)-F_m(x)\) and \(\phi_n(x)\), we need to compute the following integral for \(n=1,2,\ldots,m\): $$ \int_{a}^{b}\left(f(x)-F_{m}(x)\right) \phi_{n}(x) d x= \int_{a}^{b}\left[ f(x) - (c_1\phi_1(x) + c_2\phi_2(x)+ \cdots + c_m\phi_m(x)) \right] \phi_{n}(x) d x $$

(Expand the integrand and split the integral)

Expanding the integrand and using the linearity of the integral, we obtain $$ \int_{a}^{b} f(x)\phi_{n}(x) dx - \int_{a}^{b} \left(c_1\phi_1(x)\phi_{n}(x) + c_2\phi_2(x)\phi_{n}(x)+ \cdots + c_m\phi_m(x)\phi_{n}(x)\right) dx $$

(Use orthogonality of functions)

Now, recalling that the functions are orthogonal, we have $$ \int_{a}^{b} \phi_i(x)\phi_{j}(x) dx = 0, \quad \textrm{for} \ i \neq j $$ Hence, in the expression above, all integrals vanish except for the one with \(i=n\): $$ \int_{a}^{b} f(x)\phi_{n}(x) dx - c_n \int_{a}^{b} \phi_{n}(x)^{2} dx $$

(Substitute the definition of Fourier coefficients)

Finally, by using the definition of the Fourier coefficients (\(c_n\)), we have $$ \int_a^b f(x)\phi_n(x) dx - \frac{\int_a^b f(x) \phi_n(x) dx}{\int_a^b \phi_n^2(x) dx} \int_{a}^{b} \phi_{n}(x)^2 dx= 0 $$ So, the orthogonality between the error term and the orthogonal functions is proven. #b) Least square property of Fourier approximation#

(Compute the square error term difference)

We first compute the difference between the square error terms: $$ \int_{a}^{b}\left(f(x)-F_{m}(x)\right)^2 dx - \int_{a}^{b}\left(f(x)-P_{m}(x)\right)^2 dx $$

(Expand the square error terms)

We expand both square error terms: $$ \int_{a}^{b} \left[f^2(x) - 2f(x)F_m(x) + F_m^2(x)\right]dx - \int_{a}^{b} \left[f^2(x) - 2f(x)P_m(x) + P_m^2(x)\right]dx $$

(Combine the integrals and simplify)

Combine the two integrals and simplify the expression: $$ \int_{a}^{b} \left[2f(x)(P_m(x)-F_m(x)) + F_m^2(x)- P_m^2(x)\right] dx $$

(Substitute \(P_m\) and \(F_m\) expressions )

Recall the expressions for \(P_m\) and \(F_m\): $$ P_m = a_1\phi_1+a_2\phi_2+\cdots+a_m\phi_m, \quad F_m = c_1\phi_1+c_2\phi_2+\cdots+c_m\phi_m $$ Substitute them back into the integral: $$ \int_{a}^{b} \left[ 2f(x) \sum_{n=1}^m (a_n-c_n)\phi_n(x) + \sum_{n=1}^m (c_n^2-a_n^2)\phi_n^2(x)\right] dx $$

(Split the integral)

Split the integral into two parts: $$ 2\sum_{n=1}^m (a_n-c_n) \int_{a}^{b} f(x) \phi_n(x) dx + \sum_{n=1}^m (c_n^2-a_n^2) \int_{a}^{b} \phi_n^2(x) dx $$ Now, we proved in part (a) that \(\int_{a}^{b} (f(x)-F_m(x)) \phi_n(x)dx=0\). Therefore, the first part of the above expression is essentially the sum of zeros, which gives us $$ \sum_{n=1}^m (c_n^2-a_n^2) \int_{a}^{b} \phi_n^2(x) dx $$ Since \(\int_{a}^{b} \phi_n^2(x) dx \neq 0\) for \(n=1,2,\ldots,m\), the only way the above expression becomes zero is if \(a_n=c_n\) for all \(n=1,2,\ldots,m\). Thus, the least square property of the Fourier approximation is proven.

