WHAT STRONG MONOTONICITY CONDITION ON FOURIER COEFFICIENTS CAN MAKE THE RATIO \\f-sn(f)\\/en(f) BOUNDED? S. P. ZHOU. (Communicated by J.
|
|
- Dominick Mathews
- 5 years ago
- Views:
Transcription
1 proceedings of the american mathematical society Volume 121, Number 3, July 1994 WHAT STRONG MONOTONICITY CONDITION ON FOURIER COEFFICIENTS CAN MAKE THE RATIO \\f-sn(f)\\/en(f) BOUNDED? S. P. ZHOU (Communicated by J. Marshall Ash) Dedicated to Professor Tingfan Xiefor his sixtieth birthday Abstract. Let {</>n} JÍ, be a positive increasing sequence and <t>nf(n) decrease. We ask what exact conditions on {<j> ) make \\f Sn(f)\\/E (f) bounded? The present paper will give a complete answer to it. 1. Introduction Let C2n be the space of all real continuous functions of period 2n with norm = max_<x<00. For an even function / e C2n (we always assume that f(x) is an even function), denote its Fourier series by f(x)~yjf(n)cosnx> and the «th partial sum of the Fourier series by n S (f, x) = ^/(fc)cosa:x. k=0 In Fourier analysis, it is always an interesting question whether or not the approximation by partial sums of a continuous function can achieve the best approximation rate in uniform norm. It is well known that (cf. [Zyg]) (1) \\f-sn(f)\\ = 0(log(n + l)en(f)), where E (f) is the best approximation by trigonometric polynomials of degree n. We also know that (cf. [Zyg]), in general, the factor log(«+ 1) in (1) cannot be removed. However, people always hope that, for f(x) in some subclass of C2l[ (especially in those which can be easily characterized like continuous functions with restricted Fourier coefficients), one can obtain some better estimates, preferably, (2)_ \\f-sn(f)\\ = 0(En(f)). Received by the editors June 22, 1992 and, in revised form, October 10, Mathematics Subject Classification. Primary 42A10, 42A20, 42A32. Key words and phrases. Approximation, Fourier sums, coefficients, strong monotonicity. Supported in part by NSERC Canada American Mathematical Society /94 $ $.25 per page
2 780 S. P. ZHOU One result in this direction is due to [NeRi] showing that \\f-sn(f)\\<4een(f) holds for every function / e C2n with decreasing and logarithmically concave Fourier coefficients (that a sequence a is logarithmically concave means that it satisfies a2 >an+xan-x for every n). At the same time, we can easily see that for / e C2n with positive Fourier coefficients, consequently for / e C2n with monotone Fourier coefficients, \\f-sn(f)\\ = 0(cok(f,n-x)), k>l, obviously holds, where cok(f, t) is the modulus of smthness of order k of f(x), and the O in the above inequality depends upon k only (the same result also holds for odd continuous functions of period 2n by a little more complicated arguments). Probably from this point of view, [Xie] asked Problem XI. Does there exist a positive constant M such that, for every function / e C2n with positive Fourier coefficients and all n 1,2,..., \\f-sn(f)\\<men(f)1 Later, in a seminar, Xie asked further Problem X2. Does there exist a positive constant M such that, for every function / e C2n with decreasing Fourier coefficients and all «= 1,2,..., \\f-sn(f)\\<men(f)1 A little surprisingly, [Zho] constructed a counterexample showing that there exists a function / e C2n with strongly monotone Fourier coefficients (which means that nf(n) decreases, in symbol, nf(n) J.) such that Ums p!tftça>o. n- log ne (f) This gave a negative answer to Problem X2 as well as to Problem X1. Some natural questions now arise: What kinds of strong monotonicity conditions on Fourier coefficients can guarantee (2)? Is there any counterexample which satisfies nsf(n) j for sufficiently large s > 0 (the counterexample given in [Zho] does not satisfy this condition) but makes /- S (f)\\/e (f) unbounded? One can easily verify a trivial example that if q"f(n) j for some q > 1, then (2) holds. Therefore, a general statement of the problem must be as follows. Let O = {4>n} Li be a positive increasing sequence and 4> f(n) [. Then what are exact conditions on {</> } which make \\f - S (f)\\/en(f) bounded? The present paper will give a complete answer to this question. We always let C indicate some positive absolute constant which is not necessarily the same in every different occurrence. 2. Result and prf Given e > 0. Assume í> = {(f>n} xlx is a positive increasing sequence. Write L (0, e) to be the maximum length of string of Wk+x/MlZl not exceeding
