DOCSLIB.ORG

Fourier Series & Transform Representation of Continuous Time

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fourier Series & Transform Representation of Continuous Time UNIT II Fourier series & transform Representation of Continuous Time Signals Learning Objectives: To introduce the concept of various Fourier series To introduce the concept of convergence of Fourier series. To introduce the concept of representation of a periodic function by Fourier series To introduce the concepts of sampling of continuous time signlas. To introduce the concept of Fourier transform. Learning Outcomes: Students will be able to Apply symmetric conditions Represent the periodic function by Fourier series Obtain the relation between trigonometric and exponential Fourier series Perform transformations on signals Syllabus: Trigonometric and exponential Fourier series, relationship between trigonometric and exponential Fourier series, representation of a periodic function by the Fourier series over the entire interval, convergence of Fourier series, alternative form of trigonometric series, symmetry conditions: Even, Odd and Half-wave symmetry. Properties of Fourier series: linearity, time scaling, time shifting, time reversal, differentiation, integration, modulation, convolution and Parseval’s theorem. Complex Fourier transforms. Representation of an arbitrary function over the entire interval: Fourier transform, Existence of Fourier transform, Fourier transform of some useful functions, Fourier transform of periodic function, Properties of Fourier transform, Energy Density spectrum, Parseval’s Theorem. Introduction Many of the phenomena studied in engineering and science are periodic in nature eg. The current and voltage in an alternating current circuit. These periodic functions can be analysed into their constituent components (fundamentals and harmonics) by a process called Fourier analysis. Fourier series was developed by Joseph Fourier Definition . Fourier series is a mathematical tool which decomposes any periodic function or periodic signal into sum of set of sine and cosine terms. OR The representation of signals over a certain time interval interns of the linear composition of orthogonal functions is called Fourier series. It is sometimes called the Harmonic analysis. There are 3 important types of Fourier series. They are 1. Trigonometric form 2. Cosine form 3. Exponential form Trigonometric Fourier series The representation of signals over a certain time interval interns of the linear composition of trigonometric functions is called Trigonometric Fourier series. The trigonometric Fourier series representation of any periodic signal x (t) is defined as ∞ 푥(푡) = 푎표 + ∑푛=1[푎푛푐표푠푛푤표푡 + 푏푛푠푖푛푛푤표푡] (1) Where푎푛, 푏푛 are called constants and 푎표 is called dc component and is given by 1 푡표+푇 푎 = ∫ 푥(푡)푑푡 표 푇 푡표 2 푡표+푇 푎 = ∫ 푥(푡)푐표푠푛푤 푡 푑푡 푛 푇 표 푡표 2 푡표+푇 푏푛 = ∫ 푥(푡)푠푖푛푛푤표푡 푑푡 푇 푡표 Exponential Fourier series The representation of signals over a certain time interval interns of the linear composition of exponential functions is called Exponential