# The 8051 Microcontroller Architecture, Programming and by Delmar Thomson

By Delmar Thomson

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26), by the ∞ Weierstrass M-test, the series k=1 ejkx /k 2 is uniformly and absolutely convergent on [−π, π). 3 Hilbert Spaces, Sequence Spaces and Function Spaces Hilbert Spaces Consider a sequence of real or complex vectors, denoted by {an }∞ n=1 , in a normed vector space V. The sequence {an }∞ is said to converge in the norm n=1 or, brieﬂy, converge to a real or complex vector a ∈ V if a − an = 0. 53) Convergence in the norm is also often referred to as convergence in the mean. A sequence {an }∞ n=1 in V is called a Cauchy sequence if for every real number ε > 0 there exists an integer N such that an − am < ε for all n > m ≥ N.

Then the inverse of R can be expressed as R−1 = A−1 − A−1 B(C−1 + DA−1 B)−1 DA−1 . 6 (Woodbury’s Identity). 28) where A is a nonsingular M × M matrix, u is an M × 1 vector, and α is a scalar. Then the inverse of R can be expressed as R−1 = A−1 − αA−1 uuH A−1 . 27). Moreover, two theorems regarding partitioned matrices are stated as follows [7, p. 572], [9, pp. 5). 7. 23). 31) provided that A22 is a nonsingular square matrix. 8. 23) where A11 and A22 are also nonsingular square matrices. Then the inverse of A can be expressed as 3 For ease of later use, we give a slightly generalized statement of Woodbury’s identity by including a scalar α.

28 (Weierstrass M-Test). Suppose k=1 ak (x) is a real or complex series to be tested on an interval [xL , xU ]. If there exists a convergent ∞ series k=1 Mk such that each term Mk ≥ |ak (x)| for all x ∈ [xL , xU ], then ∞ the series k=1 ak (x) is uniformly and absolutely convergent on [xL , xU ]. Since the proof is lengthy and can be found, for instance, in [13], it is omitted here. An example using the Weierstrass M-test is provided as follows. 29 jkx Suppose ∞ /k 2 is the series to be tested on [−π, π).