3 1 + e, namely, BOUNDING \\f-s (f)\\/e (f ) 781 Ln(< >, e) = max {k-j+1: (f>j+x/(pj < 1 + e, X<j<k<n-X 4>j+2/4>j+x <l+e,..., (f>k+x/(pk < 1 + e}, in case there is no (pj+x/(pj < 1 + e for 1 < / < n- 1, we simply set Ln(<S>, e) 0. Since L +X(Q>, e) > Ln(<S>, e), we may define the limit L(<S>, e) = lim L (<D, e). n * Define J7+ = < / e C2n : f ~ ^2 f(n)cos nx ' 4nf(n) i > and &E = {fec2n: /-SB(/) We have the following result. = 0(En(f))}. Theorem 1. The necessary and sufficient condition for ^c5^e is L(&, e0) = 0(1) for some eo > 0. Lemma 1. L(0, e) = oc for all e > 0 implies that there are two increasing sequences of natural numbers {«/}/f] and {w/}g, with «/ + W/-I-1 < n +x, 1 = 1,2,..., as well as a decreasing sequence { /}/2i with lini/-^ S = 0 such that (3) (pn,+j+x I4>n,+j <1+S, j = 0,l,...,m, 1=1,2,... Prf. The argument is elementary and we omit it. D Lemma 2. Let L(0, e) = for all e > 0. Then there is a function f e J7q, such that «-» LnU ) Prf. Since <f>n is positive and increasing, there is a constant c > 0 such that (p > c for all n = 1, 2,.... Without loss of generality we therefore assume </) > 1 for all n = 1,2,...' From Lemma 1, we have three sequences {«/}, {m }, and {S } making (3) be satisfied. Then we may also assume o < 1/e2 and 3 < m < n for I = 1,2,.... We construct the required counterexample by induction. Let /(0) be any positive number and /( 1 ) = 1. Given f(k), we have the following two cases. Case 1. k ^ n, n + 1,..., n + m, I = 1,2,... In this case, we simply set (4) f(k+l) = f(k)/(2(f>k+l), in particular, f(n,) = f(nl-l)/(2<p l). Case 2. k = n for some I. Now we construct (5) /(*,+;)= hni) 2j<pni+xlo%(Mt + iy j =1,2,..., M := minl m, yjôfx \ <e, 'Otherwise write <j>$ = <j> /c. Then </>J > 1 and we use <t>'n instead.
4 782 S. P. ZHOU where [x] is the greatest integer not exceeding x, (6) f(ni + J+l)=iJ, 2<Pn,+j+X log(m/ + 1) In this way we have defined a function j = M, M, + l, m. f(x) = Y^f(n)cosnxn=0 Clearly, by condition (5), for / = 1, 2,..., n/+mi (7) g'/(fc)<^)<^l, while by conditions (4) and (6), for k ^ n + 1, n + 2,..., n + M - 1, (8) f(k+l)<\f(k). Altogether, n*lo f(n) < > which means / e C2n. We check that n +Mi f(0)-sn,(f,0)= Y f(k)+ Y k=n +X k=ni+m/+x f(k) > (pn +xlog(m, + l) logmi-o 2(j>ni+Ml+\ log(m + 1) (by (7), (8)) ^Crh <Pn,+X At the same time, En,(f) < By noting that M, M, Y Ant + ;') cos(n, + j)x - Y f(n + j) cos(n - j)x j=x ;=i + Y Ak). k=n,+m;+x M, M, Y f("i + J) cos(«, + j)x - Y f(ni + J) cos("/ - J)x j=x j=x M, f(n )sinn x ^ sinjx '(pm+xlog(mi + 1) ^ j=x j together with the well-known inequality (cf. [Zyg]) sup m>x Esmkx ~k~ k=x < 3^,
5 BOUNDING \\f-s (f)\\/en{f) 783 we have p (f^ 30t/(»/) ( f(n, + M,) \ E"7J ) ^ a 0 l+i.i«,itf.j.n log(m/ + 1) + ü \2(pni+Ul+x log(mi + 1)) = o{_m_v Therefore, I^SfMïC.o.M,. Since lim/^a// = by lim^^ôi = 0 and lim/^^an/ =, the above estimate yields _ ll/-fl»,(/)ll lim- /-co,(/) ' =. We now need to verify that / e ^>. If k n + 1, n + 2,...,n + M - 1, 1= 1,2,..., by (4), (6), When k = n + j, j = 1, 2,..., M 1, for some /,, - i., n - í/t\ /("/) (<t>n,+j+x <t>n,+j \ <pk+xf(k +1) - **/(*) = 2^+llog(M/ + 1) (-pr - J By Lemma 1, but for 1 <j <M -1, _ /(«/) ^h/+j (K+j+i_ l _ ]\ 2(j)ni+x l0g(m + 1) j + 1 V <Pn,+j j ) ' <f>ni+j+x/<t>n,+j < 1+àl, l/j>l/(ml-í)>x/jl, so that (pni+j+x/(/),+j - 1-1/7 < Si - y/ôi < 0. Altogether it follows that for k = n + j, j = 1, 2,..., M 1, Thus we are done. D <Pk+xf(k+l)-(t>kf(k)<0. Lemma 3. Assume L(0, n) = 0(1) for some en > 0. Let f e J7. Then for n>l, 00 Y f(k) = 0(f(n + l)). k=n+x Prf. Given n > 1. Since L(0, n) = 0(1), say, L(0, e0) < M, we have a sequence {«/} C {n + 1, n + 2,...} such that 0 < n - n - 1 < M, 0 < "i+x -n < M, and (9) </>,+1/</>«,> 1+eo.
