E.C. Titchmarsh. The Theory of Functions. Oxford University Press, London, 2nd edition, 1939. 15  Home/Search Document Not in Database Summary Related Articles Check |
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Algorithms for the Constrained Design of Digital Filters with.. - Lang (1999) (3 citations) (Correct)
....In Section 5.4.1, we discuss this method and its application to the unconstrained least squares IIR lter design problem. In Section 5.4.2, we will explain how to incorporate the required constraint on the pole radii. This modication of the original Gauss Newton method is based on Rouch# s theorem [131]. Section 5.4.3 presents a numerical algorithm solving the resulting subproblems of the modied Gauss Newton method. Finally, in Section 5.4.4, the complete design algorithm is summarized. 5.4.1 The Gauss Newton Method We use the polynomial coeOEcients a n and b m to represent the transfer ....
....ffi is computed such that the required constraint on the pole radii is met. 5.4.2 Rouch# s Theorem If f(z) and g(z) are analytic inside and on a closed contour C, and jg(z)j jf(z)j on C, then f(z) and f(z) g(z) have the same number of zeros inside C. A proof of this theorem is given in [131]. Let Delta(z) be dened by : ffi N z where ffi n , n = 1; 2; N , are the rst N elements of the update vector ffi in (5.15) According to (5.6) these elements update the denominator coeOEcients a n , n = 1; 2; N . Dene f(z) z A(z) z : aN ....
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E. C. Titchmarsh, The Theory of Functions, Second Edition, Oxford University Press, Oxford, 1979.
Error Terms for Closed Orbits of Hyperbolic Flows - Pollicott, Sharp (Correct)
....2 1 2N 1 2N Im(s) 2.7) This complete the proof of Proposition 3. To proceed, we need the following standard result from complex analysis which allows us to convert the bound (2.7) into a bound for # # (s) #(s) in a smaller region. Figure 1: Proof of Proposition 4 Lemma 4. [14] Suppose that #(s) is non zero and analytic on a disk D(R) s s z R . Suppose log #(s) is bounded by U 0 on D(R) Then for 0 r R and s # D(r) we have the bound # # # # 8R (R r) U log #(z) We can apply Lemma 4 in the following way. Given s # R(#) we ....
E. Titchmarsh, The theory of functions, Oxford University Press, Oxford, 1939.
Solitary Waves in the Critical Surface Tension Model - Gunney, Li, Olver (Correct)
....Lemma 2. If c 1 0, then the sum of the Dirichlet expansion (7) has a singularity at its abscissa of convergence x 0 ; in fact lim x x (x) Gamma1. Proof : If c 0, then c n = c d n 0 for all n, which implies that has a singularity at its abscissa of convergence, cf. [36]. Assume that inf x x0 (x) Gamma1. Since (x) 0 and (x) 0, is an increasing function with a lower bound on the interval (x 0 ; 1) Therefore, the limit lim x x (x) m 0 exists. When x x 0 , the derivatives of can be expressed by, 2] Z 1 K(x Gamma y) ....
Titchmarsh, E. C., Theory of Functions, Oxford University Press, London, 1968.
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....assume without restriction that (x n ) n2N is admissible again because for each x 2 C the set fz 2 C : z x = 2 Cg is bounded. Hence relation (4.13) entails lim n 1 F g (x n ; y) v g , a contradiction. The proof of Theorem 2 is based on the Phragm en Lindel of Theorems (see Section 5.6. of [21]) for the right half plane H = fz 2 C : Re z 0g. Proof of Theorem 2. We may assume that 0. For xed x; y 2 C G(z) z E k (zx; y) which is regular in H n f0g and continuous in H with G(0) 0. The integral representation (1.4) easily implies that jE k (zx; y)j max g2G e ....
....= O e jzj as z 1 within H: Next consider G along the boundary lines of H; i (t) it; t 0. According to Theorem 1, lim t 1 G(it) i v e . Moreover, G( it) G(it) for t 0 (c.f. 1.5) hence lim t 1 G( it) exists as well. Employing the Phragm en Lindel of Theorems 5.62 and 5. 64 of [21], we deduce that G is in fact bounded in H and that lim z 1; z2H G(z) i v e : 12 We nally come to the proof of Theorem 3. The key for our approach is the following simple observation: according to formula (1.4) one may write E k (x; i ) x (y) c x ( x; 2 R ) 4.14) ....
E.C. Titchmarsh, \The Theory of Functions," 2nd ed., Oxford University Press, 1950.
Asymptotic Expansion for the Lebesgue Constants of the Walsh System - Hwang (1995) (Correct)
....#(s) ds i# #(s) ds. The last integral is zero by expanding the factor (2 1) 1 in ascending powers of 2 and by applying Corollary 1 term by term. Since #( 1 it) O( t 2 ) F (u) is continuous. Note that by the functional equation for Riemann s zeta function [12], # # ( 1) 1 log 2#) 12 # # (2) 2# ) This completes the proof. # 4 Concluding remarks Let us first compare our result with the corresponding one for the sum of digits function #(n) Since = #(n) L n and = #(n) we obtain, cf. 1, 4] log 2 n nG(log 2 ....
E. C. Titchmarsh, The Theory of Functions, Oxford University Press, Second edition, 1939.
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....examples for which this approach is useful are indicated in the Appendix. Asymptotic behavior of V (oe it) We now consider the growth rate of V (oe Sigma it) for large t. Note that it follows easily from (5) that j(x) j(x) y)blog 2 yc dy = blog 2 xc bxc (y) dy: Thus (cf. [15]) 1 Gammaoe ( Gamma1 oe 1) for arbitrarily small 0. But this is far from sufficient for our purposes. Proposition 1 For Gamma1 oe 0, we have 2 oe (jtj t 0 0) Proof. By conjugacy, assume throughout this section t t 0 0. We first prove the case oe = 0, the case ....
E. C. Titchmarsh, The theory of functions, Oxford University Press, London, 2nd Ed., 1939.
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E.C. Titchmarsh. The Theory of Functions. Oxford University Press, London, 2nd edition, 1939. 15
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E. C. Titchmarsh, The Theory of Functions, Second edition, Oxford University Press, London, 1939.
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E. C. Titchmarsh, The Theory of Functions, Second edition, Oxford University Press, 1939. 41
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