Fourier series. The complex exponential form is more general and usually more convenient & more compact when compared to Trigonometric Fourier series. The exponential Fourier series of signal x (t) is given by ∞ 푗푛푤표푡 푥(푡) = ∑푛=−∞ 푐푛푒 OR −1 푗푛푤표푡 ∞ 푗푛푤표푡 푥(푡) = 푐0 + ∑푛=−∞ 푐푛푒 + ∑푛=1 푐푛푒 (2) Relationship between trigonometric and exponential Fourier series The complex exponential Fourier series is given by ∞ 푗푛푤표푡 푥(푡) = ∑푛=−∞ 푐푛푒 −1 푗푛푤표푡 ∞ 푗푛푤표푡 = 푐0 + ∑푛=−∞ 푐푛푒 + ∑푛=1 푐푛푒 ∞ −푗푛푤표푡 푗푛푤표푡 = 푐0 + ∑푛=1(푐−푛푒 + 푐푛푒 ) ∞ = 푐0 + ∑푛=1 푐−푛 (푐표푠푛푤표푡 − 푗푠푖푛푛푤표푡) + 푐푛(푐표푠푛푤표푡 + 푗푠푖푛푛푤표푡)) ∞ 푥(푡) = 푐0 + ∑푛=1(푐−푛 + 푐푛) 푐표푠푛푤표푡 + 푗(푐푛 − 푐−푛)푠푖푛푛푤표푡) (3) Compare eq(1) and eq(3) 푎0 = 푐0 푎푛 = 푐푛 + 푐−푛 푏푛 = 푗(푐푛 − 푐−푛) Similarly 푐0 = 푎0 1 푐 = (푎 − 푗푏 ) 푛 2 푛 푛 1 푐 = (푎 + 푗푏 ) −푛 2 푛 푛 Representation of a Periodic Function by the Fourier Series Over the Entire Interval\ So far we have considered the Fourier series representation of an arbitrary function over finite interval between 푡표푎푛푑 (푡표 + 푇). The function and its Fourier series may not be equal outside this interval. However, a periodic function which repeats after every T seconds is expected to have the same Fourier series for the entire interval (-∞, ∞). A periodic function has an identical Fourier series for the entire interval (-∞, ∞) as for the interval (푡표 + 푇) since same function is repeated for every T seconds. Thus for a periodic function x (t) we have ∞ 푗푛푤표푡 푥(푡) = ∑푛=−∞ 퐹푛푒 , (-∞‹t‹∞) 1 푡표+푇 퐹 = ∫ 푥(푡)푒−푗푛푤0푡푑푡 푛 푇 푡표 Dirichlets Conditions (Convergence of Fourier Series) For the Fourier series to exist for a periodic signal it must satisfy certain conditions and they are 1. Function x(t) must be a single valued function 2. Function x(t) has only finite number of maxima & minima. 3. Function x(t) has finite number of discontinuities. 푇| | 4. Function x(t) is absolutely integral over one period i,e ∫0 푥(푡) 푑푡 < ∞ Alternative form of Fourier series (Cosine Fourier series) The trigonometric Fourier series of x(t) contains “sine” and “cosine” terms of the same frequency ∞ 푥(푡) = 푎0 + ∑[푎푛푐표푠푛푤0푡 + 푏푛푠푖푛푛푤0푡] 푛=1 ∞ 푎 푏 2 2 푛 푛 푥(푡) = 푎0 + ∑ √푎푛 + 푏푛 [ 푐표푠푛푤0푡 + 푠푖푛푛푤0푡] 2 2 2 2 푛=1 √푎푛 + 푏푛 √푎푛 + 푏푛 2 2 −1 푏푛 푎푛 Substituting the values 푎0 = 퐴0 , 퐴푛 = √푎푛 + 푏푛 , 휃푛 = 1 tan , = 푐표푠휃푛 and 푎푛 2 2 √푎푛+푏푛 푏푛 = −푠푖푛휃푛 in the above equation of x(t) we get 2 2 √푎푛+푏푛 ∞ 푥(푡) = 퐴0 + ∑ 퐴푛 [푐표푠휃푛푐표푠푛푤0푡 − 푠푖푛휃푛푠푖푛푛푤0푡] 푛=1 ∞ 푥(푡) = 퐴0 + ∑푛=1 퐴푛 [cos (푛푤0푡 + 휃푛] The term 퐴0 is called the dc component. The term 퐴푛 represents the amplitude coefficients (or) harmonic amplitudes (or) spectral amplitudes of the Fourier series and 휃푛 represents the phase coefficients (or) phase angles of Fourier series. The cosine form is also called as “Harmonic Form” (or) “Polar Form” Fourier series. Symmetric conditions or Wave symmetry There are 4 types of symmetry of function x(t) can have 1. Even symmetry 2. Odd symmetry 3. Half wave symmetry Even symmetry A function x(t) is said to be even symmetry if 푥(−푡) = 푥(푡) These types of functions always symmetrical w.r.t the vertical axis 푥(푡) = 푥푒(푡) + 푥0(푡) If x(t) is even function then, 