(c) Square error term and coefficients relationship)

We want to compute the following term: $$ \int_{a}^{b}\left(f(x)-F_{m}(x)\right)^2 dx = \int_{a}^{b} \left[f^2(x) - 2f(x)F_m(x) + F_m^2(x)\right]dx $$ Expanding and using linearity of integral we get $$ \int_{a}^{b}f^2(x)dx - 2\sum_{n=1}^{m} c_n \int_{a}^{b} f(x)\phi_n(x) dx +\sum_{n=1}^{m} c_n^2 \int_{a}^{b} \phi_n^2(x)dx $$ Now, let's recall that we proved in part (a) that \(\int_{a}^{b} (f(x)-F_m(x)) \phi_n(x)dx=0\). Therefore, the first term of the above expression can be rewritten as \(\int_{a}^{b} F_m(x) \phi_n(x) dx\). So, the expression becomes: $$ \int_{a}^{b}f^2(x)dx - 2\sum_{n=1}^{m} c_n^2 \int_{a}^{b} \phi_n^2(x) dx +\sum_{n=1}^{m} c_n^2 \int_{a}^{b} \phi_n^2(x)dx $$ Simplifying we obtain, $$ \int_{a}^{b}f^2(x)dx -\sum_{n=1}^{m} c_n^2 \int_{a}^{b} \phi_n^2(x)dx $$ Thus, the relationship between the square error term and the coefficients is proven. #d) Coefficients sum bound#

(Combine square error term with relationship)

Now, we extract the bound by combining the square error term we just computed with its relationship with the coefficients from part (c). We have: $$ \int_{a}^{b}\left(f(x)-F_{m}(x)\right)^2 dx = \int_{a}^{b}f^2(x)dx -\sum_{n=1}^{m} c_n^2 \int_{a}^{b} \phi_n^2(x)dx $$ The square error term is always non-negative, so $$ \int_{a}^{b}\left(f(x)-F_{m}(x)\right)^2 dx \geq 0 $$ Then, we arrive at the required inequality: $$ \sum_{n=1}^{m} c_n^2 \int_{a}^{b} \phi_n^2(x) dx \leq \int_{a}^{b} f^2(x) dx $$ So, the bound for the sum of the square of the Fourier coefficients is proven.

Key concepts

Fourier Series

A Fourier series is a way to represent a periodic function as an infinite sum of sines and cosines. The core idea is to break down a complex periodic function into simpler sine and cosine components. This is especially useful in analyzing signals or functional behavior over an interval.
To construct a Fourier series, one needs to compute the Fourier coefficients. These coefficients represent the weights of the sine and cosine functions involved in the series.

• The coefficients are calculated using integrals over the function period.
• Each coefficient is calculated by multiplying the target function by a sine or cosine and integrating over the interval.

These coefficients help approximate a given function better as more terms of the series are summed. In scenarios where only a few terms are used, the Fourier series provides a near-enough approximation that retains essential characteristics of the original function.

Least Squares Approximation

Least Squares Approximation is a fundamental method to best fit a function in a dataset by minimizing the sum of squared differences between the data and the function's values. It's a tool frequently used in data fitting, regression analysis, and forecasting.

• In a least squares fit, the error between the data and the approximating function is minimized in terms of the sum of their squares.
• This process results in the most "optimally" aligned function, providing the smallest possible difference in a squared sense.

In the context of Fourier series, the least squares method helps ensure that the series fits the function as closely as possible within a given interval. This minimization of error is key in achieving an accurate representation of the function through orthogonal functions.

Orthogonality of Functions

Orthogonality in functions is akin to the concept of perpendicular vectors in space. Two functions are considered orthogonal over a certain interval if their inner product (or the integral of their product) over this interval equals zero.
Think of orthogonality as a form of functional independence, where one function provides no information about another. This property is crucial in simplifying complex problems, especially in the context of Fourier series and least squares approximation.

• In Fourier series, the sine and cosine functions are orthogonal to each other over their periods.
• This orthogonality ensures that each term in the series contributes uniquely to the approximation of the function.

By minimizing interference between terms, orthogonality helps in constructing an optimal approximation of functions.

Integral Calculus

Integral Calculus is a major branch of mathematics focused on the concept of integration, which involves finding the area under curves. It serves as a powerful tool for solving a variety of mathematical problems involving accumulation and area.
Integration plays a pivotal role in the calculation of Fourier coefficients, and hence in constructing Fourier series for function approximation.

• The process of integration allows us to compute areas under sine and cosine functions over an interval, which is essential in the coefficients' calculation.
• This computation helps break down intricate functional behaviors into manageable parts, assisting in the function approximation via Fourier series.

The vast applications of integral calculus extend to physics, engineering, and other sciences, making it indispensable for mathematical modeling and analysis.