APPROXIMATION ON THE SPHERE BY WEIGHTED FOURIER EXPANSIONS

A standard procedure to approximate a function f in an inner product space is to consider the Fourier series of the function with respect to an orthogonal system. The basic general results on this topic can be found in many references in the literature, for example, [2, Chapter VIII]. It is well known that even in the case in which K is a closed interval there always exists a function f in C(K) for which the corresponding Fourier series does not converge to f with respect to the uniform norm. Thus, in this and other cases, the common solution is to consider weighted expansions and to study convergence based on the choice of the weights. Here is a list of problems that emerges: how to choose the weights in order to guarantee convergence for every function in the space, to study orders of convergence, how to choose the weights so that the operators given by the truncated Fourier series inherit properties of other known operators, and so forth. In this paper, we consider some of the problems above in the case when K = Sm, the unit sphere in Rm+1. The focus is on convergence but we intend to study the analysis of convergence orders in a forthcoming paper. For functions defined on Sm, orthogonality depends upon dσm, the usual surface measure on Sm. The surface area of Sm will be written as σm. The uniform norm is then given by


Introduction and basic facts
A standard procedure to approximate a function f in an inner product space is to consider the Fourier series of the function with respect to an orthogonal system.The basic general results on this topic can be found in many references in the literature, for example, [2,Chapter VIII].
It is well known that even in the case in which K is a closed interval there always exists a function f in C(K) for which the corresponding Fourier series does not converge to f with respect to the uniform norm.Thus, in this and other cases, the common solution is to consider weighted expansions and to study convergence based on the choice of the weights.Here is a list of problems that emerges: how to choose the weights in order to guarantee convergence for every function in the space, to study orders of convergence, how to choose the weights so that the operators given by the truncated Fourier series inherit properties of other known operators, and so forth.
In this paper, we consider some of the problems above in the case when K = S m , the unit sphere in R m+1 .The focus is on convergence but we intend to study the analysis of convergence orders in a forthcoming paper.For functions defined on S m , orthogonality depends upon dσ m , the usual surface measure on S m .The surface area of S m will be written as σ m .The uniform norm is then given by and as usual, we consider the spaces L p (S m ) := L p (S m ,dσ m ), 1≤ p < ∞, with norm given by Unless stated otherwise, the letter X will denote any of the spaces introduced above and • X the corresponding norm.Orthogonality is then related to the inner product of L 2 (S m ) f ,g 2 = 1 σ m S m f g dσ m , f ,g ∈ L 2 S m . (1.3) We will write P(S m ) to denote the space formed by polynomials in m + 1 variables restricted to S m .The set Ᏼ k (S m ) of all spherical harmonics of degree k in m + 1 variables joint with the zero polynomial is a subspace of P(S m ).Let {Y k1 ,Y k2 ,...,Y kN(m,k) } be an orthonormal basis of Ᏼ k (S m ).To every function in any of the spaces above, we can associate the Fourier series in which the Fourier coefficient f (k,l) is given by The setting introduced above is, up to normalization, the same used in many standard references on analysis on the sphere.We refer the reader to [1,4,7,11,12,13,14], where the Addition Formula the Funk-Hecke Formula and other results can be found.In (1.6) and (1.7), P m k is the Legendre polynomial of degree k associated to the dimension m + 1, K : We will deal with approximations of a given function f in X by a sequence {T n ( f )} n∈N , in which the operators T n : X → X, n ∈ N := {0, 1,...}, are defined by weighted orthogonal Fourier sums of the form the weights a kl (n), n,k = 0,1,..., l = 1,2,...,N(m,k) being all real.In Section 2, we deduce some basic properties of the operators T n , including the computation of their norm in the cases X = C(S m ) and X = L 1 (S m ).In Section 3, we find necessary and sufficient conditions in order that {T n } n∈N be an approximate identity in X.Among other things, this implies that T n ( f ) converges to f in the norm of X, for every f ∈ X.In Section 4, restricting ourselves to the case X = C(S m ), we introduce the notion of localized approximate identity and study corresponding approximation properties.

The operator T n
In this section, we present some basic properties of the operator T n , some of independent interest, other to be used in the subsequent sections.That includes to decide when T n is of convolution type and the computation of its norm in some cases.
The orthonormality of the spherical harmonics yields The linearity of T n reveals that whenever Writing T n f (x) := (T n ( f ))(x) and appealing to (1.5), we deduce that (2.4) Thus, we have proved the following representation formula.
in which (2.6) Formula (2.6) defines a kernel having an expression very close to that on the left-hand side of the Addition Formula.Since that formula defines a bi-zonal polynomial kernel and this type of kernel defines spherical convolution operators on X, the goal in the next two results will thus be to verify when T n is an operator of convolution type.