푥0(푡) = 0 and 푥푒(푡) = 푥(푡) . When even symmetry exists the trigonometric Fourier series coefficients are 2 푇/2 푎표 = ∫ 푥(푡)푑푡 푇 0 4 푇/2 푎 = ∫ 푥(푡)푐표푠푛푤 푡 푑푡 푛 푇 0 표 푏푛 = 0 Odd symmetry A function x(t) is said to be odd symmetry if 푥(−푡) = −푥(푡) These types of functions always antisymmetrical w.r.t the vertical axis 푥(푡) = 푥푒(푡) + 푥0(푡) If x(t) is odd function then, 푥푒(푡) = 0 and 푥(푡) = 푥0(푡) . When odd symmetry exists the trigonometric Fourier series coefficients are 푎표 = 0 푎푛 = 0 4 푇/2 푏 = ∫ 푥(푡)푠푖푛푛푤 푡 푑푡 푛 푇 0 표 Half wave symmetry A periodic signal x(t) which satisfies the condition 푥(푡) = −푥 (푡 ± 푇) is said to be half wave symmetry. 2 The Fourier series expansion of half wave symmetry signal contains odd harmonics only (When “n” is odd). 4 푇/2 푎 = ∫ 푥(푡)푐표푠푛푤 푡 푑푡 푛 푇 0 표 4 푇/2 푏 = ∫ 푥(푡)푠푖푛푛푤 푡 푑푡 푛 푇 0 표 When “n” is even 푎푛 = 0 푏푛 = 0 Properties of Fourier Series Let us consider 푥1(푡)푎푛푑 푥2(푡) are two periodic signals with period ‘T’ and with Fourier series coefficients 퐶푛& 퐷푛. 1. Linearity Property The linearity property states that, if 퐹푆{푥1(푡)} = 퐶푛 푎푛푑 퐹푆{푥2(푡)} = 퐷푛 Then 퐹푆 {퐴 푥1(푡) + 퐵푥2(푡)} = 퐴퐶푛 + 퐵 퐷푛 Proof: From the definition of Fourier series, we have 1 푡표+푇 퐹푆{퐴 푥 (푡) + 퐵푥 (푡)} = ∫ [퐴 푥 (푡) + 퐵푥 (푡)]푒−푗푛푤0푡푑푡 1 2 푇 1 2 푡표 1 푡표+푇 1 푡표+푇 퐹푆{퐴 푥 (푡) + 퐵푥 (푡)} = 퐴 ∫ [ 푥 (푡)]푒−푗푛푤0푡푑푡 + 퐵 ∫ [푥 (푡)]푒−푗푛푤0푡푑푡 1 2 푇 1 푇 2 푡표 푡표 퐹푆{퐴 푥1(푡) + 퐵푥2(푡)} = 퐴퐶푛 + 퐵 퐷푛 2. Time Reversal The time reversal property states that, if 퐹푠{푥(푡)} = 퐶푛 Then 퐹푠{푥(−푡)} = 퐶−푛 Proof: From the dentition of Fourier series, we have ∞ 푗푛푤표푡 푥(푡) = ∑ 푐푛푒 푛=−∞ ∞ 푗푛푤표(−푡) 푥(−푡) = ∑ 푐푛푒 푛=−∞ Substitute “n = -p” in the right hand side, we get −∞ ∞ 푗(−푝)푤표(−푡) 푗푝푤표푡 푥(−푡) = ∑ 푐−푝푒 = ∑ 푐−푝푒 푝=∞ 푝=−∞ Substitute “p = n”, we get ∞ 푗푛푤표푡 −1 푥(−푡) = ∑ 푐−푛푒 = 퐹푆 {푐−푛} 푛=−∞ Therefore 퐹푠{푥(−푡)} = 퐶−푛 3. Time Shifting The time shifting property states that, if 퐹푠{푥(푡)} = 퐶푛 −푗푛푤0푡0 Then 퐹푆 {푥(푡 − 푡0) = 푒 퐶푛 Proof: From the dentition of Fourier series, we have ∞ 푗푛푤표푡 푥(푡) = ∑ 푐푛푒 푛=−∞ ∞ 푗푛푤표(푡−푡0) 푥(푡 − 푡0) = ∑ 푐푛푒 푛=−∞ ∞ −푗푛푤표푡0 푗푛푤표푡 푥(푡 − 푡0) = ∑ [푐푛푒 ] 푒 푛=−∞ −1 −푗푛푤표푡0 푥(푡 − 푡0) = 퐹푆 [푐푛푒 ] −푗푛푤0푡0 Therefore 퐹푆 {푥(푡 − 푡0) = 퐶푛푒 4. Time Scaling: The time scaling property states that, if 퐹푠{푥(푡)} = 퐶푛 Then 퐹푠{푥(푎푡)} = 퐶푛 푤푖푡ℎ 휔0 → 푎휔0 Proof: From the definition of Fourier series, we have ∞ 푗푛푤표푡 푥(푡) = ∑ 푐푛푒 푛=−∞ ∞ 푗푛푤표푎푡 푥(푎푡) = ∑ 푐푛푒 푛=−∞ ∞ 푗푛(푎푤표)푡 푥(푎푡) = ∑ 푐푛푒 푛=−∞ −1 푥(푎푡) = 퐹푆 {푐푛} Where 휔0 → 푎휔0 Therefore 퐹푠{푥(푎푡)} = 퐶푛 푤푖푡ℎ 푓푢푛푑푎푚푒푛푡푎푙 푓푟푒푞푢푒푛푐푦 표푓 푎휔0 5. Convolution Property The convolution property states that, if 퐹푆{푥1(푡)} = 퐶푛 푎푛푑 퐹푆{푥2(푡)} = 퐷푛 Then 퐹푆{푥1(푡) ∗ 푥2(푡)} = 퐶푛퐷푛 6. Time Differentiation Property The time differentiation property states that, if 퐹푠{푥(푡)} = 퐶푛 ( ) Then 퐹푆 {푑푥 푡 } = 푗푛휔 퐶 푑푡 0 푛 7. Time Integration Property The time integration property states that, if 퐹푠{푥(푡)} = 퐶푛 푡 퐶푛 Then 퐹푆 {∫ 푥(휏)푑휏} = (if 퐶0 = 0) −∞ 푗푛휔0 8.
Load more

Recommended publications
  • Blissard's Trigonometric Series with Closed-Form Sums
