1.1. General properties 5 whenever convenient. In connection with the difference expression (1.5) we also define the diagonal, upper, and lower triangular parts of R as follows (1.11) [R]0 = R(.,.), [J?]± = £ i ? ( . , . ± *)($*)*, ken implying R = [R}+ + [R]0 + [ij]_. Having these preparations out of the way, we are ready to start our investigation of second order symmetric difference expressions. To set the stage, let a,b e £(Z, M) be two real-valued sequences satisfying (1.12) a(n) G M\{0}, b(n) G K, n G Z, and introduce the corresponding second order, symmetric difference expres- sion r : £{Z) - £{Z) f(n) . - a(n)f{n + l) + a(n-l)f(n-l) + b(n)f(n)' It is associated with the tridiagonal matrix / •. •. •. \ (1.13) (1.14) a(n - 2) b{n - 1) a(n 1) a(n 1) b(n) a(n) a(n) 6 ( n + l a(n+ 1) V / and will be our main object for the rest of this section and the tools derived here - even though simple - will be indispensable for us. Before going any further, I want to point out that there is a close connection between second order, symmetric difference expressions and second order, symmet- ric differential expressions. This connection becomes more apparent if we use the difference expressions (1.15) (df)(n) = f(n+l)-f(n), (d*f)(n) = f(n-l)-f(n), (note that 9, d* are formally adjoint) to rewrite r in the following way (rf)(n) = -(d*adf)(n) + (a(n - 1) + a(n) + b(n))f(n) (1.16) = -(da-d*f)(n) + (a(n - 1) + a{n) + b(n))f(n). This form resembles very much the Sturm-Liouville differential expression, well- known in the theory of ordinary differential equations. In fact, the reader will soon realize that there are a whole lot more similarities between differentials, integrals and their discrete counterparts differences and sums. Two of these similarities are the product rules (dfg)(n) = f(n)(dg)(n) + g(n+l)(df)(n), (1.17) (d*fg)(n) = f(n)(d*g)(n) + g(n - l)(0*/)(n) and the summation by parts formula (also known as Abel transform) (1.18) Yl 9ti)(df)(j) = g(n)f(n + 1) - g(m - l)/(m) + £ (d*g)(j)f(j).
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