Lattice Functions in \(\mathbb{R}\)
The Euler summation formula expresses a finite sum of integral points in terms of the integral and derivatives of the function with explicit knowledge of the error in integral form involving the Bernoulli polynomial.
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Lattice Functions in R
The Euler summation formula expresses a finite sum of integral points in terms of the integral and derivatives of the function with explicit knowledge of the error in integral form involving the Bernoulli polynomial. The classical summation formula due to Euler (1736a,b) and MacLaurin (1742) is of great importance in many branches of periodic modeling and simulation, in lattice point summation, in constructive approximation, and in approximate integration, which motivates us to include it in this textbook. In this chapter we lay the one-dimensional foundation which is generalized to higher dimensions in Chap. 10. Our considerations show that the Euler summation formula (Sect. 9.4) and the Poisson summation formula (Sect. 9.6) on finite intervals Œa; b of R are equivalent. In consequence, periodization can be understood equivalently from both approaches. This part of the chapter is a condensed version of the material presented by Freeden (2011). The asymptotic criteria for the validity of the Poisson summation formula in all of R are strongly influenced by the notes of Mordell (1928, 1929) and their extensions to the multi-dimensional case derived in Freeden (2011). Our investigations are followed by exercises concerned with aspects in numerical integration by virtue of trapezoidal rules and constructive approximation by means of projections to polynomial spaces within one-dimensional periodic Sobolev spaces in Sect. 9.8.
9.1 Bernoulli Polynomials We start with the classical introduction of Bernoulli polynomials by the recursion relation d B .x/ D .k C 1/Bk .x/; dx kC1
x 2 Œ0; 1;
W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 9, © Springer Basel 2013
(9.1.1)
395
9 Lattice Functions in R
396
for k 2 N, corresponding to the “initial function” B0 W R ! R given by B0 .x/ D 1 ;
x 2 Œ0; 1:
(9.1.2)
It follows that Bk W Œ0; 1 ! R, k 2 N, is a polynomial of degree k that can be written in the form Bk .x/ D x k C ck1 x k1 C : : : C c0 :
(9.1.3)
We notice that, for known Bk .x/, (9.1.1) determines BkC1 .x/ up to an additive integration constant. By convention, we choose the integration constants in such a way that B2kC1 .0/ D B2kC1 .1/ D 0 (9.1.4)
for all k 2 N. In doing so we get uniqueness in the determination of all Bk (see, e.g., Magnus et al. 1966; Rademacher 1973). This fact becomes clear from the observation that B2k1 .x/ D x 2k1 C c2k2 x 2k2 C : : : C c1 x C c0
(9.1.5)
implies B2kC1 .x/ D x 2kC1 C
.2k C 1/.2k/ c2k2 x 2k C : : : C .2k C 1/cx C d; (9.1.6) .2k/.2k 1/
where the conditions B2kC1 .0/ D B2kC1 .1/ D 0 uniquely determine the constants c and d . The resulting polynomials (of degrees k D 1; 2; 3; 4) defined on the interval Œ0; 1 read as follows:
B1 .x/ D x 12 ;
(9.1.7)
B2 .x/ D x 2 x C 16 ;
(9.1.8)
B3 .x/ D x 3 32 x 2 C 12 x; B4 .x/
D x 2x C x 4
3
2
(9.1.9) 1 : 30
(9.1.10)
Definition 9.1.1. The polynomials Bk W
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