Appendix A Properties of the Gamma Functions We list here some basic properties of the Gamma function (see, e.g., Abramowitz and Stegun (1964)), defined by ∞ Γ (z)= tz−1e−tdt, ∀z ∈ C with Re(z) > 0. (A.1) 0 √ In particular, we have Γ (1)=1andΓ (1/2)= π. • Recursion formula: Γ (z + 1)=zΓ (z), Γ (n + 1)=n!, (A.2) and − / − / Γ (2z)=(2π) 1 222z 1 2Γ (z)Γ (z + 1/2). (A.3) • Connection with binomial coefficient: , - z Γ (z + 1) = . (A.4) w Γ (w + 1)Γ (z − w + 1) • Relation with Beta function: 1 Γ (x)Γ (y) B(x,y)= tx−1(1 −t)y−1dt = , x,y > 0. (A.5) 0 Γ (x + y) In particular, for α,β > −1, 1 1 (1 − x)α(1 + x)β dx = 2α+β +1 tβ (1 −t)αdt −1 0 (A.6) Γ (α + 1)Γ (β + 1) = 2α+β +1 . Γ (α + β + 2) J. Shen et al., Spectral Methods: Algorithms, Analysis and Applications, 415 Springer Series in Computational Mathematics 41, DOI 10.1007/978-3-540-71041-7, c Springer-Verlag Berlin Heidelberg 2011 416 A Properties of the Gamma Functions • Stirling’s formula: √ ! − / − 1 1 − Γ (x)= 2πxx 1 2e x 1 + + + O(x 3) , x 1. (A.7) 12x 288x2 Moreover, we have √ √ + / + / 1 2πnn 1 2 < n!en < 2πnn 1 2 1 + , n ≥ 1. (A.8) 4n Appendix B Essential Mathematical Concepts We provide here some essential mathematical concepts which have been used in the mathematical analysis throughout the book. For a more comprehensive presentation, we refer to Yosida (1980)andAdams (1975). Let (X;d) be a metric space. A sequence {xk} in X is called a Cauchy sequence,if d(xk,xl) → 0ask,l → ∞. The space (X;d) is said to be a complete space if every Cauchy sequence in X converges to an element in X. B.1 Banach Space Definition B.1. Given a (real) vector space X, a norm on X is a function ·: X → R such that •u + v≤u + v, ∀u,v ∈ X; •αu = |α|u, ∀u ∈ X and ∀α ∈ R; •u≥0, ∀u ∈ X; •u = 0 if and only if u = 0. In particular, a semi-norm on X is a function |·|: X → R satisfying the first three conditions. The space (X,·) is called a normed vector space. A Banach space is a normed vector space which is complete with respect to the metric: d(u,v)=u − v, ∀u,v ∈ X. J. Shen et al., Spectral Methods: Algorithms, Analysis and Applications, 417 Springer Series in Computational Mathematics 41, DOI 10.1007/978-3-540-71041-7, c Springer-Verlag Berlin Heidelberg 2011 418 B Essential Mathematical Concepts B.2 Hilbert Space Definition B.2. Let X be a real vector space. An inner product on X is a function (u,v) : X × X → R such that • (u,v)=(v,u), ∀u,v ∈ X; • (αu + βv,w)=α(u,w)+β(v,w), ∀u,v,w ∈ X and ∀α,β ∈ R; • (u,u) ≥ 0, ∀u ∈ X; • (u,u)=0 if and only if u = 0. Two elements u,v ∈ X are said to be orthogonal in X, if (u,v)=0. The inner product (·,·) induces a norm on X,givenby u = (u,u), ∀u ∈ X. Correspondingly, the metric on X can be defined by d(u,v)=u − v. A Hilbert space is a Banach space endowed with an inner product (i.e., every Cauchy sequence in X is convergent with respect to the induced norm). In a Hilbert space, the Cauchy–Schwarz inequality holds: |(u,v)|≤uv, ∀u,v ∈ X. (B.1) Remark B.1. If X is a complex vector space, the inner product (u,v) is a complex valued function. In the Definition B.2, the first condition should be replaced by (u,v)=(v,u), ∀u,v ∈ X. Next, we introduce the dual space of a Banach/Hilbert space X. Definition B.3. A functional F : X → R is said to be linear or continuous,ifthere exists a constant c > 0 such that |F(u)|≤cu, ∀u ∈ X. (B.2) Let X be the set of all linear functionals on X, and define the norm |F(u)| FX = sup . u∈X;u=0 u Then the space X is a Banach space, which is called the dual space of X. The bilinear form F(u)= F,u : X × X → R, is called the duality pairing on X × X. If X is a Hilbert space, then its dual space X is a Hilbert space as well. Moreover, according to the Riesz Representation Theorem, X and X are isometric, and X can be canonically identified to X. More precisely, for any