Hardy type asymptotics for cosine series in several variables with decreasing power-like coefficients

The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series in several variables have got less attention. The aim of the work is to partially fill this gap by finding the asymptotics of trigonometric series in several variables with the terms, having a form of `one minus the cosine' up to a decreasing power-like factor: \[ \sum_{z\in\mathbb{Z}^{d}\setminus\{0\}}\frac{1}{\|z\|^{d+\alpha}}\left(1-\cos\langle z,\theta\rangle\right), \qquad \theta\in\mathbb{R}^{d}, \] where $\langle\cdot,\cdot\rangle$ is the standard inner product and $\|\cdot\|$ is the max-norm on $\mathbb{R}^{d}$. The approach developed in the paper is quite elementary and essentially algebraic. It does not rely on the classic machinery of the asymptotic analysis such as slowly varying functions, Tauberian theorems or the Abel transform. However, in our case, it allows to obtain explicit expressions for the asymptotics and to extend to the general case $d\ge 1$ classical results of G. H. Hardy and other authors known for $d=1$.

The series in (1) is uniformly convergent for any α > 0 and therefore the function F d (θ) is non-negative and continuous, and F d (0) = 0. We will be interested in study of the asymptotic behavior of F d (θ) as θ → 0. An example of the function F d (θ) for d = 2 is plotted in Fig. 1.
From (4) it follows that F 1 (θ) 2H * α (θ) for each α > 0. Behavior of the functions F 1 (θ) and 2H * α (θ) is illustrated in Fig. 2. For d ≥ 2 the asymptotic behavior of the function F d (θ) at zero is much less studied. We would like to mention that for the function F 2 (θ) E. Yarovaya [12,13] obtained a lower bound of order θ α near the origin. A similar result for F 3 (θ) has been established by A. Rytova, a student of E. Yarovaya (unpublished).
In connection with this the aim of the work is to prove in Section 2 Theorems 1-4 which provide an explicit description of the asymptotic behavior of the function F d (θ) as θ → 0 for all d ≥ 2.
It is not clear whether it is possible to apply, in the case d ≥ 2, the methods developed in [1][2][3][4][5][6][7][8][9][10] for studying the asymptotics of trigonometric series due to their one-dimensional 3 specificity. Therefore in our case we use the approach of "reduction to dimension one" allowing to express the function F d (θ) as an explicit combination of the one-dimensional functions F 1 (·), or rather of the functions H α (·). This gives an opportunity to reduce the analysis of the case d ≥ 2 to the case d = 1. A similar idea was used in [12,13].
Outline the structure of the work. In this section we have explained the motivation for the analysis of asymptotic behavior of the series (1) in several variables, and have presented a concise review of the publications related to this problem. In Section 2 the main results are formulated. Theorem 1 there, about representation of the function F d (θ) in the form of a finite integral combination of the functions H α (θ) with different values of the parameter α, plays the key role. This theorem reduces the analysis of the behavior of the function F d (θ) for d ≥ 2 to the case d = 1. With the help of Theorem 1, in Theorems 2 and 3 an explicit form of the asymptotics for F d (θ) is obtained for all α > 0, and in Theorem 4 the asymptotics of the series a z (1 − cos z, θ ) with the coefficients a z asymptotically equivalent to z −(d+α) is established. The section is finalized by computation of the asymptotics of the function F d (θ) in the cases d = 2, 3. The subsequent Sections 3-6 are devoted to the proofs of Theorems 1-4.
By using asymptotic equalities (4) for the function H α (·), from the previous theorem it is easy to derive an explicit form for the asymptotics of F d (θ), for different values of the parameter α. Define 4 The set J (1) d , over which summation in (7) is made, consists of d sequences {j k }. These sequences are formed as follows: the element j 1 takes all the values 1, 2, . . . , d and, for a chosen j 1 , the sequence of values {j 2 , j 3 , . . . , j d } is obtained from {1, 2, . . . , d} by removing the member j 1 .
Theorem 2. Let d ≥ 2 and α > 0. Then where |θ| := θ 2 1 + · · · + θ 2 d is the Euclidean norm of the vector θ = {θ 1 , . . . , θ d }. The function A d (θ) is positive for θ = 0 and homogeneous of order α, that is, A d (tθ) ≡ t α A d (θ) for t ≥ 0. Therefore the first of the relations in (8) can be represented also in the form: where 0 < c ≤ A d (θ) ≤ C < ∞ for all θ satisfying |θ| = 1. The integral representation (7) for the function A d (θ) in (8) may be found inconvenient for one reason or another. It is possible to get rid of integrals in (7), and thereby in (8), by taking advantage of the fact that the multiple integral in (7) can be computed explicitly. By setting we get from (8) another form for the asymptotics of F d (θ): where the function A d (θ) admits the following alternative representation: Although the expression for the asymptotics of F d (θ) obtained in Theorem 3 is somewhat more clumsy than (8), its beauty is that it does not require to compute any integrals. Now consider the case when F (θ) is given by a series more general than (1).
By this theorem for 0 < α ≤ 2 the asymptotics of the function F (θ) can be computed explicitly by calculating the asymptotics of F d (θ) with the help of Theorem 2. As for the case α > 2, we can only assert that F (θ) has the same rate of growth at zero as If, instead of the condition a z z d+α → 1 as z → ∞, it is valid a less restrictive condition , and again we can only assert that the rate of growth of the function F (θ) for each α > 0 will be the same as the rate of growth of the function F d (θ), that is, in this case F (θ) ∼ F d (θ) as z → 0. However, explicit asymptotics for the function F (θ) are hardly can be computed in this case.
Let us finalize the section by examples of the asymptotics of the function F d (θ) for d = 2, 3 and 0 < α < 2. and