    BLISSARD’S TRIGONOMETRIC SERIES WITH CLOSED-FORM SUMS JACQUES GELINAS´ Abstract. This is a summary and verification of an elementary note written by John Blissard in 1862 for the Messenger of Mathematics. A general method of discovering trigonometric series having a closed-form sum is explained and illustrated with examples. We complete some state- ments and add references, using the summation symbol and Blissard’s own (umbral) representative notation for a more concise presentation than the original. More examples are also provided. 1. Historical examples Blissard’s well structured note [3] starts by recalling four trigonometric series which “mathemati- cal writers have exhibited as results of the differential and integral calculus” (A, B, C, G in the table below). Many such formulas had indeed been worked out by Daniel Bernoulli, Euler and Fourier. More examples can be found in 19th century textbooks and articles on calculus, in 20th century treatises on infinite series [21, 10, 19, 17] or on Fourier series [12, 23], and in mathematical tables [18, 22, 16, 11]. G.H. Hardy motivated the derivation of some simple formulas as follows [17, p. 2]. We can first agree that the sum of a geometric series 1+ x + x2 + ... with ratio x is s =1/(1 − x) because this is true when the series converges for |x| < 1, and “it would be very inconvenient if the formula varied in different cases”; moreover, “we should expect the sum s to satisfy the equation s = 1+ sx”. With x = eiθ, we obtain immediately a number of trigonometric series by separating the real and imaginary parts, by setting θ = 0, by differentiating, or by integrating [14, §13 ].
    [Show full text]
  • Power Series and Taylor's Series
    Power Series and Taylor’s Series Dr. Kamlesh Jangid Department of HEAS (Mathematics) Rajasthan Technical University, Kota-324010, India E-mail: [email protected] Dr. Kamlesh Jangid (RTU Kota) Series 1 / 19 Outline Outline 1 Introduction 2 Series 3 Convergence and Divergence of Series 4 Power Series 5 Taylor’s Series Dr. Kamlesh Jangid (RTU Kota) Series 2 / 19 Introduction Overview Everyone knows how to add two numbers together, or even several. But how do you add infinitely many numbers together ? In this lecture we answer this question, which is part of the theory of infinite sequences and series. An important application of this theory is a method for representing a known differentiable function f (x) as an infinite sum of powers of x, so it looks like a "polynomial with infinitely many terms." Dr. Kamlesh Jangid (RTU Kota) Series 3 / 19 Series Definition A sequence of real numbers is a function from the set N of natural numbers to the set R of real numbers. If f : N ! R is a sequence, and if an = f (n) for n 2 N, then we write the sequence f as fang. A sequence of real numbers is also called a real sequence. Example (i) fang with an = 1 for all n 2 N-a constant sequence. n−1 1 2 (ii) fang = f n g = f0; 2 ; 3 ; · · · g; n+1 1 1 1 (iii) fang = f(−1) n g = f1; − 2 ; 3 ; · · · g. Dr. Kamlesh Jangid (RTU Kota) Series 4 / 19 Series Definition A series of real numbers is an expression of the form a1 + a2 + a3 + ··· ; 1 X or more compactly as an, where fang is a sequence of real n=1 numbers.