linear functional F ∈ X , there exists a unique u ∈ X such that F(v)= F,v =(u,v), ∀v ∈ X and FX = u. B.3 Lax-Milgram Lemma 419 In a normed space X, a sequence {vn}⊂X is (strongly) convergent to v ∈ X, if vn − v→0asn → ∞. It is possible to introduce another type of convergence in a weaker sense. Definition B.4. A sequence {vn} in X is called weakly convergent to v ∈ X, if F(vn) → F(v) in R for all F ∈ X . If a sequence {vn} converges to v in X, it is also weakly convergent. The converse is not true unless X is a finite dimensional space. ∗ In a dual space X , a sequence of functional {Fn} in X is called weakly conver- gent to F ∈ X , if {Fn(v)} converges to F(v) for all v ∈ X. The weak convergence implies the weak∗ convergence. B.3 Lax-Milgram Lemma Definition B.5. Let X be a Hilbert space with norm ·. A functional a(·,·) : X × X → R defines a bilinear form, if for any u,v,w ∈ X and α,β ∈ R, a(αu + βv,w)=αa(u,w)+βa(v,w), a(u,αv + βw)=αa(u,w)+βa(v,w). That is, for any fixed u, both the functionals a(u,·) : X → R and a(·,u) : X → R are linear. The bilinear form is symmetric,ifa(u,v)=a(v,u) for any u,v ∈ X. Definition B.6. A bilinear form a(·,·) on a Hilbert space X is said to be continuous, if there exists a constant C > 0 such that |a(u,v)|≤Cuv, ∀u,v ∈ X, (B.3) and coercive on X, if there exists a constant α > 0 such that a(u,u) ≥ αu2, ∀u ∈ X. (B.4) It is clear that if a(·,·) is symmetric, continuous and coercive on the Hilbert space X, then a(·,·) defines an inner product on X. Theorem B.1. (Lax-Milgram lemma). Let X be a Hilbert space, let a(·,·) : X × X → R be a continuous and coercive bilinear form, and let F : X → R be a linear functional in X. Then the variational problem: Find u ∈ X such that (B.5) a(u,v)=F(v), ∀v ∈ X, has a unique solution. Moreover, we have 1 ≤ . u α F X (B.6) 420 B Essential Mathematical Concepts B.4 Lp-Space Let Ω be a Lebesgue-measurable subset of Rd (d = 1,2,3) with non-empty interior, and let u be a Lebesgue measurable function on Ω. In what follows, the integration is in the Lebesgue sense. Definition B.7. For 1 ≤ p ≤ ∞, let p L (Ω) := u : u is measurable on Ω and uLp(Ω) < ∞ , (B.7) where for 1 ≤ p < ∞, 1/p p uLp(Ω) := |u(x)| dx , (B.8) Ω and uL∞(Ω) := ess sup |u(x)|. (B.9) x∈Ω Remark B.2. Some remarks are in order. (i) The space L∞(Ω) consists all functions that are essentially bounded on Ω. A function u is said to be essentially bounded on Ω, if there exists a constant K such that |u(x)|≤Ka.e.onΩ. The greatest lower bound of such constants K is called the essential supremum of |u(x)| on Ω, denoted by uL∞(Ω). (ii) We identify functions in Lp(Ω) that are equal almost everywhere on Ω. The elements of Lp(Ω) are equivalence classes of measurable functions that satisfy (B.7) with the equivalence relation: u ≡ v, if they only differ on a measurable subset of measure zero. p Equipped with the norm ·Lp(Ω), the space L (Ω) with 1 ≤ p ≤ ∞ is a Banach space. In particular, the space L2(Ω) is a Hilbert space equipped with the inner product 2 (u,v) 2(Ω) = u(x)v(x)dx, ∀u,v ∈ L (Ω). (B.10) L Ω Definition B.8. If p and q are positive real numbers such that 1 1 p + q = pq or + = 1, (B.11) p q then we call (p,q) a pair of conjugate exponents.Asp→ 1, (B.11) forces q → ∞. Consequently, (1,∞) is also regarded as a pair of conjugate exponents. Theorem B.2. • Minkowski’s inequality.Ifu,v ∈ Lp(Ω) with 1 ≤ p ≤ ∞, then u + v ∈ Lp(Ω), and u + vLp(Ω) ≤uLp(Ω) + vLp(Ω). (B.12) B.5 Distributions and Weak Derivatives 421 • Ho¨lder’s inequality. Let p and q be conjugate exponents with 1 ≤ p ≤ ∞. If u ∈ Lp(Ω) and v ∈ Lq(Ω), then uv ∈ L1(Ω), and |u(x)v(x)|dx ≤uLp(Ω)vLq(Ω). (B.13) Ω In particular, if p = 2, the Holder’s¨ inequality reduces to the Cauchy–Schwarz inequality (B.1). It follows from (B.13)thatLq(Ω) ⊂ Lp(Ω), if p ≤ q and Ω has a finite measure.