Proof of Theorem 1
We will begin with transformation of the function F d (θ) to a form playing a key role in the subsequent constructions: and then for any complex number w then Therefore the function F d (θ) can be represented in the form So, we have proved the following where and Lemma 1 indicates that properties of the function F d (θ) are determined by properties of the function S d (ϕ, ψ). Therefore let us investigate the function S d (ϕ, ψ) in more detail.
From here it is seen that the function S + d (ϕ, ψ), as well as the function S + d (ϕ, ψ), is a sum of products of sines and cosines in the variables ϕ j and ψ j . The only difference between them is that in the representation of S + d (ϕ, ψ) all the products of sines and cosines are prepended by the plus sign while in the representation of S − d (ϕ, ψ) half of such products are prepended by the minus sign. Therefore in the case of S − d (ϕ, ψ) half of the products of sines and cosines will be mutually eliminated while in the case of S + d (ϕ, ψ) the related products will be doubled as a result of summation. More specifically, there will be doubled those products in the expansion of S − d (ϕ, ψ) that have odd number of multipliers of the form cos ϕ j sin ψ j , which is reflected in the condition of the lemma. The lemma is proved.
Example 2. Representation of the function S d (ϕ, ψ) as a sum of products of sines and cosines in the variables ϕ and ψ in dimensions d = 2, 3: Lemma 2 shows that, in the representation of the function S d (ϕ, ψ) as a sum of products of sines and cosines, each product consists of odd number of cosine multipliers in the variables ϕ j and some number of sine multipliers in the variables ϕ j . In connection with this let us find out how do the products of cosines in different variables or the products of sines preceded by one cosine look like. cos (s 1 ϕ 1 + · · · + s m ϕ m ) .
Formally, the variable ϕ 1 in (14) differs from other variables since it always appears in the related summands with the plus sign. In fact in view of evenness of cosine such a 'privileged' state may be given to any argument ϕ j by appropriate change of the signs of cosine arguments.
Lemma 4. For each d ≥ 2 the following representation is valid: Proof. In the case d = 2 the equality is verified directly. Further the proof is carried out by induction.
Example 4. Integral representations of some products of sines and cosines: If the product cos ϕ 1 · · · cos ϕ m sin ϕ m+1 · · · sin ϕ d contains not less than two cosines, that is m ≥ 2, then the following lemma holds which generalized Lemma 4.
Then, by applying to the summands in the right-hand part the integral representation from Lemma 4, we deduce the claim of the lemma.
In view of Lemma 1 the arguments of the function S d (ϕ, ψ) in the representation (12), (13) of F d (θ) have a special form: ϕ = nθ and ψ = 1 2 θ. Therefore to complete the proof of Theorem 1 we need to provide a more detailed analysis of properties of the function S d nθ, 1 2 θ .
Replacing by Lemma 5 in this last equality the products of sines and cosines in the variables nθ j by the related sums of integrals we get: cos(s 1 nθ j1 + · · · + s m nθ jm Carrying out the change of variables σ j = nη j in these integrals we achieve that cosines under the integrals will depend on n-fold arguments: cos n(s 1 θ j1 + · · · + s m θ jm Hence the term f d,n (θ) in (12), (13) can be represented in the following form cos n(s 1 θ j1 + · · · + s m θ jm From here, by subtracting unity form cosine under the integral and then again adding unity to it, we obtain 1 − cos n(s 1 θ j1 + · · · + s m θ jm + η m+1 + · · · + η d ) dη m+1 . . . dη d . Here Then 1 − cos n(s 1 θ j1 + · · · + s m θ jm 1 − cos n(s 1 θ j1 + · · · + s m θ jm Here the external sums are taken over all the sets of indices {j k } with odd number of members j 1 , . . . , j m . Then and therefore 1 − cos n(s 1 θ j1 + · · · + s m θ jm Substituting now the obtained expression in (12) and changing there the order of summation, using the fact that all the sums in (15) are finite, we obtain from which representation (6) follows. Theorem 1 is proved.

