By Emily Coddington

**Read or Download A brief account of the historical development of pseudospherical surfaces from 1827 to 1887 ... PDF**

**Similar mathematics books**

**The Electric Motor and its Applications**

In contrast to another reproductions of vintage texts (1) we've not used OCR(Optical personality Recognition), as this results in undesirable caliber books with brought typos. (2) In books the place there are photos corresponding to pics, maps, sketches and so on we've endeavoured to maintain the standard of those photos, so that they signify adequately the unique artefact.

- Understanding College and University Organization: Theories for Effective Policy and Practice; Volume II: Dynamics of the System Reprint by Bess, James L., Dee, Jay R. (2012) Paperback
- 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art
- Clifford Algebras and Lie Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)
- Introduction to Mathematical Philosophy
- Discrete integrable systems
- Multidimensional Integral Equations and Inequalities (Atlantis Studies in Mathematics for Engineering and Science) 2011 edition by Pachpatte, B.G. (2013) Paperback

**Additional info for A brief account of the historical development of pseudospherical surfaces from 1827 to 1887 ...**

**Example text**

2. We have to show that T (V ) is dense in H = 2 (N). Let f ∈ 2 (N), and for n ∈ N let fn ∈ 2 (N) be given by fn (j) = f (j) if j ≤ n, 0 if j > n. Then ||f − fn ||2 = |f (j)|2 , j>n which tends to zero as n tends to inﬁnity. So the sequence (fn ) converges to f in 2 (N). For j = 1, 2, . . , n let λj = f (j). Then fn = T (λ1 e1 + · · · + λn en ), so fn lies in the image of T , which therefore is dense in H. This concludes the existence part of the proof. For the uniqueness condition assume that there is a second isometry T : V → H onto a dense subspace.

Assume that its Fourier series converges pointwise to the function g; then ck (g)e2πikx , g(x) = k∈Z so that for x = 0 we get f (l) = g(0) = l∈Z ck (g) k∈Z 1 = g(y)e−2πiky dy k∈Z 0 1 f (y + l)e−2πiky dy. 6. THE POISSON SUMMATION FORMULA 55 Assuming that we may interchange summation and integration, this equals l+1 f (y)e−2πiky dy = k∈Z −∞ l k∈Z l∈Z ∞ f (y)e−2πiky dy = fˆ(k). k∈Z This is a formal computation, valid only under certain assumptions. We will now turn it into a theorem by giving a set of conditions that ensures the validity of those assumptions.

Proof: Since S is mapped to itself, the corollary follows from the inversion theorem. 6 Let f (x) = e−πx . Then f ∈ S and fˆ = f. 3 the function f is, up to scalar multiples, the unique solution of the diﬀerential equation f (x) = −2πxf (x). By induction one deduces that for every natural number n there is 2 2 a polynomial pn (x) such that f (n) (x) = pn (x)e−πx . Since e−πx decreases faster than any power of x as |x| → ∞, it follows that f lies in S. Then fˆ also lies in S, and we compute (fˆ) (y) = ∞ 2 (−2πix)e−πx e−2πixy dx −∞ ∞ = i 2 (e−πx ) e−2πixy dx −∞ = −2πy fˆ(y), 52 CHAPTER 3.