6 784 S. P. ZHOU Write Ef(k)=( Y+E OO / «OO "/+! \ k=n+x \k=n+x l=x k=n +X/ Evidently, for k e {n + 2, n + 3,..., n +x}, /(*) f(k) = 4>J(k)± < 4>nl+xf(n, + l)j-k < f(n, + 1), and similarly for k = n+l,n + 2,...,n, f(k) < f(n + 1). Henee / \ Y f(k)<m f(n+l) + Yf(n, + l)\ k=n+x \ l=x ) < M (f(n + 1) + (f>ní+xf(nx + 1)Y ^-) = M[f(n + l) + f(nx + l)y <Pn + 1 /=1 <t>n,+x <Mf(n + l)ll+y(l+ o)-l\ (by (9)) = 0(f(n + l)). D Prf of Theorem 1. Necessity. By Lemma 2, we have the required result. Sufficiency. From Lemma 3, \\f-sn(f)\\= Y f(k) = 0(f(n + l)). k=n+x On the other hand, En(f)> (jon\f(x)-sn(f,x)\2dx) That is, or J7 ctt^e- O ( Y f2(k)\ k=n+x 7 \ '/2 \\f-sn(f)\\ = 0(En(f)), >/(«+ i). The following are two typical applications of Theorem 1. Corollary 1. Let s > 0 be any given number. Then there exists a function f e C2n with nsf(n) { such that /i-, <n\j! Corollary 2. If f e C2n satisfies q"f(n) Tor some q > 1, then \\f-sn(f)\\ = 0(En(f)). For sine series, we have a similar result (with a similar prf).
7 Theorem 2. Let g(x) ec2n, BOUNDING \\f-s (f)\\/e (f) 785 g(x)^y^(n)sinnxn=x Then the necessary and sufficient condition for J77 c <5* is L(<b, en) = 0(1) Tor some en > 0, where J7 = \ge C2n: g ^Yê(n)sinnx, (png(n) I >. Bibliography [NeRi] D. J. Newman and T. J. Rivlin, Approximation of monomials by lower degree polynomials, Aequationes Math. 14 (1976), [Xie] T. F. Xie, Problem no. 1, Approx. Theory Appl. 3 (1987), 144. [Zho] S. P. Zhou, A problem on approximation by Fourier sums with monotone coefficients, J. Approx. Theory 62 (1990), [Zyg] A. A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, Dalhousie University, Department of Mathematics, Statistics and Computing Science, Halifax, Nova Scotia, Canada B3H 3J5 Current address: Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 address : zhouîlapprox. math. ualberta. ca
ON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT
ON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT Received: 31 July, 2008 Accepted: 06 February, 2009 Communicated by: SIMON J SMITH Department of Mathematics and Statistics La Trobe University,
More informationON THE OSCILLATION OF THE DERIVATIVES OF A PERIODIC FUNCTION
ON THE OSCILLATION OF THE DERIVATIVES OF A PERIODIC FUNCTION BY GEORGE PÖLYA AND NORBERT WIENER 1. Let f(x) be a real valued periodic function of period 2ir defined for all real values of x and possessing
More information2 K cos nkx + K si" nkx) (1.1)
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 71, Number 1, August 1978 ON THE ABSOLUTE CONVERGENCE OF LACUNARY FOURIER SERIES J. R. PATADIA Abstract. Let / G L[ it, -n\ be 27r-periodic. Noble
More informationON THE SET OF ALL CONTINUOUS FUNCTIONS WITH UNIFORMLY CONVERGENT FOURIER SERIES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 24, Number, November 996 ON THE SET OF ALL CONTINUOUS FUNCTIONS WITH UNIFORMLY CONVERGENT FOURIER SERIES HASEO KI Communicated by Andreas R. Blass)
More informationON NONNEGATIVE COSINE POLYNOMIALS WITH NONNEGATIVE INTEGRAL COEFFICIENTS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 120, Number I, January 1994 ON NONNEGATIVE COSINE POLYNOMIALS WITH NONNEGATIVE INTEGRAL COEFFICIENTS MIHAIL N. KOLOUNTZAKIS (Communicated by J. Marshall
More informationSymmetric polynomials and symmetric mean inequalities
Symmetric polynomials and symmetric mean inequalities Karl Mahlburg Department of Mathematics Louisiana State University Baton Rouge, LA 70803, U.S.A. mahlburg@math.lsu.edu Clifford Smyth Department of
More informationAbsolutely convergent Fourier series and classical function classes FERENC MÓRICZ
Absolutely convergent Fourier series and classical function classes FERENC MÓRICZ Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary, e-mail: moricz@math.u-szeged.hu Abstract.
