Mathematical problem in classical harmonic analysis
In mathematics, the question of whether the Fourier series of a periodic functionconverges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur.
Consider f an integrable function on the interval [0, 2π]. For such an f the Fourier coefficients are defined by the formula
It is common to describe the connection between f and its Fourier series by
The notation ~ here means that the sum represents the function in some sense. To investigate this more carefully, the partial sums must be defined:
The question of whether a Fourier series converges is: Do the functions (which are functions of the variable t we omitted in the notation) converge to f and in which sense? Are there conditions on f ensuring this or that type of convergence?
Before continuing, the Dirichlet kernel must be introduced. Taking the formula for , inserting it into the formula for and doing some algebra gives that
where ∗ stands for the periodic convolution and is the Dirichlet kernel, which has an explicit formula,
The Dirichlet kernel is not a positive kernel, and in fact, its norm diverges, namely
a fact that plays a crucial role in the discussion. The norm of Dn in L1(T) coincides with the norm of the convolution operator with Dn,
acting on the space C(T) of periodic continuous functions, or with the norm of the linear functional f → (Snf)(0) on C(T). Hence, this family of linear functionals on C(T) is unbounded, when n → ∞.
Magnitude of Fourier coefficients
In applications, it is often useful to know the size of the Fourier coefficient.
There are many known sufficient conditions for the Fourier series of a function to converge at a given point x, for example if the function is differentiable at x. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series converges to the average of the left and right limits (but see Gibbs phenomenon).
The Dirichlet–Dini Criterion states that:[4] if ƒ is 2π–periodic, locally integrable and satisfies
then (Snf)(x0) converges to ℓ. This implies that for any function f of any Hölder classα > 0, the Fourier series converges everywhere to f(x).
It is also known that for any periodic function of bounded variation, the Fourier series converges. In general, the most common criteria for pointwise convergence of a periodic function f are as follows:
If f satisfies a Holder condition, then its Fourier series converges uniformly.[5]
If f is of bounded variation, then its Fourier series converges everywhere. If f is additionally continuous, the convergence is uniform.[6]
If f is continuous and its Fourier coefficients are absolutely summable, then the Fourier series converges uniformly.[7]
There exist continuous functions whose Fourier series converges pointwise but not uniformly.[8]
However, the Fourier series of a continuous function need not converge pointwise. Perhaps the easiest proof uses the non-boundedness of Dirichlet's kernel in L1(T) and the Banach–Steinhaus uniform boundedness principle. As typical for existence arguments invoking the Baire category theorem, this proof is nonconstructive. It shows that the family of continuous functions whose Fourier series converges at a given x is of first Baire category, in the Banach space of continuous functions on the circle.
So in some sense pointwise convergence is atypical, and for most continuous functions the Fourier series does not converge at a given point. However Carleson's theorem shows that for a given continuous function the Fourier series converges almost everywhere.
It is also possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in [0,π] by[9]
In this example it is easy to show how the series behaves at zero. Because the function is even the Fourier series contains only cosines:
The coefficients are:
As m increases, the coefficients will be positive and increasing until they reach a value of about at for some n and then become negative (starting with a value around ) and getting smaller, before starting a new such wave. At the Fourier series is simply the running sum of and this builds up to around
in the nth wave before returning to around zero, showing that the series does not converge at zero but reaches higher and higher peaks.
Uniform convergence
Suppose , and has modulus of continuity; then the partial sums of the Fourier series converge to the function with speed[10]
for a constant that does not depend upon , nor , nor .
This theorem, first proved by D Jackson, tells, for example, that if satisfies the -Hölder condition, then
If is periodic, continuous and of bounded variation,
then the Fourier series of converges uniformly,[11]
but not necessarily absolutely,[12] to .
If this condition holds then converges absolutely for every . Conversely, for this condition to hold, it suffices that converges absolutely for some . In other words, for absolute convergence there is no issue of where the sum converges absolutely — if it converges absolutely at one point then it does so everywhere.
The family of all functions with absolutely converging Fourier series is a type of Banach algebra called the Wiener algebra, after Norbert Wiener, who proved that if ƒ has absolutely converging Fourier
series and is never zero, then 1/ƒ has absolutely converging Fourier series. A simplification of the original proof of Wiener's theorem was given by Israel Gelfand and later by Donald J. Newman in 1975.
for the constant in the
Hölder condition, a constant only dependent on ; is the norm of the Krein algebra. Notice that the 1/2 here is essential—there is an example of a 1/2-Hölder functions due to Hardy and Littlewood,[14] which do not belong to the Wiener algebra. Besides, this theorem cannot improve the best known bound on the size of the Fourier coefficient of a α-Hölder function—that is only and then not summable.
Zygmund's theorem states that, if ƒ is of bounded variationand belongs to a α-Hölder class for some α > 0, it belongs to the Wiener algebra.[15]
Norm convergence
According to the Riesz–Fischer theorem, if ƒ is square-integrable then converges to ƒ in the L2-norm, that is
The converse is also true: if the limit above is zero, then must be in .
If the partial sum SN is replaced by a suitable summability kernel (for example the Fejér sum obtained by convolution with the Fejér kernel), basic functional analytic techniques can be applied to show that norm convergence holds for 1 ≤ p < ∞.
Convergence almost everywhere
The problem whether the Fourier series of any continuous function converges almost everywhere was posed by Nikolai Lusin in the 1920s.
It was resolved positively in 1966 by Lennart Carleson. His result, now known as Carleson's theorem, tells the Fourier expansion of any function in L2 converges almost everywhere. Later on, Richard Hunt generalized this to Lp for any p > 1.
