b'@techreport{,'b'\nTITLE = {Harmonic analysis, real approximation, and the communication complexity of Boolean functions},\nAUTHOR = {Grolmusz, Vince},\nLANGUAGE = {eng},\nURL = {http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/93-161},\nNUMBER = {MPI-I-93-161},\nINSTITUTION = {Max-Planck-Institut f{\\"u}r Informatik},\nADDRESS = {Saarbr{\\"u}cken},\nYEAR = {1993},\nDATE = {1993},\nABSTRACT = {In this paper we prove several fundamental theorems, concerning the multi--party communication complexity of Boolean functions. Let $g$ be a real function which approximates Boolean function $f$ of $n$ variables with error less than $1/5$. Then --- from our Theorem 1 --- there exists a $k=O(\\log (n\\L_1(g)))$--party protocol which computes $f$ with a communication of $O(\\log^3(n\\L_1(g)))$ bits, where $\\L_1(g)$ denotes the $\\L_1$ spectral norm of $g$. We show an upper bound to the symmetric $k$--party communication complexity of Boolean functions in terms of their $\\L_1$ norms in our Theorem 3. For $k=2$ it was known that the communication complexity of Boolean functions are closely related with the {\\it rank} of their communication matrix [Ya1]. No analogous upper bound was known for the k--party communication complexity of {\\it arbitrary} Boolean functions, where $k>2$.},\nTYPE = {Research Report / Max-Planck-Institut f\xc3\xbcr Informatik},\n}\n'