    [Show full text]
  • 5. Sequences and Series of Functions in What Follows, It Is Assumed That X RN , and X a Means That the Euclidean Distance Between X and a Tends∈ to Zero,→X a 0
    32 1. THE THEORY OF CONVERGENCE 5. Sequences and series of functions In what follows, it is assumed that x RN , and x a means that the Euclidean distance between x and a tends∈ to zero,→x a 0. | − |→ 5.1. Pointwise convergence. Consider a sequence of functions un(x) (real or complex valued), n = 1, 2,.... The sequence un is said to converge pointwise to a function u on a set D if { } lim un(x)= u(x) , x D. n→∞ ∀ ∈ Similarly, the series un(x) is said to converge pointwise to a function u(x) on D RN if the sequence of partial sums converges pointwise to u(x): ⊂ P n Sn(x)= uk(x) , lim Sn(x)= u(x) , x D. n→∞ ∀ ∈ k=1 X 5.2. Trigonometric series. Consider a sequence an C where n = 0, 1, 2,.... The series { } ⊂ ± ± ∞ inx ane , x R ∈ n=−∞ X is called a complex trigonometric series. Its convergence means that sequences of partial sums m 0 + inx − inx Sm = ane , Sk = ane n=1 n=−k X X converge and, in this case, ∞ inx + − ane = lim Sm + lim Sk m→∞ k→∞ n=−∞ X ∞ ∞ Given two real sequences an and b , the series { }0 { }1 ∞ 1 a0 + an cos(nx)+ bn sin(nx) 2 n=1 X 1 is called a real trigonometric series. A factor 2 at a0 is a convention related to trigonometric Fourier series (see Remark below). For all x R for which a real (or complex) trigonometric series con- verges, the sum∈ defines a real-valued (or complex-valued) function of x.
    [Show full text]
  • IJR-1, Mathematics for All ... Syed Samsul Alam
    January 31, 2015 [IISRR-International Journal of Research ] MATHEMATICS FOR ALL AND FOREVER Prof. Syed Samsul Alam Former Vice-Chancellor Alaih University, Kolkata, India; Former Professor & Head, Department of Mathematics, IIT Kharagpur; Ch. Md Koya chair Professor, Mahatma Gandhi University, Kottayam, Kerala , Dr. S. N. Alam Assistant Professor, Department of Metallurgical and Materials Engineering, National Institute of Technology Rourkela, Rourkela, India This article briefly summarizes the journey of mathematics. The subject is expanding at a fast rate Abstract and it sometimes makes it essential to look back into the history of this marvelous subject. The pillars of this subject and their contributions have been briefly studied here. Since early civilization, mathematics has helped mankind solve very complicated problems. Mathematics has been a common language which has united mankind. Mathematics has been the heart of our education system right from the school level. Creating interest in this subject and making it friendlier to students’ right from early ages is essential. Understanding the subject as well as its history are both equally important. This article briefly discusses the ancient, the medieval, and the present age of mathematics and some notable mathematicians who belonged to these periods. Mathematics is the abstract study of different areas that include, but not limited to, numbers, 1.Introduction quantity, space, structure, and change. In other words, it is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. Mathematicians seek out patterns and formulate new conjectures. They resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity.