Contrariwise, Andrey Kolmogorov, in his first scientific work, constructed an example of a function in L1 whose Fourier series diverges almost everywhere (later improved to diverge everywhere).
Jean-Pierre Kahane and Yitzhak Katznelson proved that for any given set E of measure zero, there exists a continuous function ƒ such that the Fourier series of ƒ fails to converge on any point
of E.
Summability
Does the sequence 0,1,0,1,0,1,... (the partial sums of Grandi's series) converge to 1/2? This does not seem like a very unreasonable generalization of the notion of convergence. Hence we say that any sequence is Cesàro summable to some a if
It is not difficult to see that if a sequence converges to some a then it is also Cesàro summable to it.
To discuss summability of Fourier series, we must replace with an appropriate notion. Hence we define
and ask: does converge to f? is no longer
associated with Dirichlet's kernel, but with Fejér's kernel, namely
where is Fejér's kernel,
The main difference is that Fejér's kernel is a positive kernel. Fejér's theorem states that the above sequence of partial sums converge uniformly to ƒ. This implies much better convergence properties
If ƒ is continuous at t then the Fourier series of ƒ is summable at t to ƒ(t). If ƒ is continuous, its Fourier series is uniformly summable (i.e. converges uniformly to ƒ).
For any integrable ƒ, converges to ƒ in the norm.
There is no Gibbs phenomenon.
Results about summability can also imply results about regular convergence. For example, we learn that if ƒ is continuous at t, then the Fourier series of ƒ cannot converge to a value different from ƒ(t). It may either converge to ƒ(t) or diverge. This is because, if converges to some value x, it is also summable to it, so from the first summability property above, x = ƒ(t).
Order of growth
The order of growth of Dirichlet's kernel is logarithmic, i.e.
See Big O notation for the notation O(1). The actual value is both difficult to calculate (see Zygmund 8.3) and of almost no use. The fact that for some constant c we have
is quite clear when one examines the graph of Dirichlet's kernel. The integral over the n-th peak is bigger than c/n and therefore the estimate for the harmonic sum gives the logarithmic estimate.
This estimate entails quantitative versions of some of the previous results. For any continuous function f and any t one has
However, for any order of growth ω(n) smaller than log, this no longer holds and it is possible to find a continuous function f such that for some t,
The equivalent problem for divergence everywhere is open. Sergei Konyagin managed to construct an integrable function such that for every t one has
It is not known whether this example is best possible. The only bound from the other direction known is log n.
Multiple dimensions
Upon examining the equivalent problem in more than one dimension, it is necessary to specify the precise order of summation one uses. For example, in two dimensions, one may define
which are known as "square partial sums". Replacing the sum above with
lead to "circular partial sums". The difference between these two definitions is quite notable. For example, the norm of the corresponding Dirichlet kernel for square partial sums is of the order of while for circular partial sums it is of the order of .
Many of the results true for one dimension are wrong or unknown in multiple dimensions. In particular, the equivalent of Carleson's theorem is still open for circular partial sums. Almost everywhere convergence of "square partial sums" (as well as more general polygonal partial sums) in multiple dimensions was established around 1970 by Charles Fefferman.
Dunham Jackson The theory of Approximation, AMS Colloquium Publication Volume XI, New York 1930.
Nina K. Bary, A treatise on trigonometric series, Vols. I, II. Authorized translation by Margaret F. Mullins. A Pergamon Press Book. The Macmillan Co., New York 1964.
Antoni Zygmund, Trigonometric series, Vol. I, II. Third edition. With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002. ISBN0-521-89053-5
Yitzhak Katznelson, An introduction to harmonic analysis, Third edition. Cambridge University Press, Cambridge, 2004. ISBN0-521-54359-2
Karl R. Stromberg, Introduction to classical analysis, Wadsworth International Group, 1981. ISBN0-534-98012-0
The Katznelson book is the one using the most modern terminology and style of the three. The original publishing dates are: Zygmund in 1935, Bari in 1961 and Katznelson in 1968. Zygmund's book was greatly expanded in its second publishing in 1959, however.
Articles
Paul du Bois-Reymond, "Ueber die Fourierschen Reihen", Nachr. Kön. Ges. Wiss. Göttingen21 (1873), 571–582.
This is the first proof that the Fourier series of a continuous function might diverge. In German
Andrey Kolmogorov, "Une série de Fourier–Lebesgue divergente partout", C. R. Acad. Sci. Paris183 (1926), 1327–1328
The first is a construction of an integrable function whose Fourier series diverges almost everywhere. The second is a strengthening to divergence everywhere. In French.
Lennart Carleson, "On convergence and growth of partial sums of Fourier series", Acta Math.116 (1966) 135–157.
Richard A. Hunt, "On the convergence of Fourier series", Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), 235–255. Southern Illinois Univ. Press, Carbondale, Ill.
This is the original paper of Carleson, where he proves that the Fourier expansion of any continuous function converges almost everywhere; the paper of Hunt where he generalizes it to spaces; two attempts at simplifying the proof; and a book that gives a self contained exposition of it.
Dunham Jackson, Fourier Series and Orthogonal Polynomials, 1963
D. J. Newman, "A simple proof of Wiener's 1/f theorem", Proc. Amer. Math. Soc. 48 (1975), 264–265.
In this paper the authors show that for any set of zero measure there exists a continuous function on the circle whose Fourier series diverges on that set. In French.
Sergei Vladimirovich Konyagin, "On divergence of trigonometric Fourier series everywhere", C. R. Acad. Sci. Paris329 (1999), 693–697.
Jean-Pierre Kahane, Some random series of functions, second edition. Cambridge University Press, 1993. ISBN0-521-45602-9
The Konyagin paper proves the divergence result discussed above. A simpler proof that gives only log log n can be found in Kahane's book.
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