Pythagorean harmonic summability of Fourier series

1 Introduction

It is well known that most 2L-periodic real-valued f ( x ) ∈ C [ 0 , 2 L ] can be represented by the Fourier series

with the partial sums, see e.g. [1],

given by the Dirichlet convolution integral [1,2]

with the Dirichlet kernel

Moreover, in many applications, it is required to compare the smoothness of functional sums like ∑ k = 0 n φ k ( x ) with ∑ k = 0 n ϕ n ( k ) φ k ( x ) , where ϕ n ( k ) is a smooth function, which decays as k → n , and even vanishes as k → n → ∞ . Such sums are called smoothed sums [3]. This happens to take place with the Cesàro-Fejér sums [1,4]

associated with S n ( x ) , where

Existence of σ ( x ) = lim n → ∞ σ n ( x ) , generated by S n ( x ) , is known as C 1 summability of S ( x ) . Similarly, existence of σ ^ ( x ) , generated by σ n ( x ) , is known as C 2 summability of S ( x ) , and so on. Intuitively, one may expect a smoother behavior of the modified sums ∑ k = 0 n ϕ n ( k ) φ k ( x ) because of the averaging implied by (5) for σ n ( x ) that is more visible analytically by the convolution integral expression, which is now [3]

with the Fejér kernel

The problem of establishing a criterion for convergence of Fourier series is a rather old one and is still unresolvable in the form of a simple necessary and sufficient condition. For instance, bounded variation of f ( x ) is sufficient but not necessary for convergence of S ( x ) . Continuity of f ( x ) is neither necessary nor sufficient. Actually, there are f ( x ) whose S ( x ) converge at points of discontinuity and others whose S ( x ) diverge at points of continuity. Classical summability of Fourier series was developed during the period 1897–1957 [1] as a complementary, or even “dual,” property to their convergence. For instance, Fejér’s summability theorem [1,2] proves that at a point of continuity of f ( x ) , S ( x ) is C 1 summable, so that continuity is at any rate sufficient for this summability.

It should be noted here that all summation methods are based on averages. For example, in Cesàro-Fejér summation the average is arithmetical, while in the Abel-Poisson summation, the average is a harmonic function on the unit disk. Apart from these, the most important Fourier series summation methods are those of Riesz, Riemann, Bernstein-Rogosinski and de la Vallee-Poissin. Summation methods that are generated by a more-or-less arbitrary sequence of λ -multipliers, see e.g. [5], have also been studied.

Neoclassical summation methods for Fourier series were developed during the period 1960–1989 in [6,7, 8,9,10] and in other works. Furthermore, during the period 1990–2017, contemporary summability research has ranged from factored Fourier series [11] and product summability [12] of these series to generalizations for any orthogonal series, via sums based on Marcinkiewicz’s Θ -means [13,14]. Along these lines, we mention the recent work in [15] on the best error approximation of f ( x ) in the generalized Zygmond class, by using a matrix-Cesàro product operator of its Fourier series. Other recent generalizations include also a recent approximation [16] of f ( x , y ) functions by double Hausdorff matrix summability means of double Fourier series.

Despite the vastness of the surveyed theory for summing Fourier series, its practical applications continue to focus restrictively on the following:

1. Improving the representation of functions by Fourier series. For instance, if the σ n ( x ) sums, when n → ∞ , converge to f ( x ) at its points of continuity, then they converge moreover uniformly on [ 0 , 2 L ] if f ( x ) ∈ C [ 0 , 2 L ] . The partial sums S n ( x ) do not possess this property.

2. Fourier-represented functional or perturbational [17] analysis. For example, a function f ( x ) ≔ S ( x ) is essentially bounded iff exists a constant M such that ∣ σ n ( x ) ∣ ≤ M , ∀ n & x .

On another note, in a recent article [18] this author has reported on a minimal harmonic series for reconstructing an infinite Fourier series for f ( x ) ∈ C [ 0 , 2 L ]. A series that is constructible by a minimal series interpolation algorithm is given in [18]. The possibility for smoothing such series by linear (Cesàro-Fejér) summation has also been demonstrated in [18]. This paper is devoted to the subject of possible smoothing of Fourier series by nonlinear summations, particularly by Pythagorean harmonic and/or semi-harmonic summation.