More information1981] 209 Let A have property P 2 then [7(n) is the number of primes not exceeding r.] (1) Tr(n) + c l n 2/3 (log n) -2 < max k < ir(n) + c 2 n 2/3 (l
208 ~ A ~ 9 ' Note that, by (4.4) and (4.5), (7.6) holds for all nonnegatíve p. Substituting from (7.6) in (6.1) and (6.2) and evaluating coefficients of xm, we obtain the following two identities. (p
More informationON THE GIBBS PHENOMENON FOR HARMONIC MEANS FU CHENG HSIANG. 1. Let a sequence of functions {fn(x)} converge to a function/(se)
ON THE GIBBS PHENOMENON FOR HARMONIC MEANS FU CHENG HSIANG 1. Let a sequence of functions {fn(x)} converge to a function/(se) for x0
More informationON JAMES' QUASI-REFLEXIVE BANACH SPACE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 67, Number 2, December 1977 ON JAMES' QUASI-REFLEXIVE BANACH SPACE P. G. CASAZZA, BOR-LUH LIN AND R. H. LOHMAN Abstract. In the James' space /, there
More information252 P. ERDÖS [December sequence of integers then for some m, g(m) >_ 1. Theorem 1 would follow from u,(n) = 0(n/(logn) 1/2 ). THEOREM 2. u 2 <<(n) < c
Reprinted from ISRAEL JOURNAL OF MATHEMATICS Vol. 2, No. 4, December 1964 Define ON THE MULTIPLICATIVE REPRESENTATION OF INTEGERS BY P. ERDÖS Dedicated to my friend A. D. Wallace on the occasion of his
More informationIntroductory Analysis I Fall 2014 Homework #9 Due: Wednesday, November 19
Introductory Analysis I Fall 204 Homework #9 Due: Wednesday, November 9 Here is an easy one, to serve as warmup Assume M is a compact metric space and N is a metric space Assume that f n : M N for each
More informationTHE FIRST SIGN CHANGE OF A COSINE POLYNOMIAL
proceedings of the american mathematical society Volume 111, Number 3, March 1991 THE FIRST SIGN CHANGE OF A COSINE POLYNOMIAL JIANG ZENG (Communicated by Kenneth R. Meyer) Abstract. Nulton and Stolarsky
More informationSection 11.1 Sequences
Math 152 c Lynch 1 of 8 Section 11.1 Sequences A sequence is a list of numbers written in a definite order: a 1, a 2, a 3,..., a n,... Notation. The sequence {a 1, a 2, a 3,...} can also be written {a
More informationSTATISTICAL LIMIT POINTS
proceedings of the american mathematical society Volume 118, Number 4, August 1993 STATISTICAL LIMIT POINTS J. A. FRIDY (Communicated by Andrew M. Bruckner) Abstract. Following the concept of a statistically
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics NEWER APPLICATIONS OF GENERALIZED MONOTONE SEQUENCES L. LEINDLER Bolyai Institute, University of Szeged, Aradi vértanúk tere 1 H-6720 Szeged, Hungary
More information2. Linear recursions, different cases We discern amongst 5 different cases with respect to the behaviour of the series C(x) (see (. 3)). Let R be the
MATHEMATICS SOME LINEAR AND SOME QUADRATIC RECURSION FORMULAS. I BY N. G. DE BRUIJN AND P. ERDÖS (Communicated by Prow. H. D. KLOOSTERMAN at the meeting ow Novemver 24, 95). Introduction We shall mainly
More information(3) lk'll[-i,i] < ci«lkll[-i,i]> where c, is independent of n [5]. This, of course, yields the following inequality:
proceedings of the american mathematical society Volume 93, Number 1, lanuary 1985 MARKOV'S INEQUALITY FOR POLYNOMIALS WITH REAL ZEROS PETER BORWEIN1 Abstract. Markov's inequality asserts that \\p' \\
More informationSMOOTHNESS OF FUNCTIONS GENERATED BY RIESZ PRODUCTS
SMOOTHNESS OF FUNCTIONS GENERATED BY RIESZ PRODUCTS PETER L. DUREN Riesz products are a useful apparatus for constructing singular functions with special properties. They have been an important source
More informationError Estimates for Trigonometric Interpolation. of Periodic Functions in Lip 1. Knut Petras
Error Estimates for Trigonometric Interpolation of Periodic Functions in Lip Knut Petras Abstract. The Peano kernel method is used in order to calculate the best possible constants c, independent of M,
More informationCOMPLETENESS AND THE CONTRACTION PRINCIPLE J. M. BORWEIN'
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 87. Number 2. February IW COMPLETENESS AND THE CONTRACTION PRINCIPLE J. M. BORWEIN' Abstract. We prove (something more general than) the result that
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationVectors in Function Spaces
Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also
More informationr f l*llv»n = [/ [g*ix)]qw(x)dx < OO,
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 320, Number 2, August 1990 MAXIMAL FUNCTIONS ON CLASSICAL LORENTZ SPACES AND HARDY'S INEQUALITY WITH WEIGHTS FOR NONINCREASING FUNCTIONS MIGUEL
More informationON THE AVERAGE NUMBER OF REAL ROOTS OF A RANDOM ALGEBRAIC EQUATION
ON THE AVERAGE NUMBER OF REAL ROOTS OF A RANDOM ALGEBRAIC EQUATION M. KAC 1. Introduction. Consider the algebraic equation (1) Xo + X x x + X 2 x 2 + + In-i^" 1 = 0, where the X's are independent random
More informationCLASSES OF L'-CONVERGENCE OF FOURIER AND FOURIER-STIELTJES SERIES CASLAV V. STANOJEVIC
proceedings of the american mathematical society Volume 82, Number 2, June 1981 CLASSES OF L'-CONVERGENCE OF FOURIER AND FOURIER-STIELTJES SERIES CASLAV V. STANOJEVIC Abstract. It is shown that the Fomin
More informationPart 3.3 Differentiation Taylor Polynomials
Part 3.3 Differentiation 3..3.1 Taylor Polynomials Definition 3.3.1 Taylor 1715 and Maclaurin 1742) If a is a fixed number, and f is a function whose first n derivatives exist at a then the Taylor polynomial
More informationDefinitions & Theorems
Definitions & Theorems Math 147, Fall 2009 December 19, 2010 Contents 1 Logic 2 1.1 Sets.................................................. 2 1.2 The Peano axioms..........................................
