The One Radius Theorem for the Bessel Convolution Operator and its Applications

Let r be a fixed positive number. An obvious property of non-zero 2r-periodic functions on the real axis is that they have no antiperiod equal to 2r. In other words, if a function f on R satisfies relations f (x + r) — f (x — r) = 0,  f (x + r) + f (x — r) = 0

for x G R , then f = 0 . In terms of integral averages, this means that any continuous function on R having zero integrals over all segments [x — r,x + r] and their boundaries { x ± r }

• ^# The research was carried out within the framework of the state contract, project № 124012400352-6.

is identically zero. The above fact admits nontrivial generalizations for various multidimensional spaces. For example, if a function f G C (R n ) , n ^ 2 , has zero integrals over all balls and spheres of fixed radius, then f = 0 (see [1], the proof of Theorem 4.4). We can also use close integral conditions to characterize analytic functions on the complex plane C . Namely, a function f G C (C) is entire only if, for some r >  0 and all z G C , the equalities

У f (w) dw = 0

|w-z| = r

2П J' f ( z + re i> ) de = f (z), 0

hold (see [2, Corollary 2]). Statements of this type, called one radius theorems, are closely related to questions of injectivity of the Pompeiu transform and spectral analysis-synthesis problems for subspaces invariant under shifts [3–6]. For other variants of the one radius theorem related to the characterization of harmonic functions, the Liouville property and the so-called “freak”-theorems, see [7], Theorem 4.3.4, and [8–13].

It is more difficult to study such problems on domains other than the whole space. The main difficulties that arise in this case stem from the violation of the group structure of translations and, consequently, the inability to apply the Fourier transform. Here is a formulation of one local result established in [14].

Let n ^ 2 , Br = { x G R n : | x | < R } , and B a,b = { x G R n : a <  | x | < b } . For 0 < r < R , we put W _r ( Br ) = U r ( B r) И V _r ( Br ) , where V _r ( Br ) (respectively, U r ( Br ) ) is the set of functions f G L^1,oc( Br ) having zero integrals over all closed balls (respectively, almost all spheres) of radius r in Br . Every function f G L^1,oc ( Br ) has a Fourier series

∞ dk f (x) - ЕЕ /«(М)^’(я), where { Y(k)}, 1 C l C dk, is a fixed orthonormal basis in the space of spherical harmonics of degree k, viewed as a subspace of L2(Sn—1) (Sn—1 is a unit sphere in Rn with center at the origin).

Theorem А [14, § 5] . 1) If R > 2r and f G W r ( Br ), then f = 0 in Br .

• 2) If R C 2r, then in order that a function f G L ^1oc( Br ) to belong to the class W r ( B r), it is necessary and sufficient that

f (x) dx = 0

|x|C2r—R

and k—2

fkl(\x\) = £ Cm,k,llxl2m—n—k+2,   Cm,k,l G C m=0

for all integers k ^ 0 , 1 C l C d k , and almost all x G B 2 r - R,R , where the sum is set to be equal to zero for k = 0,1 .

If R = 2r, then the condition f G L1oc(Br) enables one to simplify the description of Wr(Br). In this case, the constants cmkj, vanish for 0 C m < k-—1, and (1) holds for each f G L1,oc(Br) (see [14, §5]). In [15, Part 2, Chap. 3, §3.4], a generalization of Theorem A to Riemannian two-point homogeneous spaces is contained. These results have led, in particular, to a solution of the support problem for some classes of functions with zero ball means, and have also found unexpected applications to overdetermined interpolation problems in the theory of entire functions (see [14] and [15, Part 2, Chap. 3, § 3.4]).

Theorem A solves the problem of describing the kernel of the operator f ^ (f * XBr,f * ■'■), f e L1oc(Br)

where * denotes the usual convolution in R ⁿ , X B r is the indicator of the ball B _r and ^ S r is the surface delta function centered on the sphere S _r = dB _r . In view of possible new applications in analysis, similar results for other types of convolution are of interest. In this paper, based on techniques related to classical orthogonal polynomials and recent research of the authors from [16], we derive an analogue of Theorem A for the Bessel convolution case, and find its applications to problems in approximation theory and the theory of partial differential equations.