This paper is organized as follows. After this introduction, Section 2 introduces the concept of a symbolic smoothing operator and applies it to the analysis of the Cesàro-Fejér C 1 linear summability. The applications cover guarantees for smoothing by the C 1 summability that reveal certain affine and Kalman filtration features of the pertaining operator and establish conditions for its contraction mapping. Section 3 is devoted to nonlinear summabilities, with results on nonlinear processing by their smoothing operators and to their contraction mapping properties. Section 4 focuses on the Pythagorean harmonic sum and reports on a sharp result on its ℐ 1 summability. In Section 5, an algorithm for a new semi-harmonic J 1 summability is advanced. Unique linear processing and contraction mapping features are identified for the smoothing operator of this summability. Then, it is demonstrated in Section 6 as to how Pythagorean summabilities fail to apply to seismic-like (with zero mean) signals. Accordingly, a regularizational asymptotic approach is developed for handling the Pythagorean harmonic summability of such signals. To lighten the reading of this paper, proofs of some results are grouped in Section 7. And, this paper concludes with Section 8.

2 Linear summation

It should be underlined, from the outset of this analysis, that the existing literature on summability of Fourier series is overwhelmingly immense, see e.g. [19,20,21, 22,23]. Moreover, nonlinear summability, that is addressed in this work, should neither be confused with nonlinear Fourier analysis, as reviewed by Semmes in [19], nor with nonlinear convergence acceleration [23,24,25].

Our starting point is accordingly an assertion that the σ n ( x ) sum of (5) can be expressed as

which is the same as

where the smoothing, of S n ( x ) , operator

(which is the author’s new concept) is symbolic (with a σ multiplication) in the sense that it acts on S n ( x ) with the map:

The quantity to the right of → above is equal to σ n ( x ) , as in (10).

Consideration of the substitutions

in (12) illustrates, via

that W σ ( n , x ) is in fact an affine transformation of S n ( x ) .

For any summation method to be applicable to divergent functional series, the method can satisfy, in a sufficient but not necessary sense, the following three properties [3]:

1. Regularity: The summation method should give correct answer for a convergent series.

2. Linearity:

   1. If ∑ k = 0 ∞ φ k ( x ) = A ( x ) and ∑ k = 0 ∞ ψ k ( x ) = B ( x ) , then ∑ k = 0 ∞ [ φ k ( x ) + ψ k ( x ) ] = A ( x ) + B ( x ) , and

   2. ∑ k = 0 ∞ c φ k ( x ) = c A ( x ) , ∀ c ∈ R .

3. Stability:

   (17) If ∑ k = 0 ∞ φ k ( x ) = A ( x ) , then ∑ k = 0 ; k ≠ j ∞ φ k ( x ) = A ( x ) − φ j ( x ) .

In this respect, Cesàro-Fejér sum, σ ( x ) , is regular, linear and stable. But not every useful method for summing Fourier series can satisfy all the previous three requirements. Recently, a technique of nonlinear summation of power series was used in [26] for solving nonlinear evolution equations. It is our intention here to focus attention on similar alternative methods to explore their possible applicability and/or limitations for summing Fourier series.

For a study of the dynamics of σ n ( x ) , with varying n , we shall suppress the x -variable by averaging over [ 0 , 2 L ], viz.,

to rewrite (16) as

Furthermore, for the real-valued f ( x ) addressed in this paper, let S n ( x ) ∈ L 2 [ 0 , 2 L ] , Hilbert space, endowed with the norm

to state the results that follow.

Note incidentally that the contraction mapping property of W σ ( n , x ) should be a guarantee for the smoothing power of the associated σ n ( x ) sum.

3 Nonlinear summation

In addition to the arithmetic mean σ n ( x ) of (5), which is linear in elements of { S k } k = 0 n , there are two other Pythagorean means [27] that happen to be nonlinear in the previous elements. These are namely the geometric mean

and the Pythagorean harmonic mean

not to be confused with the harmonic mean

of the Riesz-Nörlund ℋ 1 summability [28].

Clearly, if any S k ( x ) = 0 , then θ n ( x ) is undefined. However, if for some k , S k ( x j ) = 0 at a finite point set { x j } j = 1 J , the singularities S k − 1 ( x j ) can, nonetheless, be isolated in a way to redefine θ n ( x ) in some VP (valeur principale) sense. The situation of S k ( x ) = 0 , for some k, is avoided, though, in this paper.