More informationConvergence of sequences and series
Convergence of sequences and series A sequence f is a map from N the positive integers to a set. We often write the map outputs as f n rather than f(n). Often we just list the outputs in order and leave
More informationA CONVERGENCE CRITERION FOR FOURIER SERIES
A CONVERGENCE CRITERION FOR FOURIER SERIES M. TOMIC 1.1. It is known that the condition (1) ta(xo,h) sf(xo + h) -f(x0) = oi-r--- -V A-*0, \ log  / for a special x0 does not imply convergence of the Fourier
More informationAnalysis III. Exam 1
Analysis III Math 414 Spring 27 Professor Ben Richert Exam 1 Solutions Problem 1 Let X be the set of all continuous real valued functions on [, 1], and let ρ : X X R be the function ρ(f, g) = sup f g (1)
More informationFourier Series. 1. Review of Linear Algebra
Fourier Series In this section we give a short introduction to Fourier Analysis. If you are interested in Fourier analysis and would like to know more detail, I highly recommend the following book: Fourier
More informationHonors Advanced Algebra Unit 3: Polynomial Functions November 9, 2016 Task 11: Characteristics of Polynomial Functions
Honors Advanced Algebra Name Unit 3: Polynomial Functions November 9, 2016 Task 11: Characteristics of Polynomial Functions MGSE9 12.F.IF.7 Graph functions expressed symbolically and show key features
More informationTHE FIXED POINT PROPERTY FOR CONTINUA APPROXIMATED FROM WITHIN BY PEANO CONTINUA WITH THIS PROPERTY
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 91, Number 3, July 1984 THE FIXED POINT PROPERTY FOR CONTINUA APPROXIMATED FROM WITHIN BY PEANO CONTINUA WITH THIS PROPERTY AKIRA TOMINAGA ABSTRACT.
More informationELEMENTARY PROBLEMS AND SOLUTIONS. Edited by Russ Euler and Jawad Sadek
Edited by Russ Euler and Jawad Sadek Please submit all new problem proposals and corresponding solutions to the Problems Editor, DR. RUSS EULER, Department of Mathematics and Statistics, Northwest Missouri
More informationON LINEAR RECURRENCES WITH POSITIVE VARIABLE COEFFICIENTS IN BANACH SPACES
ON LINEAR RECURRENCES WITH POSITIVE VARIABLE COEFFICIENTS IN BANACH SPACES ALEXANDRU MIHAIL The purpose of the paper is to study the convergence of a linear recurrence with positive variable coefficients
More informationFUNCTIONAL SEQUENTIAL AND TRIGONOMETRIC SUMMABILITY OF REAL AND COMPLEX FUNCTIONS
International Journal of Analysis Applications ISSN 91-8639 Volume 15, Number (17), -8 DOI: 1.894/91-8639-15-17- FUNCTIONAL SEQUENTIAL AND TRIGONOMETRIC SUMMABILITY OF REAL AND COMPLEX FUNCTIONS M.H. HOOSHMAND
More information7: FOURIER SERIES STEVEN HEILMAN
7: FOURIER SERIES STEVE HEILMA Contents 1. Review 1 2. Introduction 1 3. Periodic Functions 2 4. Inner Products on Periodic Functions 3 5. Trigonometric Polynomials 5 6. Periodic Convolutions 7 7. Fourier
More informationMATH 1A, Complete Lecture Notes. Fedor Duzhin
MATH 1A, Complete Lecture Notes Fedor Duzhin 2007 Contents I Limit 6 1 Sets and Functions 7 1.1 Sets................................. 7 1.2 Functions.............................. 8 1.3 How to define a
More informationON THE DIVERGENCE OF FOURIER SERIES
ON THE DIVERGENCE OF FOURIER SERIES RICHARD P. GOSSELIN1 1. By a well known theorem of Kolmogoroff there is a function whose Fourier series diverges almost everywhere. Actually, Kolmogoroff's proof was
More informationUNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS
J. Korean Math. Soc. 45 (008), No. 3, pp. 835 840 UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS Paulius Drungilas Reprinted from the Journal of the Korean Mathematical Society Vol. 45, No. 3, May
More informationYunhi Cho and Young-One Kim
Bull. Korean Math. Soc. 41 (2004), No. 1, pp. 27 43 ANALYTIC PROPERTIES OF THE LIMITS OF THE EVEN AND ODD HYPERPOWER SEQUENCES Yunhi Cho Young-One Kim Dedicated to the memory of the late professor Eulyong
More informationStatic Problem Set 2 Solutions
Static Problem Set Solutions Jonathan Kreamer July, 0 Question (i) Let g, h be two concave functions. Is f = g + h a concave function? Prove it. Yes. Proof: Consider any two points x, x and α [0, ]. Let
More informationA note on unsatisfiable k-cnf formulas with few occurrences per variable
A note on unsatisfiable k-cnf formulas with few occurrences per variable Shlomo Hoory Department of Computer Science University of British Columbia Vancouver, Canada shlomoh@cs.ubc.ca Stefan Szeider Department
More informationThe Infinity Norm of a Certain Type of Symmetric Circulant Matrix
MATHEMATICS OF COMPUTATION, VOLUME 31, NUMBER 139 JULY 1977, PAGES 733-737 The Infinity Norm of a Certain Type of Symmetric Circulant Matrix By W. D. Hoskins and D. S. Meek Abstract. An attainable bound
More informationUNIFORM BOUNDS FOR BESSEL FUNCTIONS
Journal of Applied Analysis Vol. 1, No. 1 (006), pp. 83 91 UNIFORM BOUNDS FOR BESSEL FUNCTIONS I. KRASIKOV Received October 8, 001 and, in revised form, July 6, 004 Abstract. For ν > 1/ and x real we shall
More informationON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M. I. OSTROVSKII (Communicated by Dale Alspach) Abstract.