2.    Formulation of the Main Results

Everywhere below, we assume that a G (-1/2, +to) is a given number. Let L^a(lR) and ^^^(Ir), p G [1, +ro), be the classes of even complex-valued functions on the interval Ir = (-R, R) that are p-integrable and p-locally integrable with respect to the measure dpa(x) = \x\2a+1dx, respectively; let C^Ir) be the set of even k times continuously differentiable functions on Ir and D^(Ir) the family of all compactly supported functions of class C“(Ir). The set D^(Ir) is a topological vector space with the usual topology. Denote by D'(Ir) the space of all even distributions on Ir, that is, linear continuous functionals on the space D^(Ir). The entry {f, p) will mean the value of a functional f G D'(Ir) on a function p G D^(Ir). The space L^a^lR) is imbedded in D'(Ir) by identifying of a function f G lI^^IR) with the distribution

R

P ^

< f,p i = I

f (x)^(x) d^ a (x)

p ^G D^l R )

Let E ' (R) be the set of all compactly supported distributions of the class D ' (R) . For g G E ' (R) , we put

r(g) = inf {r > 0 : supp g С Ir}, where suppg is the support of the distribution g and Ir = [—r, r]. If f G D'(Ir) and R > r(g), then the Bessel convolution f * g is defined as distribution of D^(lR-r(g)) acting according to the rule

^{f ★ g ^, p ) = ff^{( x )} {g^TR'Ay}}p ^G D (IR-rg ) ),               (3)

where

T O Wy) = c a

П

J p(^x ² + y ² + 2xy cos 6 ) (sin 6?) ^{2 a} dQ, 0

c a

Г(а + 1) ^^r(^a + 2 )

and Г is the gamma function. The basic properties of this convolution are contained in [17, §§ 2, 3] and [18, Chap. 1].

For f , g ё L^^Ir ) and r(g) <  R , we have f ★ З € L^l R-rte ) ) and

^{r (} g )

(f*g)(y)= J (Tyf^xMx) d^a(x), y € IR-r(g).

In particular, if g = x _r is the characteristic function (indicator) of the segment I _r , then

r

{ ^f *Xr)($) = ^ (T y f (x) d^ a ^{( x )}, У ^€ I R - r •                     ⁽⁵⁾

Note also that

( f u ) (y)= T r ^a f (У), У € I R - r ,                           (6)

where 6 _r is the even part of the Dirac measure with a support at point r .

For 0 < r < R , we define the classes U _r,_a (l R ) , V _r,a (l R ) and W _r,_a (l R ) by equalities

U r,_a (l R ) = {f € L^I R ) ^: f *^Sr = 0 on I r - г} ,

V ra (l R ) = {f € L^(I R ) : f ★ X r =0 on I r -^ , W rA I R ) = U r,_a (l R ) П V r,_a (l R ).

The following result gives a description of the class W _r,_a (l R ) for any R > r .

Theorem 1. 1) If R ^ 2r , then W _ra (l R ) = { 0 } .

• 2)    If r < R < 2r, then f € W_r, a (l R ) if and only if f € L^°^c(Ir ) , f = 0 on (2r — R, R) and

2 r - R

J f (x) dp _a (x)=0.                               (7)

It is not difficult to establish a connection between this result and Theorem A. Indeed, let f (x) = f o (| x |) be a radial locally summable function on Br and r € (0, R) . Using the formulas for the transition to spherical coordinates and (6), we find

У f (x + ra) da =

_S n - 1

n 2n 2 r®

f 0 ²

★ 6 _r

x € Br - г .

Hence by equality

r

I f (x + y) dy = I p I

B r                   0        S ^n—1

f (x + pa) dadp

we obtain

J ^{f ( x} + y) dy =

B r

n 2n 2 Г®

(

n f0 ★ Xr

) (^|x|) ,

x € Br - г .

Relations (8) and (9) show that Theorem A for radial functions is a special case of Theorem 1 when a = Щ — 1 , n € { 2, 3,. . . } .

In the sequel, we will denote by the symbol cl x M the closure of a set M in a topological vector space X . Let also L inE be a linear span of a given set E ⊂ X .

One of the applications of Theorem 1 is the following result on approximation by generalized shifts in the space L p_a ( I r " .

Theorem 2. Let 1 <  p <  от , 0 < r < R and

M = L in { T T (X r ",T _r ^a (x t " : 0 - r } .

Then in order that

dLU'«)M = LUIR"- it is necessary and sufficient that R ^ 2r.

Note that if p = от , then equality (10) is not satisfied for any R > r . Indeed, an arbitrary function in M has a compact support on I r , so a non-zero constant cannot be approximated with any precision in the space L “ „ ( I r " by functions of M . For other results on approximation by shifts, see [6, Part 5, Chap. 2], [19, Chap. 2], and the references therein.

Theorem 1 also allow us to establish a new uniqueness theorem for solutions of the Cauchy problem related to the generalized Euler–Poisson–Darboux equation.

Theorem 3. Let f E C^(R", a > —1/2, and U be a classical solution of the Cauchy problem d2U (2a + 1" dU = d2U (2a + 3" dU x > 0, t > 0

> 0.

dx2 + x dx dt2 + t dt'

∂U

U(x, 0" = f (x) -fa (x, 0" = 0, x

For R > r >  0 , the following statements hold.

