In 1976 Fridy presented a series of theorems that chacterize
when matrices preserve the Gibbs\u27 phenomenon. In this paper we present a multidimensional extension of the results of Fridy. In particular, we prove necessary and sufficient conditions for a positive RH-matrix to preserve the double Gibbs\u27 phenomenon for positive double sequences. Other related results are also established

Billy E. Rhoades

Richard F. Patterson

In 1976 Fridy presented a series of theorems that chacterize

when matrices preserve the Gibbs' phenomenon. In this paper we present a multidimensional extension of the results of Fridy. In particular, we prove necessary and sufficient conditions for a positive RH-matrix to preserve the double Gibbs' phenomenon for positive double sequences. Other related results are also established

Patterson, Richard F.

Rhoades, Billy E.

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MATHEMATICAL COMMUNICATIONS 7Math. Commun., Vol. 14, No. 1, pp. 7-12 (2009)Four-dimensional matrix transformation and the doubleGibbs’ phenomenonRichard F.Patterson1,∗and Billy E.Rhoades21 Department of Mathematics and Statistics, University of North Florida, Jacksonville,Florida 32 224, U. S.A.2 Department of Mathematics, Indiana University, Bloomington, Indiana 47 405, U. S.A.Received January 30, 2008; accepted August 31, 2008Abstract. In 1976 Fridy presented a series of theorems that chacterize when matricespreserve the Gibbs’ phenomenon. In this paper we present a multidimensional extension ofthe results of Fridy. In particular, we prove necessary and sufficient conditions for a positiveRH-matrix to preserve the double Gibbs’ phenomenon for positive double sequences. Otherrelated results are also established.AMS subject classifications: Primary 40A99; Secondary 40A05Key words: double sequence, Pringsheim limit point, P-convergent, RH-regular, P-convergent, Gibbs’ phenomenon, double Gibbs’ phenomenon1. IntroductionIn Section 2 we present a definition of the double Gibbs’ phenomenon (Definition 5).In Section 3 we present a number of lemmas and theorems which lead to a charac-terization of those doubly infinite positive RH-matrices which preserve the Gibbs’phenomenon.2. Definitions, notations, and preliminary resultsDefinition 1 (see [6]). A double sequence x = [xk,l] has a Pringsheim limit L(denoted by P-limx = L) provided that given an  > 0 there exists an N ∈ N suchthat |xk,l − L| <  whenever k, l > N . We shall describe such an x more briefly as“P-convergent”.Definition 2 (see [6]). A double sequence x is called definite divergent, if for every(arbitrarily large) G > 0 there exist two natural numbers n1 and n2 such that |xn,k| >G for n ≥ n1, k ≥ n2.The four dimensional matrix A is said to be RH-regular if it maps every boundedP-convergent sequence into a P-convergent sequence with the same P-limit.∗Corresponding author. Email addresses: rpatters@unf.edu (R. F. Patterson),rhoades@indiana.edu (B. E.Rhoades)http://www.mathos.hr/mc c©2009 Department of Mathematics, University of Osijek8 R.F. Patterson and B.E.RhoadesThe assumption of boundedness was made because a double sequence which is P-convergent is not necessarily bounded. Using this definition Robison and Hamilton,independently, both presented the following Silverman-Toeplitz type characteriza-tion of RH-regularity.Theorem 1. The four dimensional matrix A is RH-regular if and only ifRH1 : P − limm,nam,n,k,l = 0 for each k and l;RH2 : P − limm,n∞∑k,l=0am,n,k,l = 1;RH3 : P − limm,n∞∑k=0|am,n,k,l| = 0 for each l;RH4 : P − limm,n∞∑l=0|am,n,k,l| = 0 for each k;RH5 :∞∑k,l=0|am,n,k,l| is