More informationSolutions Final Exam May. 14, 2014
Solutions Final Exam May. 14, 2014 1. (a) (10 points) State the formal definition of a Cauchy sequence of real numbers. A sequence, {a n } n N, of real numbers, is Cauchy if and only if for every ɛ > 0,
More informationMAXIMAL SEPARABLE SUBFIELDS OF BOUNDED CODEGREE
proceedings of the american mathematical society Volume 88. Number I, May 1983 MAXIMAL SEPARABLE SUBFIELDS OF BOUNDED CODEGREE JAMES K. DEVENEY AND JOHN N. MORDESON Abstract. Let L be a function field
More informationDouble Fourier series, generalized Lipschitz és Zygmund classes
Double Fourier series, generalized Lipschitz és Zygmund classes Summary of the PhD Theses Zoltán Sáfár SUPERVISOR: Ferenc Móricz DSc PROFESSOR EMERITUS UNIVERSITY OF SZEGED FACULTY OF SCIENCE AND INFORMATICS
More informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
More informationSOLUTIONS OF NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH EXCEPTIONALLY FEW ZEROS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 429 452 SOLUTIONS OF NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH EXCEPTIONALLY FEW ZEROS Gary G. Gundersen, Enid M. Steinbart, and
More informationON LARGE ZSIGMONDY PRIMES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 02, Number, January 988 ON LARGE ZSIGMONDY PRIMES WALTER FEIT (Communicated by Donald Passman) Abstract. If a and m are integers greater than, then
More informationChapter 1. Functions 1.1. Functions and Their Graphs
1.1 Functions and Their Graphs 1 Chapter 1. Functions 1.1. Functions and Their Graphs Note. We start by assuming that you are familiar with the idea of a set and the set theoretic symbol ( an element of
More informationAND RELATED NUMBERS. GERHARD ROSENBERGER Dortmund, Federal Republic of Germany (Submitted April 1982)
ON SOME D I V I S I B I L I T Y PROPERTIES OF FIBONACCI AND RELATED NUMBERS Universitat GERHARD ROSENBERGER Dortmund, Federal Republic of Germany (Submitted April 1982) 1. Let x be an arbitrary natural
More informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
More informationPM functions, their characteristic intervals and iterative roots
ANNALES POLONICI MATHEMATICI LXV.2(1997) PM functions, their characteristic intervals and iterative roots by Weinian Zhang (Chengdu) Abstract. The concept of characteristic interval for piecewise monotone
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More informationDYNAMICS OF THE ZEROS OF FIBONACCI POLYNOMIALS M. X. He Nova Southeastern University, Fort Lauderdale, FL. D, Simon
M. X. He Nova Southeastern University, Fort Lauderdale, FL D, Simon Nova Southeastern University, Fort Lauderdale, FL P. E. Ricci Universita degli Studi di Roma "La Sapienza," Rome, Italy (Submitted December
More informationTRIANGULAR TRUNCATION AND NORMAL LIMITS OF NILPOTENT OPERATORS
proceedings of the american mathematical society Volume 123, Number 6, June 1995 TRIANGULAR TRUNCATION AND NORMAL LIMITS OF NILPOTENT OPERATORS DON HADWIN (Communicated by Palle E. T. Jorgensen) Dedicated
More informationMATH NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS
MATH. 4433. NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS TOMASZ PRZEBINDA. Final project, due 0:00 am, /0/208 via e-mail.. State the Fundamental Theorem of Algebra. Recall that a subset K
More informationA Chebyshev Polynomial Rate-of-Convergence Theorem for Stieltjes Functions
mathematics of computation volume 39, number 159 july 1982, pages 201-206 A Chebyshev Polynomial Rate-of-Convergence Theorem for Stieltjes Functions By John P. Boyd Abstract. The theorem proved here extends
More informationRemarks on various generalized derivatives
Remarks on various generalized derivatives J. Marshall Ash Department of Mathematics, DePaul University Chicago, IL 60614, USA mash@math.depaul.edu Acknowledgement. This paper is dedicated to Calixto Calderón.
