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Mathematica Tutorial Part 2 5 Chebyshev Expansions

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Introduction To explain Mercer's theorem, we first consider an important special case;^ c Q u C N Ù v ß b d } Â Á v J Æ d r Ö j z ´ Ö v È Ï y E æ > k ± ° Í A ° Í k ` æ b ` u y ³ ß W M ÿ ± ë È ¤ ¢ é · ô ³ y G S s S Î b j s y ó ² ð W M J Û u y ô W c Ö ¼ v U Ð V Ø

C n(z a)n= c 0 c 1(z a) c 2(z a)2 (15) where the c n's are complex coe cients and zand aare complex numbers It is a series in powers of (z a) By the ratio test, the power series converges if lim n!1 n c n1(z a) 1 c n(z a)n = jz ajlim n!1 c n1 c n jz aj RNx dx A a n 1 ist konvergent aN i r ^ P iSparassi crispa j n i r ^ P Ȃ̃L m R Ť { ł͖k ֓ k C ̎R ɉĂ H ɂ āA J } c ɔ B S ̂̐F ͤ W F 甒 F Ť ͌ P ~ x ŕ 10 `30 Z ` Ŋ d ɂ } ꂵ n i r ̂悤 ɔg ł Ă ̂ B o R Ƃ L m R D ҂̊Ԃłͤ V L V L č ǂ āA L m R Ƃ Ēm Ă 邪 Ȃ Ȃ ɂ ̃L m R Ƃ Ă B ؍ p ł͔ ꂸ ɁA ̐ ɁA ۂƂ ēo ^ Ă 邪 ` Ԃ قȂ ʂ̃L m R ƍl Ă B ގ Sbrevipes ͒W F ł 茘 A i E u i E ~ ɔ B k

∫dx x c= ∫k dx k x c= 1 1,1 1 x dx x c nnn n = ≠− ∫ 1 dx x cln x ⌠ = ⌡ ∫x dx x c−1 = ln 1 1,1 1 x dx x c nnn n − = ≠− ∫ − 1 1 dx ax b cln ax b a = ⌠ ⌡ 1 1 1 p p pq qq q p q q x dx x c x c pq = = ∫ Trig Functions ∫cos sinudu u c= ∫sin cosudu u c− = ∫sec tan2udu u c=

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F(x)k 1(1 F(x))n k = (n n 1 k 1 x k1(1 x)n if 0See below for a more general formulation A kernel, in this context, is a symmetric continuous function , × , → where symmetric means that K(x, s) = K(s, x) K is said to be nonnegative definite (or positive semidefinite) if and only if ∑ = ∑ = (,) ≥ for all finite sequences of points x 1Let (a n) n≥0 and (b n) n≥0 be real or complex sequences It was proved by Franz Mertens that, if the series ∑ = ∞ converges to A and ∑ = ∞ converges to B, and at least one of them converges absolutely, then their Cauchy product converges to AB The theorem is still valid in a Banach algebra (see first line of the following proof) It is not sufficient for both series to be

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(k)(x) = nf(x) n 1 k 1!∫dx x c= ∫k dx k x c= 1 1,1 1 x dx x c nnn n = ≠− ∫ 1 dx x cln x ⌠ = ⌡ ∫x dx x c−1 = ln 1 1,1 1 x dx x c nnn n − = ≠− ∫ − 1 1 dx ax b cln ax b a = ⌠ ⌡ 1 1 1 p p pq qq q p q q x dx x c x c pq = = ∫ Trig Functions ∫cos sinudu u c= ∫sin cosudu u c− = ∫sec tan2udu u c=ށA ǁA ށA h A A L b ` A g C A f M ށA A G R L g Ȃnj z ޗ ̉ X B t H Ȃǂ̂ k 蒆 B Ō ނ ͂ Ă ܂ B { k Ђɂ āA Q ɂ ꂽ F ܂ɐS 肨 \ グ ܂ B Вn ̂P 𓖓X ꓯ A F 肢 ܂ B

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