Pacific Journal of Mathematics GENERALIZED WALSH TRANSFORMS RALPH GORDON SELFRIDGE Vol. 5, No. 3 November 1955 GENERALIZED WALSH TRANSFORM R. G SELFRIDGE Introduction. The Walsh functions were first defined by Walsh [6] as a completion of the Rademacher functions in the interval (0,1). As originally defined φn(x) took on the values ±1. The generalization of Chrestenson [l] 27Tnι a permits φR(x) to have the values e ' fOΓ some integer α, and also leads to a complete orthonormal system over [0,1], Fine [2] considers the original Walsh function, but with arbitrary subscript, attained by consideration of certain dyadic groups. This paper combines these two generalizations by starting with the Walsh functions as defined by Chrestenson and then using a subsidiary result of Fine to define a Walsh function φ (x) for arbitrary subscript. With φ (x) one can define a Walsh-Fourier transform for functions in Lp(0, 00 \ 1 < P < 2. Many of the ordinary Fourier transform theorems carry over, with certain modifications. For 1 < p < 2 the transform is defined as a limit in the appropriate mean, with a Plancherel theorem holding for p = 2. Since the proofs carry over from ordinary transforms, or from the L\ theory only a few theorems are stated for these cases with only brief proofs. The case of L ι re- quires considerably more preparation. Section one is devoted to definitions and obtaining certain varied but very necessary results, such as the evaluation of definite integrals of φ (x) over specified intervals, which are used constantly throughout the paper. Walshτ Fourier series are introduced, and some of the sufficient conditions for con- vergence of such a series to the generating function are listed but not proved, since the proofs are available in Chrestenson's paper [l]. Section two covers certain basic results for Lγ transforms and associated kernals. A Riemann-Lebesgue theorem follows simply from results of section one, as do sufficient conditions for the convergence to fix) of the inverse transform of the transform of / ix). The function Pβ(xQγ) is defined as Received May 26, 1953. This paper is part of a thesis, done under the direction of Professor Paul Civin, done towards the requirements for a Ph.D. at the University of Oregon. Pacific J. Math. 5 (1955), 451-480 451 452 R. G. SELFRIDGE Γ φTΓ)ψxit)dtf Jo y x and is considered in some detail. With this function it is possible to show that lim I PβixQy)f ix)dx$ under fairly general conditions. Section three is devoted almost entirely to C, 1 summability of transforms. If f{x)£zLι and has a transform F (y) then one has /(%)= lim fβ (l-[y]/β)F(y)ψT/)dγ provided / (y) is locally essentially bounded at x and fh \f{χ+t)-f(x)\dt = o(h). Jo It is not possible to remove completely the requirement of local essential boundedness, so that it cannot be said that the inverse transform is C, 1 sum- mable almost everywhere to fix). Again one has that if x has a finite expansion in powers of CC then the conditions need only be right-hand conditions. Finally section four considers transforms of functions in Lp, 1 < p < 2. For p = 2 one has a Plancherel theorem, and for all p, 1 < p < 2 one has, if fix) € Lp, then the transform F(y) G Lp/pmι and Fix) = -ί [°° fiγ)P iγ)dy, f ix) = ^- f°° F iγ )P (y )dγ. dx Jo x dx Jo X One also has C$ 1 summability of the inverse transform yielding fix) for almost every x at which f iγ) is locally essentially bounded. 1. Definitions and lemmas. Throughout this paper α is taken to be a fixed, but arbitrary integer greater than one. For such an (X each x >^ 0 has an ex- pansion x = Σ°lyy Λ^Ol"1, 0 < xι < α, that is unique if x^ j4 0 and, in case of choice, a finite expansion. Under these conditions one has: ml DEFINITION 1. Degree oί x =D(x) =-N$ where ^=2°!