More informationConvergence of greedy approximation I. General systems
STUDIA MATHEMATICA 159 (1) (2003) Convergence of greedy approximation I. General systems by S. V. Konyagin (Moscow) and V. N. Temlyakov (Columbia, SC) Abstract. We consider convergence of thresholding
More informationW. Lenski and B. Szal ON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF CONJUGATE FOURIER SERIES
F A S C I C U L I M A T H E M A T I C I Nr 55 5 DOI:.55/fascmath-5-7 W. Lenski and B. Szal ON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF CONJUGATE FOURIER SERIES Abstract. The results
More informationTHE DISTRIBUTION OF RADEMACHER SUMS S. J. MONTGOMERY-SMITH. (Communicated by William D. Sudderth)
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 109, Number 2, June 1990 THE DISTRIBUTION OF RADEMACHER SUMS S. J. MONTGOMERY-SMITH (Communicated by William D. Sudderth) Abstract. We find upper
More informationThe Distribution of Generalized Sum-of-Digits Functions in Residue Classes
Journal of umber Theory 79, 9426 (999) Article ID jnth.999.2424, available online at httpwww.idealibrary.com on The Distribution of Generalized Sum-of-Digits Functions in Residue Classes Abigail Hoit Department
More informationLimit and Continuity
Limit and Continuity Table of contents. Limit of Sequences............................................ 2.. Definitions and properties...................................... 2... Definitions............................................
More informationON THE STABILITY OF THE MONOMIAL FUNCTIONAL EQUATION
Bull. Korean Math. Soc. 45 (2008), No. 2, pp. 397 403 ON THE STABILITY OF THE MONOMIAL FUNCTIONAL EQUATION Yang-Hi Lee Reprinted from the Bulletin of the Korean Mathematical Society Vol. 45, No. 2, May
More information1 More concise proof of part (a) of the monotone convergence theorem.
Math 0450 Honors intro to analysis Spring, 009 More concise proof of part (a) of the monotone convergence theorem. Theorem If (x n ) is a monotone and bounded sequence, then lim (x n ) exists. Proof. (a)
More informationINTEGRAL ORTHOGONAL BASES OF SMALL HEIGHT FOR REAL POLYNOMIAL SPACES
INTEGRAL ORTHOGONAL BASES OF SMALL HEIGHT FOR REAL POLYNOMIAL SPACES LENNY FUKSHANSKY Abstract. Let P N (R be the space of all real polynomials in N variables with the usual inner product, on it, given
More informationMathematica Slovaca. Jaroslav Hančl; Péter Kiss On reciprocal sums of terms of linear recurrences. Terms of use:
Mathematica Slovaca Jaroslav Hančl; Péter Kiss On reciprocal sums of terms of linear recurrences Mathematica Slovaca, Vol. 43 (1993), No. 1, 31--37 Persistent URL: http://dml.cz/dmlcz/136572 Terms of use:
More informationd(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N
Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x
More information(2) fv*-t)2v(»-«)=(*-a>:+(*-*)2
proceedings of the american mathematical society Volume 23, Number 2, December 995 OSTROWSKI TYPE INEQUALITIES GEORGE A. ANASTASSIOU (Communicated by Andrew M. Bruckner) Abstract. Optimal upper bounds
More informationGeometric Series and the Ratio and Root Test
Geometric Series and the Ratio and Root Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 5, 2018 Outline 1 Geometric Series
More informationLogical Connectives and Quantifiers
Chapter 1 Logical Connectives and Quantifiers 1.1 Logical Connectives 1.2 Quantifiers 1.3 Techniques of Proof: I 1.4 Techniques of Proof: II Theorem 1. Let f be a continuous function. If 1 f(x)dx 0, then
More informationNON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXVI, 2 (2017), pp. 287 297 287 NON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL PINGPING ZHANG Abstract. Using the piecewise monotone property, we give a full description
More informationSequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.
Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence
More informationMath 321 Final Examination April 1995 Notation used in this exam: N. (1) S N (f,x) = f(t)e int dt e inx.