Lemma 2.2. The kernel K n is bi-zonal if and only if
..,n, the Addition Formula implies that for some function .8) while the previous theorem and relation (2.1) imply that (2.9) Thus, (2.11) The spherical convolution defined by K is the operator Basic properties of this operator along with some information about bi-zonal kernels are to be found in [4,7,8,14] and the references therein.We adopt here the most common notation for the convolution operator, that is, Theorem 2.3.The operator T n is of convolution type if and only if K n is bi-zonal.
V. A. Menegatto and A. C. Piantella 325 that is, In particular, (2.15) This leads to K n (x, y) = L n ( x, y ), x, y ∈ S m , because K n is polynomial.
Next, we introduce some notation.Given a kernel K : S m × S m → C, we will write K x e K y to denote the functions y ∈ S m → K(x, y) and x ∈ S m → K(x, y), respectively.It is easily seen that these two functions are in L 1 (S m ) when K is polynomial.In just one step of the proof of Theorem 2.4 below we use the space L ∞ (S m ) as defined in [3, page 184].
We would like to observe that the second half of the proof above is very close to that of [3,Theorem 6.18].The upper bound given above coincides with T n X in at least two cases.
V. A. Menegatto and A. C. Piantella 327 Theorem 2.5.For X = C(S m ) and X = L 1 (S m ), it holds (2.26) Proof.Due to Theorem 2.4, we need to prove that (2.27) Using the formula [3, page 189] we obtain and, consequently, (2.30) To finish the proof, first observe that (2.31) Finally, using the formula [3, page 223] we obtain (2.33) This completes the proof of the theorem.