^ χi& 9 %N ^0. GENERALIZED WALSH TRANSFORMS 453 For convenience in what follows it will be supposed that, by addition of dummy coefficients if necessary, x = Σ°l^ Xi&~K M < 0, so that D(x) is not necessarily — M. In all that follows the use of a subscript on a variable will mean the cor- responding coefficient of the CC-expansion. All intervals will be closed on the left and open on the right unless otherwise stated, and any number that has a finite CX-expansion will be called an Cί-adic rational. Now if Cύ = e πι/a one defines in sucession: 1 DEFINITION 2. (a) φo(χ) = ω* n (b) Φn(x) = φo(d x) n >_0 ί (c) IfB=Σ?-Jvn<α- , ώ U) = Π?-0U.(*))""* it ii it i υ i (d) φ (x) - φv i(x) ΦΓ i(y)t [x] = greatest integer in x. Since the addition of an integer will not change Xι it is easy to see that n φQ(x) has period one, and hence φn(x) has period 0L~ . Further one has that φ \x) — CU and ψ \x) = ω § z = >. ._Λ n .%.. ,, so that ψ \x) is ot period one. φQ(x) compares with the original Rademacher function, and for the case (X = 2 the φ ix) reduce to the ordinary Walsh functions. For 0C >_ 2 the com- pleteness and orthonormality of φn(x) over [0, l] have been shown by Chresten- son [l]. Since k{ = 0, i > 0, if k is an integer, it is clear that Φn(k) = 1. Thus Defi- nition 2d) extending the Φn(x) to arbitrary subscript is consistent with Defi- nition 2c). Further, by symmetry, one has φ (x) = φχ(γ). Before defining transforms for functions there are a number of preliminary steps and additional terminology that can be used to advantage. LEMMA 1. // i*-N i=~M then 454 R. G. SELFREDGE Proof. From Definitions (2a, b, c) it is easy to see that φr Ά(X) = ωμ where x an μ=*Σ,i=Q ymi i + i* d similarly for 0rj(y). Combining the two yields the result. DEFINITION 3. // ι=-iV ι=~M then z -x@y is defined as oo z— ^"^ Zj Cί"1, where zι = x± + y. (mod Cί), i=-Γ provided that zt ^ (X — 1 for i >_ K9 and Γ = max (M, N). Similarly z = xQ γ is defined as z - Σ zi&"1 where Zj = xι -y. (mod α). Clearly for each x9 z = x@y or z - x Qy is defined for almost every y. The following lemmas then follow quite simply from Lemma 1 and Definition 3. LEMMA 2. (a) φA(λnx) = φ(any) -oo<rc<oo y x (b) φAx) φAz) = φAx © z ) each %, a.e.z. 7 7 7 (c) ψ^ix) ψAz) = ώ (Λ; Q Z) = ^v(z Q %), each %, a.e. z. 7 7 7 7 The last identity of Lemma 2 can be written in a slightly different form, which is useful in case of integration with respect to y. LEMMA 3. // 0 <y < β then there is a q such that ψ (x) φ (z) = φAq). Proof. It is only necessary to define q in case x Q z is undefined. Take n n such that β < 0L , and define q.^x.-z. (mod α) for i < n$ and then set q ± Σ "s-^ q CC\ Clearly this definition will suffice. DEFINITION 4. (a) Dn(ί) = Σ?:J φXt) (b) Cb(t) = 1, 0 < t < 6, =0 b < t. GENERALIZED WALSH TRANSFORMS 455 N LEMMA 4. (a) D^it) = OL CamN(it!), where it ! = t - [ί]; Dn(ί) =Dn({ί 1) N (b) Dn(t)=Dp(a t)DaN(t) + 0 aN(t)Dq(t) where rc = pCί^ + 7, 0 < 9 < otN. (c) |Dπ(ί)| < α/2{ί}. Proof, (a) Let ίn be the first non-zero coefficient in the expansion of ί. If n ^ N then by Lemma 1 φk(t) = 1 and D N(t) = α^. If w < /V, consider the range pαn </c < (p + 1) CCn. In this range one has N z k 0Γ imntn ψk(t) = ω , z= Σ -ih + i Ψk(*) = Aω 9 withίn^0. Thus one has (p+l)α"-l cu 1 kt tk Now summing on p yields the desired result. The second part is immediate since Ψk(t) = ψk({t\). (b) Since q < OC one has rc = pα © 7 and n-1 pα^-l τι-1 p-l α^-l i=0 ^ A;=0 j = 0 ^ ί=0 t)Ό (ί) + N = Σ Φk^ yv *A αN(ί)^(ί)=Z)p(α ί)DαN(ί) + tA aNit)Dq(t). /c=0 α pα α pα (c) Let D({ί}) = Λ^. Then n paN ^ aNq ^ α/ Now |D^(ί)| < α^/2 < α/2{ί|, since Dq{t) = Dq(t) - D N(t). 1.0 COROLLARY, (a) D. 456 R.G. SELFRIDGE n n (b) £ Ψk(*)ψk(y)=O, unless [θL χ] = [θi y]9 Sα»-1 n n or Σ Φk^)'^py')^^ι([a χ] Q[a γ]). (c) IfD({x\)=N then D (x) = 0 for n > N. Corresponding to Lemma 4 there are equivalent results with respect to the integral of φ (x). b DEFINITION 5. P&U) = fo ψt(χ)dx. n n LEMMA 5. (a) Pb(t) = a Pb^n{a t) (b) Pι(χ)=Pi([χ'\) = Ci (c) P (x) = OLnC (x) (d) Proof, (a) This follows immediately by change of variable of integration.