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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.

Periodic Extension

A function f(x) is called periodic if there exists a constant T > 0 for which
\[ f(x+T) = f(x) \]
for all x in the domain of definition of f(x). (It is understod that both x and x + T lie in the domain.) Such a constant T is called a period of the function f(x). Periodic functions arise in many problems of physics and engineering. For example, a simple periodic function of great importance for applications is
\[ y = A\,\sin \left( \omega t + \phi \right) , \]
where A. ω, and ϕ are known constants. This function is called a harmonic of amplitude |A|, (angular) frequency ω, and initial phase ϕ. The period of such a harmonic function is T = 2π/ω. To define a periodic function it is sufficient to define it on any interval of length T. Now we give aalgorithmic definition of periodic function:

It is clear that sum, difference, product, or quotient of two periodic functions of the same period is again a periodic function of the same period T. If T is a period of the function f(x), then the number 2T, 3T, 4T, … are also periods.

\[ f(x) = f(x+T) = f(x+ 2T) = f(x+3T) = \cdots . \]
If function f(x) of period T is integrable on an interval of length T, then it is integrable on any other interval of the same length, and the value of the integral is the same:
\[ \int_a^{a+T} f(x) \,{\text d}x = \int_b^{b+T} f(x) \,{\text d}x , \]
for any real numbers 𝑎 and b.