By N. K. Bary

ISBN-10: 1483199169

ISBN-13: 9781483199160

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Additional info for A Treatise on Trigonometric Series. Volume 1

Example text

We know that the concept of an integral, starting with Cauchy, has widened, so that an increasingly large class of integrable functions has developed. In this book we will always understand by the class of "integrable functions" those integrable according to Lebesgue. These functions, as is known, are called summable; the series set up for them are named the FourierLebesgue series. For brevity's sake we shall simply say "Fourier series" but at the same time realise that the series being considered are always summable.

4). 3), the coefficients of which are any complex numbers. 2) it is seen that the numbers cn and c_n are conjugate complex numbers, that is, c_n = ~c~n (the symbol ä always indicates the number conjugate to a). § 3. A brief historical synopsis The possibility of representing a function by a trigonometric series was first considered by Euler in 1753 in connection with the work by Daniel Bernoulli on "Vibrating Strings" which had appeared at that time. If a string, fixed at both ends, is disturbed from its state of equilibrium and is allowed to vibrate freely without being given any initial velocity, then Bernoulli affirmed that the position of the string at time t is determined by the formula oo TtX y = Σ&ρ sinp—r-cospkt, * P=I where /is the length of the string and k is some coefficient which depends on the density and tension of the string.

4) we can consider both the case when x -» + oo (or x -* — oo) and the case x -> x0, where x0 is some fixed number. An analogous statement to the preceding can be made, namely: LEMMA. 1. If F(x) = jf(t)dt, 0{x) = a thenfrom f(x) \cp{t)dt9 a in a < x < b = O (φ(χ)) it follows that F(x) = O [Φ(χ)] in a oo at x -► b; the following statement would be more exact.