Some Background
In this chapter, we present in a very concise way some fundamental aspects of linear spaces, orthogonal functions, and orthogonal polynomials. Three family of orthogonal polynomials (Legendre, Chebyshev, and Jacobi), of great importance to us, are discuss
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Some Background
In this chapter, we present in a very concise way some fundamental aspects of linear spaces, orthogonal functions, and orthogonal polynomials. Three family of orthogonal polynomials (Legendre, Chebyshev, and Jacobi), of great importance to us, are discussed. Most of the presented results are taken from [1–3]; so all the stated theorems are given without proofs. The interested reader who wants to go deeper into these topics is invited to check the mentioned references and the very rich literature on this subject. The results presented here serve as a preparation for differential beamforming in forthcoming chapters.
3.1 Linear Spaces A linear space is defined as a class of functions, all having the same domain, with the properties [1]: (i) if two functions f and g belong to the class, their sum f + g does also, and (ii) if f is a member of the class, every scalar multiple of f is also. For example, the collection of all continuous functions is a linear space, since the sum of two continuous functions is continuous and any scalar multiple of a continuous function is continuous. We will only consider functions whose domain is [−1, 1]. The class of all functions defined and continuous in the interval [−1, 1] is a linear space. This space is denoted by C[−1, 1]. A finite set of functions f 1 , f 2 , . . . , f n belonging to C[−1, 1] is linearly dependent if and only if one of the functions is a linear combination of the others. Otherwise, it is said to be linearly independent. An infinite sequence of functions is said to be linearly independent if and only if every finite set of functions in the sequence is linearly independent. Otherwise, it is said to be linearly dependent. This is valid for infinite sequences as well as for finite © The Author(s) 2016 J. Benesty et al., Fundamentals of Differential Beamforming, SpringerBriefs in Electrical and Computer Engineering, DOI 10.1007/978-981-10-1046-0_3
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3 Some Background
sequences. For example, the infinite sequence 1, x 2 , x 3 , . . . is linearly independent. It is clear that every linear combination of terms in this sequence is a polynomial. A collection of elements of a linear space is said to span the space if every element of the linear space is a linear combination of elements in this collection. A collection of elements of a linear space is called a basis for the linear space if it spans the space and is linearly independent. If a space has a finite basis, then it is said to be finite-dimensional. If it has an infinite basis, it is infinite-dimensional. An inner product in a linear space is a number, denoted by f, g, which is a function of pairs of elements of the space satisfying the following axioms [1]: (i) (ii) (iii) (iv)
f, f ≥ 0, with equality if and only if f = 0, f, g = g, f ∗ , f, g + h = f, g + f, h, and α f, g = α f, g, where α is an arbitrary number.
The norm or length of f , denoted by f , is defined as f =
f, f .
(3.1)
Any norm has the three fundamental properties: (i) f ≥ 0, with
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