By Rajendra Bhatia
This can be a concise creation to Fourier sequence protecting historical past, significant issues, theorems, examples, and purposes. it may be used for self learn, or to complement undergraduate classes on mathematical research. starting with a quick precis of the wealthy heritage of the topic over 3 centuries, the reader will savor how a mathematical idea develops in levels from a realistic challenge (such as conduction of warmth) to an summary conception facing recommendations reminiscent of units, capabilities, infinity, and convergence. The summary conception then offers unexpected functions in various parts. routines of various hassle are integrated all through to check knowing. A vast diversity of functions also are coated, and instructions for additional studying and learn are supplied, besides a bankruptcy that offers fabric at a extra complex point compatible for graduate scholars.
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Extra info for Fourier Series (Mathematical Association of America Textbooks)
Example text
Since for r < I the series I::x rl 11 lei 11 H. being absolutely and unifonnly convergent. can be differentiated tem1 by term. this shows that ll(r. 8) as dcl1ned above is a solution of the Laplace equation. 2). 4). 5) in the sense that ll(r. 0) tends to {(0) uniformly as r - I. Let f and g be two periodic functions on~ with period 2rr. Iff and g arc continuous. f * g by 24) is not quite a Fourier series because of the presence of the first term. Subtracting this term we get the Fourier series for the periodic function F(x ). How about term by term differentiation? 2. Let f be a continuous and piecewise C 1 function on T Show that if f' is piecewise C 1, then the Fourier series f (t) = j (n )ei111 can be differentiated term by term. and the series so obtained converges pointwise to [f' (t +) +f' (t _) ]. (Use Dirichlet's L ! 2 are satisfied by functions like f (t) = It I and f (t) = I sin tl. This apparently anomalous situation-the possibility of expanding any function as a sum of periodic functions and further the possibility of expanding the same function as both a sum of even functions and a sum of odd functions-was a part of the raging controversy at the beginning of the subject. See Chapter 0. Examples in the next section will clarify further how this apparent anomaly is neatly resolved by our analysis. 1. Let f be the "pulse function'' defined as j(x) = l -1 I if- Tl