Approximate identities
In this section, keeping the notation in (1.9), we will search for conditions in order that the following approximation property holds: As usual, a sequence {T n } n∈N satisfying (3.1) is called an approximate identity in X.We refer the reader to [8] and some references therein, for nice examples of approximate identities in X, including examples involving spherical convolution, spherical shifting, and so forth.Since spherical convolution operators belong to the class of operators we are dealing with here, many results in this section may be considered as generalizations of those corresponding to approximation by spherical convolution.For results dealing with approximate identities and spherical convolution, see [5,6,8,9,10].
Theorem 3.1.If {T n } n∈N is an approximate identity in X then the following properties hold: Proof.Assume {T n } n∈N is an approximate identity in X.Since every sequence for some nonnegative constant C. Since Y kl ≡ 0, we can choose y 0 ∈ S m such that Y kl (y 0 ) = 0. Thus, (3.5) and (3.6) lead to where σ m (Λ) = 0. Taking y 1 ∈ S m \ Λ such that Y kl (y 1 ) = 0, as we certainly can, we conclude that In both cases the conclusion is lim n→∞ a kl (n) = 1.
Next, we search for some converse results.
Theorem 3.2.Let f ∈ C(S m ) be a function fulfilling the following condition: if q ∈ P(S m ) then there exists a constant Proof.Let > 0. We use the Weierstrass approximation theorem to select q ∈ P(S m ) such that f − q ∞ < and write q in the form defining B 1 := max r kl : k = 0,1,...,M, l = 1,2,...,N(m,k) , it is clear that Consequently, ..,N(m,k), then lim n→∞ T n (q) − q ∞ = 0 and there is an N( ) ∈ N such that completing the proof of the theorem.
) be a function fulfilling the following condition: if q ∈ P(S m ) then there exists a constant V. A. Menegatto and A. C. Piantella 331 Proof.Let > 0. Since C(S m ) is dense in L p (S m ), there exists a function g ∈ C(S m ) such that f − g p < /2.The Weierstrass approximation theorem provides a polynomial q such that q − g ∞ < /2.Hence, As in the proof of the previous theorem, if lim n→∞ a kl (n) = 1, k ∈ N, l = 1,2,...,N(m,k) then lim n→∞ T n (q) − q ∞ = 0. Thus, the inequality T n (q) − q p ≤ T n (q) − q ∞ implies that lim n→∞ T n (q) − q p = 0. Now, there exists N( Therefore, This completes the proof.
Combining the previous three theorems we have the following.
Theorem 3.4.The sequence {T n } n∈N is an approximate identity in X if and only if the following conditions hold: Next, we investigate the possibility of changing condition (i) by another involving the kernel K n .This is suggested by Theorem 2.1.
for some positive constant C, not depending on n, then Theorem 2.4 implies that (3.21) The previous theorem closes the proof.
The converse of this result holds in at least two cases.
Theorem 3.6.Let either X = C(S m ) or X = L 1 (S m ).Then {T n } n∈N is an approximate identity in X if and only if the following properties hold: Proof.One implication is consequence of the previous theorem.As for the other, assume {T n } n∈N is an approximate identity in X. Theorem 3.1 yields the existence of a positive constant C such that (3.22) Hence, Under the conditions in the statement of the theorem, Theorem 2.5 is applicable.Hence, and, therefore, This takes care of condition (i).Condition (ii) follows directly from Theorem 3.1.
If K n is positive, that is, K n (x, y) ≥ 0, x, y ∈ S m , then we can sharpen Theorem 3.6 as follows.
Theorem 3.7.Let either X = C(S m ) or X = L 1 (S m ).Assume that every K n is positive.Then {T n } n∈N is an approximate identity in X if and only if lim n→∞ a kl (n) = 1, k ∈ N, l = 1,2,...,N(m,k).
Proof.It is sufficient to prove that (ii) implies (i) in Theorem 3.6.The positivity of K n and the orthonormality of the spherical harmonics yield Thus, the result follows from the previous theorem.
These are elementary consequences of the above results.
Theorem 3.8.Let either ...,N(m,k), then the following assertions are equivalent: If {T n } n∈N is an approximate identity in C(S m ) then it is an approximate identity in L p (S m ), 1 ≤ p < ∞ (Theorems 3.5 and 3.6).Thus, it is quite obvious that the case X = C(S m ) is the most important among all.
We close the section, giving a method to construct approximate identities.It is based on a corresponding property of spherical convolution.Theorem 3.10.Let {S n } n∈N be another sequence having a representation as in (1.9).If both {T n } n∈N and {S n } n∈N are approximate identities in X then {T n • S n } n∈N is an approximate identity in X.
Proof.Let {T n } n∈N be as in (1.9) and represent {S n } n∈N in the form with c i j (n) ∈ R, i = 0,1,..., j = 1,2,...,N(m,i).Using (2.5), (2.6) and arranging we have that Now, Theorem 3.4 guarantees that {T n • S n } n∈N is an approximate identity in X.
An important consequence of Theorem 3.10 is this: if {T n } n∈N is an approximate identity in X and k ∈ N then {T k n } n∈N is an approximate identity in X.

Localized approximate identities
In this section, we deal with the space C(S m ) only and investigate approximation at one fixed point.The term localized refers to this and nothing else.To motivate the main definition in this section, let {T n } be as in (1.9) and assume that {T n } n∈N is an approximate identity in C(S m ).Since (n)Y kl (x)Y kl (y) (n) f (i, j)Y i j (y)dσ m (y), (n)c i j (n) f (i, j)Y kl (x) 1 σ m S m Y i j (y)Y kl (y)dσ m (y) (n)c kl (n) f (k,l)Y kl (x), x ∈ S m , f ∈ X. (3.32)If {T n } n∈N and {S n } n∈N are approximate identities in X, Theorem 3.1 implies that lim n→∞ a kl(n) = lim n→∞ c kl (n) = 1, k ∈ N, l = 1,2,...,N(m,k).(n)c kl (n) = 1, k ∈ N, l = 1,2,...,N(m,k).(3.34)That same theorem produces constants C 1 ,C 2 > 0 such that lim n→∞ T n f (y) = f (y), y ∈ S m , f ∈ C(S m ), (n)Y kl (y)Y kl (x) f (x)dσ m (x) (n)Y kl (x) f (x)dσ m (x), y ∈ S m , f ∈ C(S m ),(4.2) in which c y kl (n) := a kl (n)Y kl (y).
V. A. Menegatto and A. C. Piantella 329 a clear contradiction.Thus, C > 0 and (i) is proved.To prove (ii) fix k ∈ N and l ∈ {1, 2,...,N(m,k)}.SinceT n Y kl (y) = a kl (n)Y kl (y), y ∈ S m , n ≥ k,We split the proof in two cases.If X = C(S m ), our assumption on {T n } n∈N implies that lim n→∞T n Y kl (y) = Y kl (y), y ∈ S m .(3.6)