1) If R ^ 2r , then U = 0 on the triangle T 1 = { 0 < x < R, 0 — x } if and only if

U(x, r" = 0 for x E (0, R — r" and   U(r, t" = 0 for t E (0, R — r".        (11)

• 2)    If R < 2r, then in order for condition (11) to be fulfilled, it is necessary and sufficient that U = 0 on the triangle T2 = {2r — R < x < R, 0 < t < min{R — x,x + R — 2r}} and

3.    Auxiliary Statements

2 r - R

J U(x, 0" d^ a (x" = 0.

For various applications of the usual spherical means in R ⁿ to partial differential equations, see [6, Part 5, Chap. 6], [15, Part 2, Chap. 7, § 7.5 B], [20, Chap. 4–7] and [21, Chap. 6, § 13]. The validity of Theorems 1–3 will be established in § 4 below. In § 3 the auxiliary statements necessary to prove the main results are given.

As usual, the symbols N and Z ₊ denote the sets of all positive integers and nonnegative integers, respectively.

For X E C, m E Z+, we put (A"m = A(A + 1"... (A + m — 1" if m E N, and (A"o = 1. The classical Gegenbauer polynomials Ca(x" can be defined on the segment [—1,1] by the equalities n ca(x" = £ m=0

(a"m (a"n-m    // m!(n — m"! cos ^('

-

n — 2m" arccos x),   if a >  — 1/2, a = 0,

СП(x) = -Tn(x),   if n G N,   C0(x) = To(x) = 1, nn where Tn(x) = cos(n arccos x) is the Chebyshev polynomial of the first kind of degree n (see [22, Chap. 10, §§ 10.9 and 10.11]).

The following two statements are well-known theorems on the expansion of a function into a Fourier series of Gegenbauer and Chebyshev polynomials (see, for example, [23, Chap. 3 and 7]).

Lemma 1. Let a > -1/2, a = 0, f G C“[-1,1] and a.«(f) = ^nrFTTTr I f (x)Ca(xK1 - x2)"-2 dx, n G Z+.

vn(2a) n Г (a + 2) J

Then

∞∞ f (x) = E ana(f )Ca(x), f‘(x) = -a £ ana(f)C—1 (x), n=0

where the series converge absolutely and uniformly on [ - 1,1] .

Lemma 2. Let f G C“[-1,1] and ano(f ) = -^—2----7 I x(x) Tn(x) dx, n G Z+.

n(2 - sign n) J     V1 - x ²

- 1

Then

∞∞

f(x) = £ anof)Tn(x),  f‘(x) = £ nan,o(f)СП-1(x), n=0                        n=1

where the series converge absolutely and uniformly on [ - 1 , 1] .

Let us now establish additional information about the coefficients of such expansions for functions of a special kind depending on a parameter.

Lemma 3. Let 1 < R< 2, t G IR-1, f G C “(2 - R, R) and fn(t) = an,aFM), n G Z+,                          (14)

where

F(x,t) = f (V 1 + 2xt + t ² ) , x G [ - 1,1].

Then:

• 1)    if a >  - 1 / 2 , a = 0 , then

• -¹— ^fn ^{( t ) - nf}n ^{( t )}) ^{- 2 tf}n +1 ^{( t )}

; + a

= — --—f+f(+2(t) + (n + 2a + 2)fn+2(t)), n G Z+;

n + a + 2'

• 2)    if a = 0 , then

(2 - sign n)tf n (t) - nf n (t) - 2(n + 1)tf n ₊i (t)

= tfa+2(t) + (n + 2)fn+2(t), n G Z+;

• 3)    if a >  - 1 / 2 and f o = f 1 = 0 on I R _- 1 , then f = 0 on (2 - R, R ) .

  < 1) It follows from the definition of the function F that tdF = (t + x) dF.

  ∂t∂x

  On the other hand, by virtue of Lemma 1 we have

  ∞∞

  = Ef (t)C a (x),      = 2a£f n (t)C a +i ¹ (x),

  ∂t∂x

  n=0

  Substituting (18) into (17), we arrive at the equality

  
  

  x G [ - 1,1].

  
  

  ∞

  
  

  ∞

  
  

  £ tf (t)C a (x) = 2a £(t + x)f „ (t)C a +1 (x), x G [ — 1,1].

  
  

  n =0

  
  

  n =1

  
  
  
  
  
  
  
  

We transform the left-hand side of (19) using the formula

^C » ^{- 1} =    + .

₁ ( C * (x) — C i- , (x) ) ,  n >  2, A > 2, A = 1

(see [22, Chap. 10, § 10.9, formula (3.6)]), and the right-hand side using the relation xc£(x) =

^{( n} + 1)C n ₊i ( ^x ) + (n + 2A — 1И' П Jx।

2(n + A)

, n G N, A = 0

(see [22, Chap. 10, § 10.9, formula (13)]). Then changing the summation indices in the sums and collecting the coefficients for the polynomials C a ⁺¹ (x) , we get (15).