Math 321 Final Examination April 1995 Notation used in this exam: N 1 π (1) S N (f,x) = f(t)e int dt e inx. 2π n= N π (2) C(X, R) is the space of bounded real-valued functions on the metric space X, equipped
More informationON THE DENSITY OF SOME SEQUENCES OF INTEGERS P. ERDOS
ON THE DENSITY OF SOME SEQUENCES OF INTEGERS P. ERDOS Let ai
More information3.2 Bases of C[0, 1] and L p [0, 1]
70 CHAPTER 3. BASES IN BANACH SPACES 3.2 Bases of C[0, ] and L p [0, ] In the previous section we introduced the unit vector bases of `p and c 0. Less obvious is it to find bases of function spaces like
More informationMath 341: Convex Geometry. Xi Chen
Math 341: Convex Geometry Xi Chen 479 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca CHAPTER 1 Basics 1. Euclidean Geometry
More information1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),
Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer
More informationCommentationes Mathematicae Universitatis Carolinae
Commentationes Mathematicae Universitatis Carolinae Bogdan Szal Trigonometric approximation by Nörlund type means in L p -norm Commentationes Mathematicae Universitatis Carolinae, Vol. 5 (29), No. 4, 575--589
More informationGlobal Asymptotic Stability of a Nonlinear Recursive Sequence
International Mathematical Forum, 5, 200, no. 22, 083-089 Global Asymptotic Stability of a Nonlinear Recursive Sequence Mustafa Bayram Department of Mathematics, Faculty of Arts and Sciences Fatih University,
More informationA note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series
Publ. Math. Debrecen 73/1-2 2008), 1 10 A note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series By SOO HAK SUNG Taejon), TIEN-CHUNG
More informationOn the Irrationality of Z (1/(qn+r))
JOURNAL OF NUMBER THEORY 37, 253-259 (1991) On the Irrationality of Z (1/(qn+r)) PETER B. BORWEIN* Department of Mathematics, Statistics, and Computing Science, Dalhousie University, Halifax, Nova Scotia,
More informationThe Hilbert Transform and Fine Continuity
Irish Math. Soc. Bulletin 58 (2006), 8 9 8 The Hilbert Transform and Fine Continuity J. B. TWOMEY Abstract. It is shown that the Hilbert transform of a function having bounded variation in a finite interval
More informationHyers-Ulam Rassias stability of Pexiderized Cauchy functional equation in 2-Banach spaces
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 5 (202), 459 465 Research Article Hyers-Ulam Rassias stability of Pexiderized Cauchy functional equation in 2-Banach spaces G. Zamani Eskandani
More informationON A PARTITION PROBLEM OF CANFIELD AND WILF. Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia
#A11 INTEGERS 12A (2012): John Selfridge Memorial Issue ON A PARTITION PROBLEM OF CANFIELD AND WILF Željka Ljujić Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia z.ljujic20@uniandes.edu.co
More informationOn Some Estimates of the Remainder in Taylor s Formula
Journal of Mathematical Analysis and Applications 263, 246 263 (2) doi:.6/jmaa.2.7622, available online at http://www.idealibrary.com on On Some Estimates of the Remainder in Taylor s Formula G. A. Anastassiou
More informationOn Existence of Positive Solutions for Linear Difference Equations with Several Delays
Advances in Dynamical Systems and Applications. ISSN 0973-5321 Volume 1 Number 1 (2006), pp. 29 47 c Research India Publications http://www.ripublication.com/adsa.htm On Existence of Positive Solutions
More informationfn" <si> f2(i) i + [/i + [/f] (1.1) ELEMENTARY SEQUENCES (1.2)
Internat. J. Math. & Math. Sci. VOL. 15 NO. 3 (1992) 581-588 581 ELEMENTARY SEQUENCES SCOTT J. BESLIN Department of Mathematics Nicholls State University Thibodaux, LA 70310 (Received December 10, 1990
More informationUNBOUNDED DISCREPANCY IN FROBENIUS NUMBERS
#A2 INTEGERS 11 (2011) UNBOUNDED DISCREPANCY IN FROBENIUS NUMBERS Jeffrey Shallit School of Computer Science, University of Waterloo, Waterloo, ON, Canada shallit@cs.uwaterloo.ca James Stankewicz 1 Department
More informationTHE PRESERVATION OF CONVERGENCE OF MEASURABLE FUNCTIONS UNDER COMPOSITION
THE PRESERVATION OF CONVERGENCE OF MEASURABLE FUNCTIONS UNDER COMPOSITION ROBERT G. BARTLE AND JAMES T. JOICHI1 Let / be a real-valued measurable function on a measure space (S,, p.) and let
More informationCOMBINATORIAL DIMENSION OF FRACTIONAL CARTESIAN PRODUCTS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 120, Number 1, January 1994 COMBINATORIAL DIMENSION OF FRACTIONAL CARTESIAN PRODUCTS RON C. BLEI AND JAMES H. SCHMERL (Communicated by Jeffry N.
More informationDYNAMIC ONE-PILE NIM Arthur Holshouser 3600 Bullard St., Charlotte, NC, USA, 28208
Arthur Holshouser 3600 Bullard St., Charlotte, NC, USA, 28208 Harold Reiter Department of Mathematics, UNC Charlotte, Charlotte, NC 28223 James Riidzinski Undergraduate, Dept. of Math., UNC Charlotte,
More informationAn idea how to solve some of the problems. diverges the same must hold for the original series. T 1 p T 1 p + 1 p 1 = 1. dt = lim
An idea how to solve some of the problems 5.2-2. (a) Does not converge: By multiplying across we get Hence 2k 2k 2 /2 k 2k2 k 2 /2 k 2 /2 2k 2k 2 /2 k. As the series diverges the same must hold for the
More informationUpper Bounds for Partitions into k-th Powers Elementary Methods
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 9, 433-438 Upper Bounds for Partitions into -th Powers Elementary Methods Rafael Jaimczu División Matemática, Universidad Nacional de Luján Buenos Aires,
More informationDefining the Integral
Defining the Integral In these notes we provide a careful definition of the Lebesgue integral and we prove each of the three main convergence theorems. For the duration of these notes, let (, M, µ) be
More information