    [Show full text]
  • Symmetric Integrals and Trigonometric Series
    SYMMETRIC INTEGRALS AND TRIGONOMETRIC SERIES George E. Cross and Brian S. Thomson 1 Introduction The first example of a convergent trigonometric series that cannot be ex- pressed in Fourier form is due to Fatou. The series ∞ sin nx (1) log(n + 1) nX=1 converges everywhere to a function that is not Lebesgue, or even Perron, integrable. It follows that the series (1) cannot be represented in Fourier form using the Lebesgue or Perron integrals. In fact this example is part of a whole class of examples as Denjoy [22, pp. 42–44] points out: if bn 0 ∞ ց and n=1 bn/n = + then the sum of the everywhere convergent series ∞ ∞ n=1Pbn sin nx is not Perron integrable. P The problem, suggested by these examples, of defining an integral so that the sum function f(x) of the convergent or summable trigonometric series ∞ (2) a0/2+ (an cos nx + bn sin nx) nX=1 is integrable and so that the coefficients, an and bn, can be written as Fourier coefficients of the function f has received considerable attention in the lit- erature (cf. [10], [11], [21], [22], [26], [32], [34], [37], [44] and [45].) For an excellent survey of the literature prior to 1955 see [27]; [28] is also useful. (In 1AMS 1980 Mathematics Subject Classification (1985 Revision). Primary 26A24, 26A39, 26A45; Secondary 42A24. 1 some of the earlier works ([10] and [45]) an additional condition on the series conjugate to (2) is imposed.) In addition a secondary literature has evolved devoted to the study of the properties of and the interrelations between the several integrals which have been constructed (for example [3], [4], [5], [6], [7], [8], [12], [13], [14], [15], [16], [17], [18], [19], [23], [29], [30], [31], [36], [38], [39], [40], [42] and [43]).
    [Show full text]
  • Trigonometric Series with General Monotone Coefficients
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 326 (2007) 721–735 www.elsevier.com/locate/jmaa Trigonometric series with general monotone coefficients ✩ S. Tikhonov Centre de Recerca Matemàtica (CRM), Bellaterra (Barcelona) E-08193, Spain Received 3 October 2005 Available online 18 April 2006 Submitted by D. Waterman Abstract We study trigonometric series with general monotone coefficients. Convergence results in the different metrics are obtained. Also, we prove a Hardy-type result on the behavior of the series near the origin. © 2006 Elsevier Inc. All rights reserved. Keywords: Trigonometric series; Convergence; Hardy–Littlewood theorem 1. Introduction In this paper we consider the series ∞ an cos nx (1) n=1 and ∞ an sin nx, (2) n=1 ✩ The paper was partially written while the author was staying at the Centre de Recerca Matemàtica in Barcelona, Spain. The research was funded by a EU Marie Curie fellowship (Contract MIF1-CT-2004-509465) and was partially supported by the Russian Foundation for Fundamental Research (Project 06-01-00268) and the Leading Scientific Schools (Grant NSH-4681.2006.1). E-mail address: [email protected]. 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.02.053 722 S. Tikhonov / J. Math. Anal. Appl. 326 (2007) 721–735 { }∞ where an n=1 is given a null sequence of complex numbers. We define by f(x) and g(x) the sums of the series (1) and (2) respectively at the points where the series converge.
    [Show full text]
  • Chapter 1 Fourier Series
    Chapter 1 Fourier Series ßb 7W<ïtt|d³ ¯öóÍ ]y÷HÕ_ &tt d³Äó*× *y ÿ|1Òr÷bpÒr¸Þ ÷ßi;?÷ß×2 7W×ôT| T¹yd³JXà²ô :K» In this chapter we study Fourier series. Basic definitions and examples are given in Section 1. In Section 2 we prove the fundamental Riemann-Lebesgue lemma and discuss the Fourier series from the mapping point of view. Pointwise and uniform convergence of the Fourier series of a function to the function itself under various regularity assumptions are studied in Section 3. In Section 1.5 we establish the L2-convergence of the Fourier series without any additional regularity assumption. There are two applications. In Section 1.4 it is shown that every continuous function can be approximated by polynomials in a uniform manner. In Section 1.6 a proof of the classical isoperimetric problem for plane curves is presented. 1.1 Definition and Examples The concept of series of functions and their pointwise and uniform convergence were discussed in Mathematical Analysis II. Power series and trigonometric series are the most important classes of series of functions. We learned power series in Mathematical Analysis II and now we discuss Fourier series. First of all, a trigonometric series on [−π; π] is a series of functions of the form 1 X (an cos nx + bn sin nx); an; bn 2 R: n=0 1 2 CHAPTER 1. FOURIER SERIES As cos 0x = 1 and sin 0x = 0, we always set b0 = 0 and express the series as 1 X a0 + (an cos nx + bn sin nx): n=1 It is called a cosine series if all bn vanish and sine series if all an vanish.