Proof of Theorem 2
Let us note that the function F d (θ) is even and it does not change values under any permutation of the coordinates of the vector θ = {θ 1 , θ 2 , . . . , θ d }, and also under any change of sign of this vector. In particular, and behavior of the function F d (θ) is completely determined by its values in the "first quadrant": (16) 14 Therefore, in what follows in this section, without loss in generality the vector θ can be taken as belonging to the "first quadrant".
First show that the asymptotic behavior of the function F d (θ) is defined actually only by those summands in (6) which relate to the case m = 1.
Lemma 6. For d ≥ 2 and 0 < α ≤ 2 the following equality is valid: Proof. Let us estimate the rate of growth of different summands in (6).
First sum. We start with the estimation of the first external sum in (6):
Case m = 3, 5, . . . . In this case m + α − 1 > 2 and then by (4) 0 ≤ H m+α−1 (x) x 2 . Therefore 0 ≤ H m+α−1 (s 1 θ j1 + · · · + s m θ jm + η m+1 + · · · + η d ) Then for all sufficiently small values of θ we have 0 ≤ R m,j1,...,j d ,s1,...,sm (θ) cot 1 2 since each factor 2θ j k cot 1 2 θ j k is bounded for all sufficiently small values of θ j k . Hence 0 ≤ R m,j1,...,j d ,s1,...,sm (θ) Case m = 1. By verbatim repetition of the estimates from the previous case with usage of the representation (4) we obtain, for 0 < α ≤ 2, that Let us show that the inverse estimate is also valid: By (4) there exists a q > 0 such that H α (x) ≥ qH * α (x) for all sufficiently small values of x ≥ 0. Then, taking into account that by assumption (16) all the coordinates of the vector θ are nonnegative, we obtain If here s 1 = 1 then s 1 can be thrown off in the formula. If s 1 = −1 then in view of evenness of the function H * α (x) we first can make change of sign for all summands in the expression H * α (s 1 θ j1 + η 2 + · · · + η d ) and then make change of sign for the variables of integration (with the appropriate change of the limits of integration). As a result in both cases, s 1 = 1 and s 1 = −1, we obtain Since the integrand here is nonnegative then, by appropriate reducing the region of integration, we can obtain the following inequality: from which, in view of monotone non-decreasing of the function H * α (x) for small x ≥ 0, it follows that because η k ≥ 1 2 θ j k , k = 2, . . . , d, for η k ∈ 1 2 θ j k , θ j k . Then which, for small values of θ , implies (23).
Let us complete the proof of theorem. Case 0 < α < 2. To obtain the required asymptotic representation (8) in this case it suffices to substitute the asymptotic expression (4) for the function H α (·) in equality (17) from Lemma 6. However one should bear in mind that formula (4) does not provide any estimates for smallness of the "reminder terms" in the asymptotic representation of the integrand function H α (·). Owing to this we cannot affirm that the reminder terms in the representation (8) are of the order O(|θ| 2 ) and would have to prove a weaker asymptotic equality.

20
Substituting the obtained expression in (11) and making there summation over the terms up to the second order of smallness inclusive we obtain the required representation (8). Theorem 2 is proved.

Proof of Theorem 3
Let us note that for each integer m ≥ 1 and real α > 0 the following equality holds: To prove this equality it suffices to represent the factor |x| α as |x| α = (x 2 ) α 2 and then make termwise differentiation of the obtained expression routinely.

Proof of Theorem 4
Given arbitrary ε > 0, by the condition of Theorem 4 such a ρ = ρ(ε) can be found that 1 − a z z d+α ≤ ε for z ≥ ρ.
Here the first summand in the right-hand part tends to zero as θ → 0 since by Theorem 1 F d (θ) ∼ θ α in the case 0 < α ≤ 2 and F d (θ) ∼ ln 1 θ θ 2 in the case α = 2 for small values of θ , whereas all the terms under the summation sign, the number of which is finite, have quadratic order of smallness at zero.
The second summand does not exceed ε by absolute value since by the choice of ρ we have: Hence lim sup and in view of arbitrariness of ε we obtain that F (θ) F d (θ). Theorem 4 is proved.