By the way, despite the above linearity of the σ n ( x ) summation, its pertaining smoothing operator W σ ( n , x ) is, by (15), nonlinear (affine). In fact, the same can also be said about most summations ς n ( x ) , which can be linked to a corresponding ς symbolic smoothing operator W ς ( n , x ) , viz.,

In particular, both Pythagorean mean symbolic operators

and

act on the same S n ( x ) , but with the distinct corresponding maps, viz.,

4 Pythagorean harmonic summability

It should be noted here that for any sequence ( S n ) = ( S k ) k = 0 n , with S n > 0 , ∀ n , the following famous inequality [27],

always holds. Incidentally, the inequality (44), which favors the harmonic mean, can be invoked to prove a lemma that follows.

Furthermore, it is useful to underline, at this point, three features that add promise to employing the θ n ( x ) harmonic mean to summability of S ( x ) .

1. The harmonic mean is often used in other disciplines, particularly in statistics [27] to compute average ratios. Ratios that can incidentally be seen in the Fejér kernel (8) of the convolution integrals (7).

2. This mean is less sensitive, than γ n ( x ) or σ n ( x ) , to possible extremely large spikes in some S k ( x ) ′ s (like in point-wise divergence) of the S ( x ) series.

3. The apparent restriction of S k ( x ) ≠ 0 , ∀ k , and particularly of S 0 ( x ) ≠ 0 , on direct application of θ n ( x ) does not exclude however, its application in regularized form is shown later in Section 6.1. Moreover, θ n ( x ) has a more obvious relation to the summability of S ( x ) than the alternative Ω .

For these reasons, we shall focus attention, in the rest of this paper, on the θ n ( x ) mean and, in the future, on a pertaining minimal, in the sense of [18], ℐ 1 summation series of S ( x ) . For an aim for this work, it perhaps suffices to merely establish relevance, robustness and/or possible limitations of this alternative new technique of summability.

The nonlinear dynamics of (42), with varying n , cannot be freed of the x -dependence by an averaging process similar to (18). Nonetheless, for a representative fixed x = x 0 , one can define

to analyze the recurrence relation associated with (42)

This is easily rewritten as

Now, regardless of the magnitude and sign of θ ˜ n − 1 S ˜ n ∈ R , ∃ an N ∈ N , after which

Consequently, for n ≥ N , equation (48) tends to the generalized logistic equation

which is transformable, via the map

to a standard logistic equation

with a unit growth rate r .

The trajectories of (52) are known, see e.g. [29], to be generated by two fixed points (the first is at u = 0 ). Therefore, one of the trajectories should necessarily satisfy lim n → ∞ u n = 0 . The other trajectory, namely due to r = 1 , is fortunately, however, neither bifurcative nor chaotic. The same can, undoubtedly, be said about the trajectories of the precursor equation (50).

Despite the rather complicated analysis of the solution to (52), it is nonetheless not impossible to attempt to linearize it. In this respect, relation (50) can readily be put into the following form:

In the special case of (64), when θ ˜ n − 1 S ˜ n = κ > 1 , 1 − κ = − α , α > 0 , we have

Further consideration of (54) in (53) converts it to

where e ˜ n is a certain error induced by the approximation (54). In particular, if κ = 2 , i.e., α = 1 , then

which is another kind of Kalman filtration, quite similar to (16), with its familiar smoothing power.

Moreover, one can always try to verify the existence of some reasonably representative trajectories for (47). Toward this end, let us rewrite (47) in the following form:

Since the number of φ ˜ -terms in S ˜ 0 is 1, in S ˜ 1 is 2, and in S ˜ n is ( n + 1 ) , we may consider a pair of extreme cases for such trajectories.

1. If φ ˜ n φ ˜ k ≈ 1 , ∀ n , k , then

   S ˜ n S ˜ 0 = n + 1 , S ˜ n S ˜ 1 = n + 1 2 , S ˜ n S ˜ 2 = n + 1 3 , … , S ˜ n S ˜ k = n + 1 k + 1 ,

   and

   (58) ∑ k = 0 n − 1 S ˜ n S ˜ k = ( n + 1 ) 1 + 1 2 + 1 3 + ⋯ + 1 n = ( n + 1 ) ln ( 2 n + 1 ) .

   Eventually, for large enough n , we have

   (59) θ ˜ n = S ˜ n ln ( 2 n + 1 ) ,

   indicating that lim n → ∞ θ ˜ n = 0 , ∀ S ˜ n , a trajectory apparently associated with the first fixed point of (50).

2. If S ˜ n S ˜ k ≈ 1 , ∀ n , k , then

   (60) θ ˜ n = S ˜ n ,

   is a trajectory generated by the second fixed point of (50), which is reasonably possible, despite its deficiency in any S ˜ n smoothing features.