    [Show full text]
  • Note on the Lerch Zeta Function
    Pacific Journal of Mathematics NOTE ON THE LERCH ZETA FUNCTION F. OBERHETTINGER Vol. 6, No. 1 November 1956 NOTE ON THE LERCH ZETA FUNCTION F. OBERHETTINGER l Introduction, The functional equation for the Riemann zeta function and Hurwitz's series were derived by a formal application of Poisson's summation formula in some previous papers [9], [10], [12, p. 17]. The results are correct although obtained by equating two infinite series whose regions of convergence exclude each other. It was shown later that this difficulty can be overcome if a generalized form of Poisson's formula is used [6]. It is the purpose of this note to show that the application of Poisson's summation formula in its ordinary form to the more general case of Lerch's zeta function [1], [2], [3], [5, pp. 27-30], [8] does not present the difficulties with respect to convergence which arise in its special cases. Thus a very simple and direct method for developing the theory of this function can be given. It may be noted that Hurwitz's series can be obtained immediately upon specializing one of the parameters, (cf. also [1]). 2. Lerch's zeta function Φ(z, s, v) is usually defined by the power series (la) Φ(z,8,v)= Σzm(v + mys \z\<l, vφO, -1, -2, . m=0 Equivalent to this is the integral representation [5, pp. 27-30] (lb) Φ(z, β, v)=— v~s + \Γ(v 2 Jo + 2 ("sin ( Jo \ \z\<\, which can be written as V]os (1c) Φ(z, 8, v)=—v- / iv- A- τΐ :-Γ(l-s)(logi) -^-V-5 Σ V; r^4- 4- 2 f °° sin f s tan-1-^— t log ^Vv2 4- t2)-sIZ(eZiίt - iyιdt Jo V v / upon replacing the first integral on the right of (lb) by the second and Received February 11, 1954.
    [Show full text]
  • INFINITE SERIES 1. Introduction the Two Basic Concepts of Calculus
    INFINITE SERIES KEITH CONRAD 1. Introduction The two basic concepts of calculus, differentiation and integration, are defined in terms of limits (Newton quotients and Riemann sums). In addition to these is a third fundamental limit process: infinite series. The label series is just another name for a sum. An infinite series is a \sum" with infinitely many terms, such as 1 1 1 1 (1.1) 1 + + + + ··· + + ··· : 4 9 16 n2 The idea of an infinite series is familiar from decimal expansions, for instance the expansion π = 3:14159265358979::: can be written as 1 4 1 5 9 2 6 5 3 5 8 π = 3 + + + + + + + + + + + + ··· ; 10 102 103 104 105 106 107 108 109 1010 1011 so π is an “infinite sum" of fractions. Decimal expansions like this show that an infinite series is not a paradoxical idea, although it may not be clear how to deal with non-decimal infinite series like (1.1) at the moment. Infinite series provide two conceptual insights into the nature of the basic functions met in high school (rational functions, trigonometric and inverse trigonometric functions, exponential and logarithmic functions). First of all, these functions can be expressed in terms of infinite series, and in this way all these functions can be approximated by polynomials, which are the simplest kinds of functions. That simpler functions can be used as approximations to more complicated functions lies behind the method which calculators and computers use to calculate approximate values of functions. The second insight we will have using infinite series is the close relationship between functions which seem at first to be quite different, such as exponential and trigonometric functions.
