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Introduction to Linear Algebra with Mathematica

Glossary

Preface

This section is devoted to periodic extensions of functions defined on some finite interval. Since classical Fourier series provide periodic extensions, we need to compare it with corresponding periodic extension of the original function.

Mean Square Error

Let f(x) be an arbitrary square integrable function defined on the interval [𝑎, b], and let {ϕn(x)}n ≥ 0 be a system of orthogonal functions from 𝔏²([𝑎, b]). So
\[ \int_a^b \phi_n (x)^{\ast} \phi_k (x)\,{\text d}x = 0 \qquad\mbox{for}\quad n\ne k , \]
and
\[ \int_a^b \left\vert \phi_n (x) \right\vert^2 {\text d}x = \| \phi_n \|^2 > 0 \qquad\mbox{for}\quad n=0,1,2,\ldots . \]
Let SN(x) be a linear combination of the first N + 1 functions of the given orthogonal system:
\[ S_N (x) = \gamma_0 \hi_0 (x) + \gamma_1 \hi_1 (x) + \gamma_2 \hi_2 (x) + \cdots + \gamma_N \hi_N (x) , \]
where γ0, γ1, γ2, … , γN are constants. Since all members of the orthogonal system are square integrable functions, their linear combination SN(x) is also square integrable. Hence the difference f(x) − SN(x) is also square integrable. Now we consider the quantity
\[ \delta_N = \int_a^b \left\vert f(x) - S_N (x) \right\vert^2 {\text d}x , \]
which we call the mean square error in approximating f(x) by SN(x).

We now pose the problem of choosing the coefficients γ0, γ1, γ2, … , γN (for a given N) in such a way that the deviation δN is a minimum. Expanding the square, we get

\[ \delta_N = \int_a^b \left\vert f(x) \right\vert^2 {\text d}x -2 \int_a^b f(x)^{\ast} \, S_N (x)\, {\text d}x + \int_a^b \left\vert S_N (x) \right\vert^2 {\text d}x \]
The second integral becomes
\[ \int_a^b f(x)^{\ast} \, S_N (x)\, {\text d}x = \sum_{n=0}^N \gamma_n \int_a^b f(x)^{\ast} \, \phi_n (x)\, {\text d}x = \sum_{n=0}^N \gamma_n \left\langle f \,|\, \phi_n \right\rangle . \]
We know that
\[ c_n \| \phi_n \|^2 = \left\langle f \,|\, \phi_n \right\rangle , \qquad n=0,1,2,\ldots , N ; \]
are Fourier coefficients of function f. Moreover, we have

Mean Square Convergence