P-convergent;RH6 : there exist positive numbers A and B such that∑k,l>B|am,n,k,l| < A.Definition 3 (see [5]). The double sequence [y] is a double subsequence of the se-quence [x] provided that there exist two increasing double index sequences {nj} and{kj} such that if zj = xnj ,kj , then y is formed byz1 z2 z5 z10z4 z3 z6 −z9 z8 z7 −− − − −.Definition 4 (see [5]). A number β is called a Pringsheim limit point of the doublesequence [x] provided that there exists a subsequence [y] of [x] that has Pringsheimlimit β : P-lim[y] = β.Let {xk,l} be a double sequence of real numbers and, for each n, let αn =supn{xk,l : k, l ≥ n}. The Pringsheim limit superior of [x] is defined as follows:1. if α = +∞ for each n, then P-lim sup[x] := +∞;2. if α <∞ for some n, then P-lim sup[x] := infn{αn}.Similarly, let βn = infn{xk,l : k, l ≥ n}. Then the Pringsheim limit inferior of [x] isdefined as follows:1. if βn = −∞ for each n, then P-lim inf[x] := −∞;Four-dimensional matrix transformation and the double Gibbs’ phenomenon 92. if βn > −∞ for some n, then P-lim inf[x] := supn{βn}.We now present a definition of the double Gibbs phenomenon.Definition 5. Let f be a double real-valued sequence {fk,l} which is P-convergentto a function φ at each point of a deleted neighborhood D of the point (x0, y0) =α. If there exist subsequences {fki,lj} of {fk,l} and a double sequence {xk,l} withP− limi,j xi,j = α and eitherP − limi,jfki,lj (xi,j) > P − lim supi,jφ(xi,j) (1)orP − limi,jfki,lj (xi,j) < P − lim infi,jφ(xi,j) (2)then f is said to possess the double Gibbs phenomenon at α.3. Main resultsThis presentation shall examine real matrix summability methods that map doublesequences into double sequences. The transformation A that transforms f intoAf(x, y) for each (x, y) ∈ D where (Af)(x, y) = {(Af)m,n(x, y)}∞m,n=0 is defined by(Af)m,n(x, y) =∞∑k,l=1am,n,k,lfk,l(x, y).Observe that if the limit function φ has limit α, then (1) and (2) imply that theGibbs phenomenon is equivalent to nonuniform P-convergence. This observationyields the following lemma.Lemma 1. The double real-valued sequence {fk,l} is uniformly P-convergent on Dif and only if f is uniformly P-convergent on every countable subset of D.Let {fk,l} be double sequence of real-valued functions and x a double sequencein D. We can now consider the following four-dimensional matrix of functions:Fm,n,k,l = fm,n(xk,l)− φ(xk,l).Note that each pairwise column of {Fm,n} P-converges to 0 and thus Lemma 3 canbe used to describe the connection between uniformly P-convergence and the doubleGibbs Phenomenon in the following sense.Lemma 2. The double real-valued sequence {fk,l} is uniformly P-convergent on Dif and only if for every double sequence x in D the corresponding four dimensionmatrix F has the property that its pairwise column double sequences P-convergesuniformly to 0.Lemma 3. If f P-converges (point-wise) in a deleted neighborhood D of α, then thefollowing are equivalent:10 R.F. Patterson and B.E.Rhoades1. f displays the double Gibbs phenomenon at α;2. there is a double number sequence x P-converging to α for which the corre-sponding functions matrix F is such thatlimm,n,k,lFm,n,k,l 6= 0.The notation limk,l,i,jFk,l,i,j = λ shall meanP − limm,n( supk,l>m,n;i,j>m,nFk,l,i,j − λ) = 0.Let G denote the collection of matrices F which have 0 as a pairwise Pringsheimcolumn limit and which violates (1) or (2).Theorem 2. If A is GP-preserving and {ki}∞i=0 and {lj}∞j=0 is any infinite subsetof pairwise column indices, thenP − limm,n(supi,j|am,n,ki,lj |) > 0.Proof. Suppose that A has a pairwise column such thatP − limm,n(supi,j|am,n,ki,lj |) = 0,and consider the following:Fk,l,i,j :={1, (k, l) = (ki, lj)0, (k, l) 6= (ki, lj).Since {ki} and {lj} are divergent single subsequences, F is in G. Then(AF )m,n,i,j = am,n,ki,lj for all (m,n) and (i, j),and limm,n,i,j(AF )m,n,i,j = 0. (i.e. AF is not in G). Hence A is not GP-preserving.For RH-regular matrices both pairwise columns and rows are P-null double se-quences. In this case Theorem 2 can be simplified as follows:Theorem 3. If A is an RH-regular matrix and (ki, lj) a doubly infinite set of pair-wise column indices, thenP − limm,n(supi,j|am,n,ki,lj |) > 0 (3)holds if and only if there exists a positive number  and infinitely many termsam,n,ki,lj such that |am,n,ki,lj | ≥ .Corollary 1. If A is RH-regular and GP-preserving, thenP − limm,n( sup(k,l)>(m,n);(i,j)>(m,n)|ak,l,i,j |) > 0.Four-dimensional matrix transformation and the double Gibbs’ phenomenon 11This immediately follows from (3).This result has an invariant Pringsheim core type flavor, which was suggestedby both Fridy ([2, Theorem 3]) and [5, Theorem 3.2], with a proof that is similar toTheorem 2, and is therefore omitted.Theorem 4. If A is GP-preserving and (ki, lj) a doubly infinite set of pairwisecolumn indices, thenP − lim supm,n|∞∑i,j=1am,n,ki,lj | > 0. (4)Thus far all lemmas and theorems have provided only sufficient conditions toensure that A is GP-preserving. However, note thatA(x)m,n =x2m,2n + x2m+1,2n+12is an RH-regular transformation that satisfies the conditions that there exists apositive number  and infinitely many terms am,n,ki,lj such that |am,n,ki,lj | ≥  andP − limm,n( sup(k,l)>(m,n);(i,j)>(m,n)|ak,l,i,j |) > 0hold.Consider the following four dimensional matrixFm,n,k,l := 1, if k = 2i, l = 2j−1, if k = 2i, l = 2j0, otherwise.Observe that F ∈ G and (AF )m,n,k,l = 0, so that A is not GP-preserving. Inaddition, let G+ denote the collection of nonnegative matrices F which have 0 as apairwise Pringsheim column limit and fail to satisfy (1) or (2).To obtain necessary conditions we need to place suitable restriction on A and F .The following is a partial converse of Theorem 3.Theorem 5. If A is an RH-regular nonnegative four dimensional matrix, then Amaps G+ into G+ if and only if there exists a positive number  and infinitely manyterms am,n,ki,lj such that |am,n,ki,lj | ≥ .Proof. If F is in G+, then there exists a positive number δ and two double Pring-sheim sequences (mi, nj) and (ki, lj) such that for each pair (i, j), Fmi,nj ,ki,lj ≥δ. By (3) there are infinitely many Pringsheim order terms am,n,ki,lj such thatam,n,ki,lj ≥  > 0. For each am,n,ki,lj we have(AF )m,n,ki,lj ≥ am,n,ki,ljFmi,nj ,ki,lj ≥ δ.Thus AF satisfies (3), and, since AF is nonnegative, it is clearly in G+. Note thatthe converse clearly follows from Theorem 2.12 R.F. Patterson and B.E.RhoadesReferences[1] C.R.Adams, On Summability of Double Series, Trans. Amer. Math. Soc. 34(1932),215-230.[2] J.A. Fridy, Matrix summability and a generalized Gibbs phenomenon, AequationsMath. 14(1976), 405-412.[3] H. J.Hamilton, Transformations of Multiple Sequences, Duke Math. Jour. 2(1936),29-60.[4] G.H.Hardy, Divergent Series, Oxford Univ. Press, London, 1949.[5] R.F. Patterson Analogues of some Fundamental Theorems of Summability Theory,Internat. J. Math. & Math. Sci. 23(2000), 1-9.[6] A.Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, MathematischeAnnalen 53(1900), 289-321[7] G.M.Robison, Divergent Double Sequences and Series, Amer. Math. Soc. Trans.28(1926), 50-73.

Four-dimensional matrix transformation and the double Gibbs\u27 phenomenon

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