    [Show full text]
  • Summable Trigonometric Series
    Pacific Journal of Mathematics SUMMABLE TRIGONOMETRIC SERIES RALPH D. JAMES Vol. 6, No. 1 November 1956 SUMMABLE TRIGONOMETRIC SERIES R. D. JAMES l Introduction. One of the problems in the theory of trigono- metric series in the form (1.1) -—OD + Σ {an cos nx + bn sin nx) = Σ «n(aθ is that of suitably defining a process of integration such that, if the series (1.1) converges to a function f(x), then f(x) is integrable and the coefficients anf bn are given in Fourier form. The problem has been solved by Denjoy [3], Verblunsky [10], Marcinkiewicz and Zygmund [8], Burkill [1], [2], and James [6]. In Verblunsky's paper and in BurkilΓs first paper, additional hypotheses other than the convergence of (1.1) were made, and in all the papers some modification of the form of the Fourier formulas was necessary: An extension of the problem is to consider series that are summable (C, k), &2>1. This has been solved by Wolf [11] when the sum func- tion is Perron integrable. The problem of defining a process of inte- gration which may be applied to any series summable (C, k) may be solved if an additional condition involving the conjugate series a (1.2) Σ ( n sin nx - bn cos nx) = - is imposed. With this extra condition, it is proved, in § 2, that the formal product of cos px or sin px and a series summable (C, k) to f(x) is also summable (C, k) to f{x)c,o$px or /(a?) sin pa:.
    [Show full text]
  • On the Class of Limits of Lacunary Trigonometric Series
    On the class of limits of lacunary trigonometric series Christoph Aistleitner∗ Abstract Let (nk)k≥1 be a lacunary sequence of positive integers, i.e. a sequence satisfying 1 n +1/n >q> 1, k 1, and let f be a “nice” 1-periodic function with f(x) dx = 0. k k ≥ 0 Then the probabilistic behavior of the system (f(n x)) ≥1 is very similar to the behavior k k R of sequences of i.i.d. random variables. For example, Erd˝os and G´al proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos2πx and lacunary (nk)k≥1: N −1/2 lim sup(2N log log N) f(nkx)= f 2 (1) N→∞ =1 k k Xk 1/2 1 2 for almost all x (0, 1), where f 2 = f(x) dx is the standard deviation of ∈ k k 0 the random variables f(nkx). If (nk)k≥1hasR certain number-theoretic properties (e.g. n +1/n ), a similar LIL holds for a large class of functions f, and the constant on k k → ∞ the right-hand side is always f 2. For general lacunary (nk)k≥1 this is not necessarily true: Erd˝os and Fortet constructedk k an example of a trigonometric polynomial f and a lacunary sequence (nk)k≥1, such that the lim sup in the LIL (1) is not equal to f 2 and not even a constant a.e. In this paper show that the class of possible functionsk k on the right-hand side of (1) can be very large: we give an example of a trigonometric polynomial f, such that for any function g(x) with sufficiently small Fourier coefficients there exists 2 a lacunary sequence (nk)k≥1 such that (1) holds with f 2 + g(x) instead of f 2 on the right-hand side.
    [Show full text]
  • 1 Taylor-Maclaurin Series
    1 Taylor-Maclaurin Series 2 00 2 Writing x = x0 + n4x; x1 = (n − 1)4x; ::, we get, (4y0) = y0 (4x) ::: and letting n ! 1, a leap of logic reduces the interpolation formula to: 1 y = y + (x − x )y0 + (x − x )2 y00 + ::: 0 0 0 0 2! 0 Definition 1.0.1. A function f is said to be an Cn function on (a; b) if f is n-times differentiable and the nth derivative f (n) is continuous on (a; b) and f is said to belong to C1 if every derivative of f exists (and is continuous) on (a; b). Taylor’s Formula: Suppose f belongs to C1 on (−R; R). Then for every n 2 N, and x 2 (−R; R) we have: 1 1 f(x) = f(0) + f 0(0)x + f (2)(0) x2 + ::: + f (n)(0) xn + R (x) 2! n! n f is said to be analytic at 0 if the remainder Rn(x) ! 0 as n ! 1. There are two standard forms for the remainder. 1. Integral form: Z x 1 (n+1) n Rn(x) = f (t)(x − t) dt n! 0 Proof. Integrating by parts, 1 Z x 1 1 Z x f (n+1)(t)(x − t)ndt = − f (n)(0)xn + f (n)(t)(x − t)n−1dt n! 0 n! (n − 1)! 0 Proof is completed by induction on n on observing that the result holds for n = 0 by the Funda- mental theorem of Calculus. 2. Lagrange’s form: There exists c 2 (0; x) such that: 1 R (x) = f (n+1)(c)xn+1 n (n + 1)! This form is easily derived from the integral form using intermediate value theorem of integral calculus.
    [Show full text]