• 2)    This case is treated similarly to the previous one (see Lemma 2). In this case, instead of (20) and (21) we should use the formulas

Tn(x) = 2 (Cn(x) — Cn-2(X)), n > 2, xCn(x) = 2(Cn+1(x) + с^-^х^ n G N

(see [22, Chap. 10, § 10.11, formula (3), (6)], and equality (21) for A = 1 ).

• 3)    It is easy to see that if f _n = f n +1 = 0 on I R- 1 for some n G Z + , then f n +2 = 0 on I R _- 1 . Indeed, under the above condition relations (15) and (16) imply the equalities

f n +2 (0)=0 and dt (| t | ⁿ + ² « + ² f n ₊2 (t) ) =0, t G I r - 1 , t = 0.

Hence, taking into account the fact that fn+2(t) is bounded as t ^ 0 (see (14), (12) and (13)), we obtain fn+2(t) = 0. Thus, all coefficients fo,f1,f2,... vanish on IR-1 and f = 0 on (2 — R,R). >

Let в G (— 1, + w ) ,

Ie (z) = -J where Jβ is the Bessel function of the first kind of order β. It is well known (see [22, Chap. 7, § 7.9]) that the function Iβ has infinitely many zeros, all its zeros are real, simple and

The following statement is a special case of the Theorem 1 established by the authors in [16].

Lemma 4. Let R >  1 , f £ V^c ^ Ir ) A C “ ( Ir ) and

21-а          Г cA(f) = q“^q2 QQ J f (x)Ia(Ax) d^a(x), A £ NA+l-л+2^ о

Then f (x) = E CXf) ММ, X £ IR,

AG N a +1

where c \ (f ) = O(A -N ) as A ^ + w for any fixed N >  0 and the series converges in the space C “ (R) .

Corollary 1. Let R >  1 , t £ i R — i and f £ V i ,_a (I R ) A C “( Ir ) . Then

У f ^V 1 + 2xt + t ² ) x ( 1 — x ² ) ^{a 2} dx = 0.

- 1

< For A > 0 , we have (see [22, Chap. 7, § 7.6, formula (14)])

У I _a ( a ^ 1 + 2xt + 1 ² ^ x ( 1 — x ² ) ^{a 2} dx = -V n2 ^a r ^a + ^^A 2 t I _{a +1} (A) I _{a +1} (At).

- 1

Hence by Lemma 4 we obtain the required statement. ⊲

Let f £ D^(Ir), A £ R \ {0}. We define the distribution fA £ D^(Ir/|a|) by equality f ’^   |A|2a+2

У ^{f ( x} )’^ ( x ) У ^ ^£ D ^{( I} R/|A| )•

For f £ L^^Ir ) , the distribution f ^A belongs to l J^ C I r/ i a i ) and f \x ) = f (Ax) (see (2)). In addition,

T“(fA) = (TAXf Л(23)

Lemma 5. Let f £ D^ ( I r ) , g £ E ^ (R) and A £ R \ { 0 } . If R | A | >  r(g), then

(f *gA) 1 = |A|21a+2f 1 ★g on IRIXI-r(g).

<1 For any functions ^ £ D^(IR|A|-r(g)), ^ £ D^(Ir—r(g)/|A|), from (3) and (22) we have f1 ★g,^) = |A|2a+2 f(x), (g(y)’ TAX ^(y))}’

{f ★gA^} = |A|21a+2 (f(x)’ {g^T^^y/^}}-

Using (22), (23) and (26), we find

(( f ★ g ^A ) i a^) = ^{|A| 2 a +2 f} * g ^A ’^ ^A ) = ^{f ( x )}’ (g⁽y^{) ,T a} _x ^(y) )} .

Comparing this relation with (25), we arrive at equality (24). ⊲

Corollary 2. In order that f G W _r,_a (l R ), it is necessary and sufficient that f ^r G W 1 ,a ^{( I} R/r ).

• < Formula (24) shows that (f *5_rУ = f^r*51, (f *xrУ = r^2a+2f^r*x1. These relations yield the required statement. >

4. Proof of Theorems 1–3

• <1 Proof Of Theorem 1. In view of Corollary 2, we can assume that r = 1 .

Let us prove assertion 2) . Let 1 < R < 2 and f G W i ,_a (l R ) . We fix e G (0, R — 1) and consider a function ш _£ G D ^ (R) with the following properties: supp w _£ С I _s , ш _£ ^ 0 and

ε

У u _£ (x) d^ _a (x) = 1.