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CMSC FMM 878R/AMSC 698R Lecture 6 Outline Representation of functions in the space of coefficients Matrix representation of operators Truncation and truncated operators Translation operator Reexpansion coefficients R|R and S|S translation operators Examples S|R and R|S translation operators Properties of translation operators Why do we need represent functions in different spaces? Functions should be efficiently summed up; Sums of functions should be compressed; Error bounds should be established; Functions should be translated and expanded over different bases; For computations we need discrete and finite function representations. Some functions measured experimentally or approximated by splines, and there is no explicit analytical representation in the whole space. Function Representation in the Space of Coefficients Function Representation in the Space of Coefficients (2) C() F() R A() p-Truncated Vectors Dense in F() F() Fp() A() Rp Matrix Representation of Linear Operators F() F(') A() F A(') Representation of a Linear Operator p-Truncation (Projection) Operator Pr(p) Fp() F() A() Rp() Norm of p-Truncation Operator (important for error bounds) p-Truncated Operator Norm of p-Truncated Operator (important for error bounds) Translation Operator t y+t y Example of Translation Operator T(t) (y+t) (y) t y R|R-reexpansion Example of R|R-reexpansion R|R-translation operator Why the same operator named differently? The first letter shows the basis for (y) The second letter shows the basis for (y +t) Needed only to show the expansion basis (for operator representation) Matrix representation of R|R-translation operator Consider Coefficients of shifted function Coefficients of original function Reexpansion the of same function over shifted basis We have: R|R-reexpansion of the same function over shifted basis (2) r1(x*+t) y x R R t 2 x*x *2 +t (R|R) Original expansion Is valid only here! r1 |y - x* - t| < r 1 = r - |t| x *1 * r r(x*) Since r1(x*+t) r(t) ! 1 Example of power series reexpansion Example of power series reexpansion (2). Relation to Taylor series. Let's check this for Taylor series, when expansion coefficients are For A0 this yields Taylor series again! Check for Al For Al we obtained Taylor series for the l-th derivative! Wow! S|S-reexpansion S|S-translation operator S|S and R|R-translation operators are very similar, (actually, this is just two representations of the same translation operator in different domains and bases) But picture is different... r1(x*+t) r(x*) y Original expansion Is valid only here! |y - x* - t| > r 1 = r + |t| 2 r1 S S r x*1 t * (S|S) x*+tx*2 Since r1(x*+t) r(t) ! Also |xi - x* | < r singular point ! 1 xi S|R-reexpansion Does R|S reexpansion exist? Theoretically yes (in some cases, e.g. analytical continuation); In practice, since the domain of S-expansion is larger then the domain of R-expansion, this either not useful (due to error bounds), or can be avoided in algorithms; We will not use R|S-reexpansions in the FMM algorithms. S|R-translation operator S|R-operator has almost the same properties as S|S and R|R (t cannot be zero) Picture is different... r1(x*+t) x*+t r1 y Original expansion Is valid only here! r(x*) |y - x* - t| < r 1 = |t| - r Since r1(x*+t) r(t) ! Also |xi - x* | < r singular point ! (S|R) S t S x*1 * (S|S) x*2 r 1 xi Properties of the translation operator Spectrum of the translation operator eigen value eigen function derivative in direction s

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