Index formula for convolution type operators with data functions in alg(SO,PC)

We establish an index formula for the Fredholm convolution type operators A=∑k=1makW0(bk) acting on the space L2(R)L2(R), where akak, bkbk belong to the C⁎C⁎-algebra alg(SO,PC)alg(SO,PC) of piecewise continuous functions on RR that admit finite sets of discontinuities and slowly oscillate at ±∞, first in the case where all akak or all bkbk are continuous on RR and slowly oscillating at ±∞, and then assuming that ak,bk∈alg(SO,PC)ak,bk∈alg(SO,PC) satisfy an extra Fredholm type condition. The study is based on a number of reductions to operators of the same form with smaller classes of data functions akak, bkbk, which include applying a technique of separation of discontinuities and eventually lead to the so-called truncated operators Ar=∑k=1mak,rW0(bk,r) for sufficiently large r>0r>0, where the functions ak,r,bk,r∈PCak,r,bk,r∈PC are obtained from ak,bk∈alg(SO,PC)ak,bk∈alg(SO,PC) by extending their values at ±r   to all ±t≥r±t≥r, respectively. We prove that indA=limr→∞⁡indAr although A=s-limr→∞Ar only.